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Tensor bundle
In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To do calculus on the tensor bundle a connection is needed, except for the special case of the exterior derivative of antisymmetric tensors.
Definition
See also: Tensor field § Tensor bundles
A tensor bundle is a fiber bundle where the fiber is a tensor product of any number of copies of the tangent space and/or cotangent space of the base space, which is a manifold. As such, the fiber is a vector space and the tensor bundle is a special kind of vector bundle.
References
• Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218 (Second ed.). New York London: Springer-Verlag. ISBN 978-1-4419-9981-8. OCLC 808682771.
• Saunders, David J. (1989). The Geometry of Jet Bundles. London Mathematical Society Lecture Note Series. Vol. 142. Cambridge New York: Cambridge University Press. ISBN 978-0-521-36948-0. OCLC 839304386.
• Steenrod, Norman (5 April 1999). The Topology of Fibre Bundles. Princeton Mathematical Series. Vol. 14. Princeton, N.J.: Princeton University Press. ISBN 978-0-691-00548-5. OCLC 40734875.
See also
• Fiber bundle – Continuous surjection satisfying a local triviality condition
• Spinor bundle
• Tensor field – Assignment of a tensor continuously varying across a mathematical space
Manifolds (Glossary)
Basic concepts
• Topological manifold
• Atlas
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Tensors
Glossary of tensor theory
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Tensor
definitions
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• Tensor operator
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• Two-point tensor
Operations
• Covariant derivative
• Exterior covariant derivative
• Exterior derivative
• Exterior product
• Hodge star operator
• Lie derivative
• Raising and lowering indices
• Symmetrization
• Tensor contraction
• Tensor product
• Transpose (2nd-order tensors)
Related
abstractions
• Affine connection
• Basis
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• Connection form
• Covariance and contravariance of vectors
• Differential form
• Dimension
• Exterior form
• Fiber bundle
• Geodesic
• Levi-Civita connection
• Linear map
• Manifold
• Matrix
• Multivector
• Pseudotensor
• Spinor
• Vector
• Vector space
Notable tensors
Mathematics
• Kronecker delta
• Levi-Civita symbol
• Metric tensor
• Nonmetricity tensor
• Ricci curvature
• Riemann curvature tensor
• Torsion tensor
• Weyl tensor
Physics
• Moment of inertia
• Angular momentum tensor
• Spin tensor
• Cauchy stress tensor
• stress–energy tensor
• Einstein tensor
• EM tensor
• Gluon field strength tensor
• Metric tensor (GR)
Mathematicians
• Élie Cartan
• Augustin-Louis Cauchy
• Elwin Bruno Christoffel
• Albert Einstein
• Leonhard Euler
• Carl Friedrich Gauss
• Hermann Grassmann
• Tullio Levi-Civita
• Gregorio Ricci-Curbastro
• Bernhard Riemann
• Jan Arnoldus Schouten
• Woldemar Voigt
• Hermann Weyl
| Wikipedia |
Tensor calculus
In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime).
Part of a series of articles about
Calculus
• Fundamental theorem
• Limits
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• Inverse function theorem
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Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita,[1] it was used by Albert Einstein to develop his general theory of relativity. Unlike the infinitesimal calculus, tensor calculus allows presentation of physics equations in a form that is independent of the choice of coordinates on the manifold.
Tensor calculus has many applications in physics, engineering and computer science including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory, and machine learning.
Working with a main proponent of the exterior calculus Elie Cartan, the influential geometer Shiing-Shen Chern summarizes the role of tensor calculus:[2]
In our subject of differential geometry, where you talk about manifolds, one difficulty is that the geometry is described by coordinates, but the coordinates do not have meaning. They are allowed to undergo transformation. And in order to handle this kind of situation, an important tool is the so-called tensor analysis, or Ricci calculus, which was new to mathematicians. In mathematics you have a function, you write down the function, you calculate, or you add, or you multiply, or you can differentiate. You have something very concrete. In geometry the geometric situation is described by numbers, but you can change your numbers arbitrarily. So to handle this, you need the Ricci calculus.
Syntax
Tensor notation makes use of upper and lower indexes on objects that are used to label a variable object as covariant (lower index), contravariant (upper index), or mixed covariant and contravariant (having both upper and lower indexes). In fact in conventional math syntax we make use of covariant indexes when dealing with Cartesian coordinate systems $(x_{1},x_{2},x_{3})$ frequently without realizing this is a limited use of tensor syntax as covariant indexed components.
Tensor notation allows upper index on an object that may be confused with normal power operations from conventional math syntax.
Key concepts
Vector decomposition
Tensors notation allows a vector (${\vec {V}}$) to be decomposed into an Einstein summation representing the tensor contraction of a basis vector (${\vec {Z}}_{i}$ or ${\vec {Z}}^{i}$) with a component vector ($V_{i}$ or $V^{i}$).
${\vec {V}}=V^{i}{\vec {Z}}_{i}=V_{i}{\vec {Z}}^{i}$
Every vector has two different representations, one referred to as contravariant component ($V^{i}$) with a covariant basis (${\vec {Z}}_{i}$), and the other as a covariant component ($V_{i}$) with a contravariant basis (${\vec {Z}}^{i}$). Tensor objects with all upper indexes are referred to as contravariant, and tensor objects with all lower indexes are referred to as covariant. The need to distinguish between contravariant and covariant arises from the fact that when we dot an arbitrary vector with its basis vector related to a particular coordinate system, there are two ways of interpreting this dot product, either we view it as the projection of the basis vector onto the arbitrary vector, or we view it as the projection of the arbitrary vector onto the basis vector, both views of the dot product are entirely equivalent, but have different component elements and different basis vectors:
${\vec {V}}\cdot {\vec {Z}}_{i}=V_{i}={\vec {V}}^{T}{\vec {Z}}_{i}={\vec {Z}}_{i}^{T}{\vec {V}}={\mathrm {proj} _{{\vec {Z}}^{i}}({\vec {V}})}\cdot {\vec {Z}}_{i}={\mathrm {proj} _{\vec {V}}({\vec {Z}}^{i})}\cdot {\vec {V}}$
${\vec {V}}\cdot {\vec {Z}}^{i}=V^{i}={\vec {V}}^{T}{\vec {Z}}^{i}={{\vec {Z}}^{i}}^{T}{\vec {V}}={\mathrm {proj} _{{\vec {Z}}_{i}}({\vec {V}})}\cdot {\vec {Z}}^{i}={\mathrm {proj} _{\vec {V}}({\vec {Z}}_{i})}\cdot {\vec {V}}$
For example, in physics you start with a vector field, you decompose it with respect to the covariant basis, and that's how you get the contravariant coordinates. For orthonormal cartesian coordinates, the covariant and contravariant basis are identical, since the basis set in this case is just the identity matrix, however, for non-affine coordinate system such as polar or spherical there is a need to distinguish between decomposition by use of contravariant or covariant basis set for generating the components of the coordinate system.
Covariant vector decomposition
${\vec {V}}=V^{i}{\vec {Z}}_{i}$
variable description Type
${\vec {V}}$ vector invariant
$V^{i}$ contravariant components (ordered set of scalars) variant
${\vec {Z}}_{i}$ covariant bases (ordered set of vectors) variant
Contravariant vector decomposition
${\vec {V}}=V_{i}{\vec {Z}}^{i}$
variable description type
${\vec {V}}$ vector invariant
$V_{i}$ covariant components (ordered set of scalars) variant
${\vec {Z}}^{i}$ contravariant bases (ordered set of covectors) variant
Metric tensor
The metric tensor represents a matrix with scalar elements ($Z_{ij}$ or $Z^{ij}$) and is a tensor object which is used to raise or lower the index on another tensor object by an operation called contraction, thus allowing a covariant tensor to be converted to a contravariant tensor, and vice versa.
Example of lowering index using metric tensor:
$T_{i}=Z_{ij}T^{j}$
Example of raising index using metric tensor:
$T^{i}=Z^{ij}T_{j}$
The metric tensor is defined as:
$Z_{ij}={\vec {Z}}_{i}\cdot {\vec {Z}}_{j}$
$Z^{ij}={\vec {Z}}^{i}\cdot {\vec {Z}}^{j}$
This means that if we take every permutation of a basis vector set and dotted them against each other, and then arrange them into a square matrix, we would have a metric tensor. The caveat here is which of the two vectors in the permutation is used for projection against the other vector, that is the distinguishing property of the covariant metric tensor in comparison with the contravariant metric tensor.
Two flavors of metric tensors exist: (1) the contravariant metric tensor ($Z^{ij}$), and (2) the covariant metric tensor ($Z_{ij}$). These two flavors of metric tensor are related by the identity:
$Z_{ik}Z^{jk}=\delta _{i}^{j}$
For an orthonormal Cartesian coordinate system, the metric tensor is just the kronecker delta $\delta _{ij}$ or $\delta ^{ij}$, which is just a tensor equivalent of the identity matrix, and $\delta _{ij}=\delta ^{ij}=\delta _{j}^{i}$.
Jacobian
In addition a tensor can be readily converted from an unbarred($x$) to a barred coordinate(${\bar {x}}$) system having different sets of basis vectors:
$f(x^{1},x^{2},\dots ,x^{n})=f{\bigg (}x^{1}({\bar {x}}),x^{2}({\bar {x}}),\dots ,x^{n}({\bar {x}}){\bigg )}={\bar {f}}({\bar {x}}^{1},{\bar {x}}^{2},\dots ,{\bar {x}}^{n})={\bar {f}}{\bigg (}{\bar {x}}^{1}(x),{\bar {x}}^{2}(x),\dots ,{\bar {x}}^{n}(x){\bigg )}$
by use of Jacobian matrix relationships between the barred and unbarred coordinate system (${\bar {J}}=J^{-1}$). The Jacobian between the barred and unbarred system is instrumental in defining the covariant and contravariant basis vectors, in that in order for these vectors to exist they need to satisfy the following relationship relative to the barred and unbarred system:
Contravariant vectors are required to obey the laws:
$v^{i}={\bar {v}}^{r}{\frac {\partial x^{i}({\bar {x}})}{\partial {\bar {x}}^{r}}}$
${\bar {v}}^{i}=v^{r}{\frac {\partial {\bar {x}}^{i}(x)}{\partial x^{r}}}$
Covariant vectors are required to obey the laws:
$v_{i}={\bar {v}}_{r}{\frac {\partial {\bar {x}}^{i}(x)}{\partial x^{r}}}$
${\bar {v}}_{i}=v_{r}{\frac {\partial x^{r}({\bar {x}})}{\partial {\bar {x}}^{i}}}$
There are two flavors of Jacobian matrix:
1. The J matrix representing the change from unbarred to barred coordinates. To find J, we take the "barred gradient", i.e. partial derive with respect to ${\bar {x}}^{i}$:
$J={\bar {\nabla }}f(x({\bar {x}}))$
2. The ${\bar {J}}$ matrix, representing the change from barred to unbarred coordinates. To find ${\bar {J}}$, we take the "unbarred gradient", i.e. partial derive with respect to $x^{i}$:
${\bar {J}}=\nabla {\bar {f}}({\bar {x}}(x))$
Gradient vector
Tensor calculus provides a generalization to the gradient vector formula from standard calculus that works in all coordinate systems:
$\nabla F=\nabla _{i}F{\vec {Z}}^{i}$
Where:
$\nabla _{i}F={\frac {\partial F}{\partial Z^{i}}}$
In contrast, for standard calculus, the gradient vector formula is dependent on the coordinate system in use (example: Cartesian gradient vector formula vs. the polar gradient vector formula vs. the spherical gradient vector formula, etc.). In standard calculus, each coordinate system has its own specific formula, unlike tensor calculus that has only one gradient formula that is equivalent for all coordinate systems. This is made possible by an understanding of the metric tensor that tensor calculus makes use of.
See also
• Vector analysis
• Matrix calculus
• Ricci calculus
• Curvilinear coordinates
• Tensors in curvilinear coordinates
• Multilinear subspace learning
• Multilinear algebra
• Differential geometry
References
1. Ricci, Gregorio; Levi-Civita, Tullio (March 1900). "Méthodes de calcul différentiel absolu et leurs applications" [Methods of the absolute differential calculus and their applications]. Mathematische Annalen (in French). Springer. 54 (1–2): 125–201. doi:10.1007/BF01454201. S2CID 120009332.
2. "Interview with Shiing Shen Chern" (PDF). Notices of the AMS. 45 (7): 860–5. August 1998.
Further reading
• Dimitrienko, Yuriy (2002). Tensor Analysis and Nonlinear Tensor Functions. Springer. ISBN 1-4020-1015-X.
• Sokolnikoff, Ivan S (1951). Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua. Wiley. ISBN 0471810525.
• Borisenko, A.I.; Tarapov, I.E. (1979). Vector and Tensor Analysis with Applications (2nd ed.). Dover. ISBN 0486638332.
• Itskov, Mikhail (2015). Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics (2nd ed.). Springer. ISBN 9783319163420.
• Tyldesley, J. R. (1973). An introduction to Tensor Analysis: For Engineers and Applied Scientists. Longman. ISBN 0-582-44355-5.
• Kay, D. C. (1988). Tensor Calculus. Schaum’s Outlines. McGraw Hill. ISBN 0-07-033484-6.
• Grinfeld, P. (2014). Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer. ISBN 978-1-4614-7866-9.
External links
• Dullemond, Kees; Peeters, Kasper (1991–2010). "Introduction to Tensor Calculus" (PDF). Retrieved 17 May 2018.
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definitions
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• Tensor density
• Tensors in curvilinear coordinates
• Mixed tensor
• Antisymmetric tensor
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• Tensor operator
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• Two-point tensor
Operations
• Covariant derivative
• Exterior covariant derivative
• Exterior derivative
• Exterior product
• Hodge star operator
• Lie derivative
• Raising and lowering indices
• Symmetrization
• Tensor contraction
• Tensor product
• Transpose (2nd-order tensors)
Related
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• Affine connection
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• Fiber bundle
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Notable tensors
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| Wikipedia |
Tensor contraction
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by applying the summation convention to a pair of dummy indices that are bound to each other in an expression. The contraction of a single mixed tensor occurs when a pair of literal indices (one a subscript, the other a superscript) of the tensor are set equal to each other and summed over. In Einstein notation this summation is built into the notation. The result is another tensor with order reduced by 2.
For the module-theoretic construction of tensor fields and their contractions, see tensor product of modules § Example from differential geometry: tensor field.
Tensor contraction can be seen as a generalization of the trace.
Abstract formulation
Let V be a vector space over a field k. The core of the contraction operation, and the simplest case, is the natural pairing of V with its dual vector space V∗. The pairing is the linear transformation from the tensor product of these two spaces to the field k:
$C:V\otimes V^{*}\rightarrow k$
corresponding to the bilinear form
$\langle v,f\rangle =f(v)$
where f is in V∗ and v is in V. The map C defines the contraction operation on a tensor of type (1, 1), which is an element of $V\otimes V^{*}$. Note that the result is a scalar (an element of k). Using the natural isomorphism between $V\otimes V^{*}$ and the space of linear transformations from V to V,[1] one obtains a basis-free definition of the trace.
In general, a tensor of type (m, n) (with m ≥ 1 and n ≥ 1) is an element of the vector space
$V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*}$
(where there are m factors V and n factors V∗).[2][3] Applying the natural pairing to the kth V factor and the lth V∗ factor, and using the identity on all other factors, defines the (k, l) contraction operation, which is a linear map which yields a tensor of type (m − 1, n − 1).[2] By analogy with the (1, 1) case, the general contraction operation is sometimes called the trace.
Contraction in index notation
In tensor index notation, the basic contraction of a vector and a dual vector is denoted by
${\tilde {f}}({\vec {v}})=f_{\gamma }v^{\gamma }$
which is shorthand for the explicit coordinate summation[4]
$f_{\gamma }v^{\gamma }=f_{1}v^{1}+f_{2}v^{2}+\cdots +f_{n}v^{n}$
(where vi are the components of v in a particular basis and fi are the components of f in the corresponding dual basis).
Since a general mixed dyadic tensor is a linear combination of decomposable tensors of the form $f\otimes v$, the explicit formula for the dyadic case follows: let
$\mathbf {T} =T_{j}^{i}\mathbf {e} _{i}\otimes \mathbf {e} ^{j}$
be a mixed dyadic tensor. Then its contraction is
$T_{j}^{i}\mathbf {e} _{i}\cdot \mathbf {e} ^{j}=T_{j}^{i}\delta _{i}{}^{j}=T_{j}^{j}=T_{1}^{1}+\cdots +T_{n}^{n}$.
A general contraction is denoted by labeling one covariant index and one contravariant index with the same letter, summation over that index being implied by the summation convention. The resulting contracted tensor inherits the remaining indices of the original tensor. For example, contracting a tensor T of type (2,2) on the second and third indices to create a new tensor U of type (1,1) is written as
$T^{ab}{}_{bc}=\sum _{b}{T^{ab}{}_{bc}}=T^{a1}{}_{1c}+T^{a2}{}_{2c}+\cdots +T^{an}{}_{nc}=U^{a}{}_{c}.$
By contrast, let
$\mathbf {T} =\mathbf {e} ^{i}\otimes \mathbf {e} ^{j}$
be an unmixed dyadic tensor. This tensor does not contract; if its base vectors are dotted, the result is the contravariant metric tensor,
$g^{ij}=\mathbf {e} ^{i}\cdot \mathbf {e} ^{j}$,
whose rank is 2.
Metric contraction
See also: Raising and lowering indices § An example from Minkowski spacetime
As in the previous example, contraction on a pair of indices that are either both contravariant or both covariant is not possible in general. However, in the presence of an inner product (also known as a metric) g, such contractions are possible. One uses the metric to raise or lower one of the indices, as needed, and then one uses the usual operation of contraction. The combined operation is known as metric contraction.[5]
Application to tensor fields
Contraction is often applied to tensor fields over spaces (e.g. Euclidean space, manifolds, or schemes). Since contraction is a purely algebraic operation, it can be applied pointwise to a tensor field, e.g. if T is a (1,1) tensor field on Euclidean space, then in any coordinates, its contraction (a scalar field) U at a point x is given by
$U(x)=\sum _{i}T_{i}^{i}(x)$
Since the role of x is not complicated here, it is often suppressed, and the notation for tensor fields becomes identical to that for purely algebraic tensors.
Over a Riemannian manifold, a metric (field of inner products) is available, and both metric and non-metric contractions are crucial to the theory. For example, the Ricci tensor is a non-metric contraction of the Riemann curvature tensor, and the scalar curvature is the unique metric contraction of the Ricci tensor.
One can also view contraction of a tensor field in the context of modules over an appropriate ring of functions on the manifold[5] or the context of sheaves of modules over the structure sheaf;[6] see the discussion at the end of this article.
Tensor divergence
As an application of the contraction of a tensor field, let V be a vector field on a Riemannian manifold (for example, Euclidean space). Let $V^{\alpha }{}_{\beta }$ be the covariant derivative of V (in some choice of coordinates). In the case of Cartesian coordinates in Euclidean space, one can write
$V^{\alpha }{}_{\beta }={\partial V^{\alpha } \over \partial x^{\beta }}.$
Then changing index β to α causes the pair of indices to become bound to each other, so that the derivative contracts with itself to obtain the following sum:
$V^{\alpha }{}_{\alpha }=V^{0}{}_{0}+\cdots +V^{n}{}_{n}$
which is the divergence div V. Then
$\operatorname {div} V=V^{\alpha }{}_{\alpha }=0$
is a continuity equation for V.
In general, one can define various divergence operations on higher-rank tensor fields, as follows. If T is a tensor field with at least one contravariant index, taking the covariant differential and contracting the chosen contravariant index with the new covariant index corresponding to the differential results in a new tensor of rank one lower than that of T.[5]
Contraction of a pair of tensors
One can generalize the core contraction operation (vector with dual vector) in a slightly different way, by considering a pair of tensors T and U. The tensor product $T\otimes U$ is a new tensor, which, if it has at least one covariant and one contravariant index, can be contracted. The case where T is a vector and U is a dual vector is exactly the core operation introduced first in this article.
In tensor index notation, to contract two tensors with each other, one places them side by side (juxtaposed) as factors of the same term. This implements the tensor product, yielding a composite tensor. Contracting two indices in this composite tensor implements the desired contraction of the two tensors.
For example, matrices can be represented as tensors of type (1,1) with the first index being contravariant and the second index being covariant. Let $\Lambda ^{\alpha }{}_{\beta }$ be the components of one matrix and let $\mathrm {M} ^{\beta }{}_{\gamma }$ be the components of a second matrix. Then their multiplication is given by the following contraction, an example of the contraction of a pair of tensors:
$\Lambda ^{\alpha }{}_{\beta }\mathrm {M} ^{\beta }{}_{\gamma }=\mathrm {N} ^{\alpha }{}_{\gamma }$.
Also, the interior product of a vector with a differential form is a special case of the contraction of two tensors with each other.
More general algebraic contexts
Let R be a commutative ring and let M be a finite free module over R. Then contraction operates on the full (mixed) tensor algebra of M in exactly the same way as it does in the case of vector spaces over a field. (The key fact is that the natural pairing is still perfect in this case.)
More generally, let OX be a sheaf of commutative rings over a topological space X, e.g. OX could be the structure sheaf of a complex manifold, analytic space, or scheme. Let M be a locally free sheaf of modules over OX of finite rank. Then the dual of M is still well-behaved[6] and contraction operations make sense in this context.
See also
• Tensor product
• Partial trace
• Interior product
• Raising and lowering indices
• Musical isomorphism
• Ricci calculus
Notes
1. Let L(V, V) be the space of linear transformations from V to V. Then the natural map
$V^{*}\otimes V\rightarrow L(V,V)$
is defined by
$f\otimes v\mapsto g,$
where g(w) = f(w)v. Suppose that V is finite-dimensional. If {vi} is a basis of V and {fi} is the corresponding dual basis, then $f^{i}\otimes v_{j}$ maps to the transformation whose matrix in this basis has only one nonzero entry, a 1 in the i,j position. This shows that the map is an isomorphism.
2. Fulton, William; Harris, Joe (1991). Representation Theory: A First Course. GTM. Vol. 129. New York: Springer. pp. 471–476. ISBN 0-387-97495-4.
3. Warner, Frank (1993). Foundations of Differentiable Manifolds and Lie Groups. GTM. Vol. 94. New York: Springer. pp. 54–56. ISBN 0-387-90894-3.
4. In physics (and sometimes in mathematics), indices often start with zero instead of one. In four-dimensional spacetime, indices run from 0 to 3.
5. O'Neill, Barrett (1983). Semi-Riemannian Geometry with Applications to Relativity. Academic Press. p. 86. ISBN 0-12-526740-1.
6. Hartshorne, Robin (1977). Algebraic Geometry. New York: Springer. ISBN 0-387-90244-9.
References
• Bishop, Richard L.; Goldberg, Samuel I. (1980). Tensor Analysis on Manifolds. New York: Dover. ISBN 0-486-64039-6.
• Menzel, Donald H. (1961). Mathematical Physics. New York: Dover. ISBN 0-486-60056-4.
Tensors
Glossary of tensor theory
Scope
Mathematics
• Coordinate system
• Differential geometry
• Dyadic algebra
• Euclidean geometry
• Exterior calculus
• Multilinear algebra
• Tensor algebra
• Tensor calculus
• Physics
• Engineering
• Computer vision
• Continuum mechanics
• Electromagnetism
• General relativity
• Transport phenomena
Notation
• Abstract index notation
• Einstein notation
• Index notation
• Multi-index notation
• Penrose graphical notation
• Ricci calculus
• Tetrad (index notation)
• Van der Waerden notation
• Voigt notation
Tensor
definitions
• Tensor (intrinsic definition)
• Tensor field
• Tensor density
• Tensors in curvilinear coordinates
• Mixed tensor
• Antisymmetric tensor
• Symmetric tensor
• Tensor operator
• Tensor bundle
• Two-point tensor
Operations
• Covariant derivative
• Exterior covariant derivative
• Exterior derivative
• Exterior product
• Hodge star operator
• Lie derivative
• Raising and lowering indices
• Symmetrization
• Tensor contraction
• Tensor product
• Transpose (2nd-order tensors)
Related
abstractions
• Affine connection
• Basis
• Cartan formalism (physics)
• Connection form
• Covariance and contravariance of vectors
• Differential form
• Dimension
• Exterior form
• Fiber bundle
• Geodesic
• Levi-Civita connection
• Linear map
• Manifold
• Matrix
• Multivector
• Pseudotensor
• Spinor
• Vector
• Vector space
Notable tensors
Mathematics
• Kronecker delta
• Levi-Civita symbol
• Metric tensor
• Nonmetricity tensor
• Ricci curvature
• Riemann curvature tensor
• Torsion tensor
• Weyl tensor
Physics
• Moment of inertia
• Angular momentum tensor
• Spin tensor
• Cauchy stress tensor
• stress–energy tensor
• Einstein tensor
• EM tensor
• Gluon field strength tensor
• Metric tensor (GR)
Mathematicians
• Élie Cartan
• Augustin-Louis Cauchy
• Elwin Bruno Christoffel
• Albert Einstein
• Leonhard Euler
• Carl Friedrich Gauss
• Hermann Grassmann
• Tullio Levi-Civita
• Gregorio Ricci-Curbastro
• Bernhard Riemann
• Jan Arnoldus Schouten
• Woldemar Voigt
• Hermann Weyl
| Wikipedia |
Tensor decomposition
In multilinear algebra, a tensor decomposition[1][2] [3] is any scheme for expressing a "data tensor" (M-way array) as a sequence of elementary operations acting on other, often simpler tensors. Many tensor decompositions generalize some matrix decompositions.[4]
Tensors are generalizations of matrices to higher dimensions (or rather to higher orders, i.e. the higher number of dimensions) and can consequently be treated as multidimensional fields.[1][5] The main tensor decompositions are:
• Tensor rank decomposition;[6]
• Higher-order singular value decomposition;[7]
• Tucker decomposition;
• matrix product states, and operators or tensor trains;
• Online Tensor Decompositions[8][9][10]
• hierarchical Tucker decomposition;[11]
• block term decomposition[12][13][11][14]
Notation
This section introduces basic notations and operations that are widely used in the field.
Table of symbols and their description.
SymbolsDefinition
${a,{\bf {a}},{\bf {a}}^{T},\mathbf {A} ,{\mathcal {A}}}$scalar, vector, row, matrix, tensor
${\bf {a}}={vec(.)}$vectorizing either a matrix or a tensor
${\bf {A}}_{[m]}$matrixized tensor ${\mathcal {A}}$
$\times _{m}$mode-m product
Introduction
A multi-way graph with K perspectives is a collection of K matrices ${X_{1},X_{2}.....X_{K}}$ with dimensions I × J (where I, J are the number of nodes). This collection of matrices is naturally represented as a tensor X of size I × J × K. In order to avoid overloading the term “dimension”, we call an I × J × K tensor a three “mode” tensor, where “modes” are the numbers of indices used to index the tensor.
References
1. Vasilescu, MAO; Terzopoulos, D. "Multilinear (tensor) image synthesis, analysis, and recognition [exploratory dsp]". IEEE Signal Processing Magazine. 24 (6): 118–123.
2. Kolda, Tamara G.; Bader, Brett W. (2009-08-06). "Tensor Decompositions and Applications". SIAM Review. 51 (3): 455–500. Bibcode:2009SIAMR..51..455K. doi:10.1137/07070111X. ISSN 0036-1445. S2CID 16074195.
3. Sidiropoulos, Nicholas D.; De Lathauwer, Lieven; Fu, Xiao; Huang, Kejun; Papalexakis, Evangelos E.; Faloutsos, Christos (2017-07-01). "Tensor Decomposition for Signal Processing and Machine Learning". IEEE Transactions on Signal Processing. 65 (13): 3551–3582. arXiv:1607.01668. Bibcode:2017ITSP...65.3551S. doi:10.1109/TSP.2017.2690524. ISSN 1053-587X. S2CID 16321768.
4. Bernardi, A.; Brachat, J.; Comon, P.; Mourrain, B. (2013-05-01). "General tensor decomposition, moment matrices and applications". Journal of Symbolic Computation. 52: 51–71. arXiv:1105.1229. doi:10.1016/j.jsc.2012.05.012. ISSN 0747-7171. S2CID 14181289.
5. Rabanser, Stephan; Shchur, Oleksandr; Günnemann, Stephan (2017). "Introduction to Tensor Decompositions and their Applications in Machine Learning". arXiv:1711.10781 [stat.ML].
6. Papalexakis, Evangelos E. (2016-06-30). "Automatic Unsupervised Tensor Mining with Quality Assessment". Proceedings of the 2016 SIAM International Conference on Data Mining. Society for Industrial and Applied Mathematics: 711–719. arXiv:1503.03355. doi:10.1137/1.9781611974348.80. ISBN 978-1-61197-434-8. S2CID 10147789.
7. Vasilescu, M.A.O.; Terzopoulos, D. (2002). Multilinear Analysis of Image Ensembles: TensorFaces (PDF). Lecture Notes in Computer Science; (Presented at Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark). Vol. 2350. Springer, Berlin, Heidelberg. doi:10.1007/3-540-47969-4_30. ISBN 978-3-540-43745-1.
8. Gujral, Ekta; Pasricha, Ravdeep; Papalexakis, Evangelos E. (7 May 2018). Ester, Martin; Pedreschi, Dino (eds.). "SamBaTen: Sampling-based Batch Incremental Tensor Decomposition". Proceedings of the 2018 SIAM International Conference on Data Mining. doi:10.1137/1.9781611975321. hdl:10536/DRO/DU:30109588. ISBN 978-1-61197-532-1. S2CID 21674935.
9. Gujral, Ekta; Papalexakis, Evangelos E. (9 October 2020). "OnlineBTD: Streaming Algorithms to Track the Block Term Decomposition of Large Tensors". 2020 IEEE 7th International Conference on Data Science and Advanced Analytics (DSAA). pp. 168–177. doi:10.1109/DSAA49011.2020.00029. ISBN 978-1-7281-8206-3. S2CID 227123356.
10. Gujral, Ekta (2022). "Modeling and Mining Multi-Aspect Graphs With Scalable Streaming Tensor Decomposition". arXiv:2210.04404 [cs.SI].
11. Vasilescu, M.A.O.; Kim, E. (2019). Compositional Hierarchical Tensor Factorization: Representing Hierarchical Intrinsic and Extrinsic Causal Factors. In The 25th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD’19): Tensor Methods for Emerging Data Science Challenges. arXiv:1911.04180.
12. De Lathauwer, Lieven (2008). "Decompositions of a Higher-Order Tensor in Block Terms—Part II: Definitions and Uniqueness". SIAM Journal on Matrix Analysis and Applications. 30 (3): 1033–1066. doi:10.1137/070690729.
13. Vasilescu, M.A.O.; Kim, E.; Zeng, X.S. (2021), "CausalX: Causal eXplanations and Block Multilinear Factor Analysis", Conference Proc. of the 2020 25th International Conference on Pattern Recognition (ICPR 2020), pp. 10736–10743, arXiv:2102.12853, doi:10.1109/ICPR48806.2021.9412780, ISBN 978-1-7281-8808-9, S2CID 232046205
14. Gujral, Ekta; Pasricha, Ravdeep; Papalexakis, Evangelos (2020-04-20). "Beyond Rank-1: Discovering Rich Community Structure in Multi-Aspect Graphs". Proceedings of the Web Conference 2020. Taipei Taiwan: ACM. pp. 452–462. doi:10.1145/3366423.3380129. ISBN 978-1-4503-7023-3. S2CID 212745714.
| Wikipedia |
Ricci calculus
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection.[lower-alpha 1][1][2][3] It is also the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus), developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900.[4] Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework, and made contributions to the theory, during its applications to general relativity and differential geometry in the early twentieth century.[5]
"Tensor index notation" redirects here. For a summary of tensors in general, see Glossary of tensor theory.
A component of a tensor is a real number that is used as a coefficient of a basis element for the tensor space. The tensor is the sum of its components multiplied by their corresponding basis elements. Tensors and tensor fields can be expressed in terms of their components, and operations on tensors and tensor fields can be expressed in terms of operations on their components. The description of tensor fields and operations on them in terms of their components is the focus of the Ricci calculus. This notation allows an efficient expression of such tensor fields and operations. While much of the notation may be applied with any tensors, operations relating to a differential structure are only applicable to tensor fields. Where needed, the notation extends to components of non-tensors, particularly multidimensional arrays.
A tensor may be expressed as a linear sum of the tensor product of vector and covector basis elements. The resulting tensor components are labelled by indices of the basis. Each index has one possible value per dimension of the underlying vector space. The number of indices equals the degree (or order) of the tensor.
For compactness and convenience, the Ricci calculus incorporates Einstein notation, which implies summation over indices repeated within a term and universal quantification over free indices. Expressions in the notation of the Ricci calculus may generally be interpreted as a set of simultaneous equations relating the components as functions over a manifold, usually more specifically as functions of the coordinates on the manifold. This allows intuitive manipulation of expressions with familiarity of only a limited set of rules.
Notation for indices
Space and time coordinates
Where a distinction is to be made between the space-like basis elements and a time-like element in the four-dimensional spacetime of classical physics, this is conventionally done through indices as follows:[6]
• The lowercase Latin alphabet a, b, c, ... is used to indicate restriction to 3-dimensional Euclidean space, which take values 1, 2, 3 for the spatial components; and the time-like element, indicated by 0, is shown separately.
• The lowercase Greek alphabet α, β, γ, ... is used for 4-dimensional spacetime, which typically take values 0 for time components and 1, 2, 3 for the spatial components.
Some sources use 4 instead of 0 as the index value corresponding to time; in this article, 0 is used. Otherwise, in general mathematical contexts, any symbols can be used for the indices, generally running over all dimensions of the vector space.
Coordinate and index notation
The author(s) will usually make it clear whether a subscript is intended as an index or as a label.
For example, in 3-D Euclidean space and using Cartesian coordinates; the coordinate vector A = (A1, A2, A3) = (Ax, Ay, Az) shows a direct correspondence between the subscripts 1, 2, 3 and the labels x, y, z. In the expression Ai, i is interpreted as an index ranging over the values 1, 2, 3, while the x, y, z subscripts are only labels, not variables. In the context of spacetime, the index value 0 conventionally corresponds to the label t.
Reference to basis
Indices themselves may be labelled using diacritic-like symbols, such as a hat (ˆ), bar (¯), tilde (˜), or prime (′) as in:
$X_{\hat {\phi }}\,,Y_{\bar {\lambda }}\,,Z_{\tilde {\eta }}\,,T_{\mu '}$
to denote a possibly different basis for that index. An example is in Lorentz transformations from one frame of reference to another, where one frame could be unprimed and the other primed, as in:
$v^{\mu '}=v^{\nu }L_{\nu }{}^{\mu '}.$
This is not to be confused with van der Waerden notation for spinors, which uses hats and overdots on indices to reflect the chirality of a spinor.
Upper and lower indices
Ricci calculus, and index notation more generally, distinguishes between lower indices (subscripts) and upper indices (superscripts); the latter are not exponents, even though they may look as such to the reader only familiar with other parts of mathematics.
In the special case that the metric tensor is everywhere equal to the identity matrix, it is possible to drop the distinction between upper and lower indices, and then all indices could be written in the lower position. Coordinate formulae in linear algebra such as $a_{ij}b_{jk}$ for the product of matrices may be examples of this. But in general, the distinction between upper and lower indices should be maintained.
Covariant tensor components
A lower index (subscript) indicates covariance of the components with respect to that index:
$A_{\alpha \beta \gamma \cdots }$
Contravariant tensor components
An upper index (superscript) indicates contravariance of the components with respect to that index:
$A^{\alpha \beta \gamma \cdots }$
Mixed-variance tensor components
A tensor may have both upper and lower indices:
$A_{\alpha }{}^{\beta }{}_{\gamma }{}^{\delta \cdots }.$
Ordering of indices is significant, even when of differing variance. However, when it is understood that no indices will be raised or lowered while retaining the base symbol, covariant indices are sometimes placed below contravariant indices for notational convenience (e.g. with the generalized Kronecker delta).
Tensor type and degree
The number of each upper and lower indices of a tensor gives its type: a tensor with p upper and q lower indices is said to be of type (p, q), or to be a type-(p, q) tensor.
The number of indices of a tensor, regardless of variance, is called the degree of the tensor (alternatively, its valence, order or rank, although rank is ambiguous). Thus, a tensor of type (p, q) has degree p + q.
Summation convention
The same symbol occurring twice (one upper and one lower) within a term indicates a pair of indices that are summed over:
$A_{\alpha }B^{\alpha }\equiv \sum _{\alpha }A_{\alpha }B^{\alpha }\quad {\text{or}}\quad A^{\alpha }B_{\alpha }\equiv \sum _{\alpha }A^{\alpha }B_{\alpha }\,.$
The operation implied by such a summation is called tensor contraction:
$A_{\alpha }B^{\beta }\rightarrow A_{\alpha }B^{\alpha }\equiv \sum _{\alpha }A_{\alpha }B^{\alpha }\,.$
This summation may occur more than once within a term with a distinct symbol per pair of indices, for example:
$A_{\alpha }{}^{\gamma }B^{\alpha }C_{\gamma }{}^{\beta }\equiv \sum _{\alpha }\sum _{\gamma }A_{\alpha }{}^{\gamma }B^{\alpha }C_{\gamma }{}^{\beta }\,.$
Other combinations of repeated indices within a term are considered to be ill-formed, such as
$A_{\alpha \alpha }{}^{\gamma }\qquad $(both occurrences of $\alpha $ are lower; $A_{\alpha }{}^{\alpha \gamma }$ would be fine)
$A_{\alpha \gamma }{}^{\gamma }B^{\alpha }C_{\gamma }{}^{\beta }$($\gamma $ occurs twice as a lower index; $A_{\alpha \gamma }{}^{\gamma }B^{\alpha }$ or $A_{\alpha \delta }{}^{\gamma }B^{\alpha }C_{\gamma }{}^{\beta }$ would be fine).
The reason for excluding such formulae is that although these quantities could be computed as arrays of numbers, they would not in general transform as tensors under a change of basis.
Multi-index notation
If a tensor has a list of all upper or lower indices, one shorthand is to use a capital letter for the list:[7]
$A_{i_{1}\cdots i_{n}}B^{i_{1}\cdots i_{n}j_{1}\cdots j_{m}}C_{j_{1}\cdots j_{m}}\equiv A_{I}B^{IJ}C_{J}$
where I = i1 i2 ⋅⋅⋅ in and J = j1 j2 ⋅⋅⋅ jm.
Sequential summation
A pair of vertical bars | ⋅ | around a set of all-upper indices or all-lower indices (but not both), associated with contraction with another set of indices when the expression is completely antisymmetric in each of the two sets of indices:[8]
$A_{|\alpha \beta \gamma |\cdots }B^{\alpha \beta \gamma \cdots }=A_{\alpha \beta \gamma \cdots }B^{|\alpha \beta \gamma |\cdots }=\sum _{\alpha <\beta <\gamma }A_{\alpha \beta \gamma \cdots }B^{\alpha \beta \gamma \cdots }$
means a restricted sum over index values, where each index is constrained to being strictly less than the next. More than one group can be summed in this way, for example:
${\begin{aligned}&A_{|\alpha \beta \gamma |}{}^{|\delta \epsilon \cdots \lambda |}B^{\alpha \beta \gamma }{}_{\delta \epsilon \cdots \lambda |\mu \nu \cdots \zeta |}C^{\mu \nu \cdots \zeta }\\[3pt]={}&\sum _{\alpha <\beta <\gamma }~\sum _{\delta <\epsilon <\cdots <\lambda }~\sum _{\mu <\nu <\cdots <\zeta }A_{\alpha \beta \gamma }{}^{\delta \epsilon \cdots \lambda }B^{\alpha \beta \gamma }{}_{\delta \epsilon \cdots \lambda \mu \nu \cdots \zeta }C^{\mu \nu \cdots \zeta }\end{aligned}}$
When using multi-index notation, an underarrow is placed underneath the block of indices:[9]
$A_{\underset {\rightharpoondown }{P}}{}^{\underset {\rightharpoondown }{Q}}B^{P}{}_{Q{\underset {\rightharpoondown }{R}}}C^{R}=\sum _{\underset {\rightharpoondown }{P}}\sum _{\underset {\rightharpoondown }{Q}}\sum _{\underset {\rightharpoondown }{R}}A_{P}{}^{Q}B^{P}{}_{QR}C^{R}$
where
${\underset {\rightharpoondown }{P}}=|\alpha \beta \gamma |\,,\quad {\underset {\rightharpoondown }{Q}}=|\delta \epsilon \cdots \lambda |\,,\quad {\underset {\rightharpoondown }{R}}=|\mu \nu \cdots \zeta |$
Raising and lowering indices
By contracting an index with a non-singular metric tensor, the type of a tensor can be changed, converting a lower index to an upper index or vice versa:
$B^{\gamma }{}_{\beta \cdots }=g^{\gamma \alpha }A_{\alpha \beta \cdots }\quad {\text{and}}\quad A_{\alpha \beta \cdots }=g_{\alpha \gamma }B^{\gamma }{}_{\beta \cdots }$
The base symbol in many cases is retained (e.g. using A where B appears here), and when there is no ambiguity, repositioning an index may be taken to imply this operation.
Correlations between index positions and invariance
This table summarizes how the manipulation of covariant and contravariant indices fit in with invariance under a passive transformation between bases, with the components of each basis set in terms of the other reflected in the first column. The barred indices refer to the final coordinate system after the transformation.[10]
The Kronecker delta is used, see also below.
Basis transformation Component transformation Invariance
Covector, covariant vector, 1-form $\omega ^{\bar {\alpha }}=L_{\beta }{}^{\bar {\alpha }}\omega ^{\beta }$ $a_{\bar {\alpha }}=a_{\gamma }L^{\gamma }{}_{\bar {\alpha }}$ $a_{\bar {\alpha }}\omega ^{\bar {\alpha }}=a_{\gamma }L^{\gamma }{}_{\bar {\alpha }}L_{\beta }{}^{\bar {\alpha }}\omega ^{\beta }=a_{\gamma }\delta ^{\gamma }{}_{\beta }\omega ^{\beta }=a_{\beta }\omega ^{\beta }$
Vector, contravariant vector $e_{\bar {\alpha }}=e_{\gamma }L_{\bar {\alpha }}{}^{\gamma }$ $u^{\bar {\alpha }}=L^{\bar {\alpha }}{}_{\beta }u^{\beta }$ $e_{\bar {\alpha }}u^{\bar {\alpha }}=e_{\gamma }L_{\bar {\alpha }}{}^{\gamma }L^{\bar {\alpha }}{}_{\beta }u^{\beta }=e_{\gamma }\delta ^{\gamma }{}_{\beta }u^{\beta }=e_{\gamma }u^{\gamma }$
General outlines for index notation and operations
Tensors are equal if and only if every corresponding component is equal; e.g., tensor A equals tensor B if and only if
$A^{\alpha }{}_{\beta \gamma }=B^{\alpha }{}_{\beta \gamma }$
for all α, β, γ. Consequently, there are facets of the notation that are useful in checking that an equation makes sense (an analogous procedure to dimensional analysis).
Free and dummy indices
Indices not involved in contractions are called free indices. Indices used in contractions are termed dummy indices, or summation indices.
A tensor equation represents many ordinary (real-valued) equations
The components of tensors (like Aα, Bβγ etc.) are just real numbers. Since the indices take various integer values to select specific components of the tensors, a single tensor equation represents many ordinary equations. If a tensor equality has n free indices, and if the dimensionality of the underlying vector space is m, the equality represents mn equations: each index takes on every value of a specific set of values.
For instance, if
$A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D^{\alpha }{}_{\beta }{}E_{\delta }=T^{\alpha }{}_{\beta }{}_{\delta }$
is in four dimensions (that is, each index runs from 0 to 3 or from 1 to 4), then because there are three free indices (α, β, δ), there are 43 = 64 equations. Three of these are:
${\begin{aligned}A^{0}B_{1}{}^{0}C_{00}+A^{0}B_{1}{}^{1}C_{10}+A^{0}B_{1}{}^{2}C_{20}+A^{0}B_{1}{}^{3}C_{30}+D^{0}{}_{1}{}E_{0}&=T^{0}{}_{1}{}_{0}\\A^{1}B_{0}{}^{0}C_{00}+A^{1}B_{0}{}^{1}C_{10}+A^{1}B_{0}{}^{2}C_{20}+A^{1}B_{0}{}^{3}C_{30}+D^{1}{}_{0}{}E_{0}&=T^{1}{}_{0}{}_{0}\\A^{1}B_{2}{}^{0}C_{02}+A^{1}B_{2}{}^{1}C_{12}+A^{1}B_{2}{}^{2}C_{22}+A^{1}B_{2}{}^{3}C_{32}+D^{1}{}_{2}{}E_{2}&=T^{1}{}_{2}{}_{2}.\end{aligned}}$
This illustrates the compactness and efficiency of using index notation: many equations which all share a similar structure can be collected into one simple tensor equation.
Indices are replaceable labels
Replacing any index symbol throughout by another leaves the tensor equation unchanged (provided there is no conflict with other symbols already used). This can be useful when manipulating indices, such as using index notation to verify vector calculus identities or identities of the Kronecker delta and Levi-Civita symbol (see also below). An example of a correct change is:
$A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D^{\alpha }{}_{\beta }{}E_{\delta }\rightarrow A^{\lambda }B_{\beta }{}^{\mu }C_{\mu \delta }+D^{\lambda }{}_{\beta }{}E_{\delta }\,,$
whereas an erroneous change is:
$A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D^{\alpha }{}_{\beta }{}E_{\delta }\nrightarrow A^{\lambda }B_{\beta }{}^{\gamma }C_{\mu \delta }+D^{\alpha }{}_{\beta }{}E_{\delta }\,.$
In the first replacement, λ replaced α and μ replaced γ everywhere, so the expression still has the same meaning. In the second, λ did not fully replace α, and μ did not fully replace γ (incidentally, the contraction on the γ index became a tensor product), which is entirely inconsistent for reasons shown next.
Indices are the same in every term
The free indices in a tensor expression always appear in the same (upper or lower) position throughout every term, and in a tensor equation the free indices are the same on each side. Dummy indices (which implies a summation over that index) need not be the same, for example:
$A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D^{\alpha }{}_{\delta }E_{\beta }=T^{\alpha }{}_{\beta }{}_{\delta }$
as for an erroneous expression:
$A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D_{\alpha }{}_{\beta }{}^{\gamma }E^{\delta }.$
In other words, non-repeated indices must be of the same type in every term of the equation. In the above identity, α, β, δ line up throughout and γ occurs twice in one term due to a contraction (once as an upper index and once as a lower index), and thus it is a valid expression. In the invalid expression, while β lines up, α and δ do not, and γ appears twice in one term (contraction) and once in another term, which is inconsistent.
Brackets and punctuation used once where implied
When applying a rule to a number of indices (differentiation, symmetrization etc., shown next), the bracket or punctuation symbols denoting the rules are only shown on one group of the indices to which they apply.
If the brackets enclose covariant indices – the rule applies only to all covariant indices enclosed in the brackets, not to any contravariant indices which happen to be placed intermediately between the brackets.
Similarly if brackets enclose contravariant indices – the rule applies only to all enclosed contravariant indices, not to intermediately placed covariant indices.
Symmetric and antisymmetric parts
Symmetric part of tensor
Parentheses, ( ), around multiple indices denotes the symmetrized part of the tensor. When symmetrizing p indices using σ to range over permutations of the numbers 1 to p, one takes a sum over the permutations of those indices ασ(i) for i = 1, 2, 3, …, p, and then divides by the number of permutations:
$A_{(\alpha _{1}\alpha _{2}\cdots \alpha _{p})\alpha _{p+1}\cdots \alpha _{q}}={\dfrac {1}{p!}}\sum _{\sigma }A_{\alpha _{\sigma (1)}\cdots \alpha _{\sigma (p)}\alpha _{p+1}\cdots \alpha _{q}}\,.$
For example, two symmetrizing indices mean there are two indices to permute and sum over:
$A_{(\alpha \beta )\gamma \cdots }={\dfrac {1}{2!}}\left(A_{\alpha \beta \gamma \cdots }+A_{\beta \alpha \gamma \cdots }\right)$
while for three symmetrizing indices, there are three indices to sum over and permute:
$A_{(\alpha \beta \gamma )\delta \cdots }={\dfrac {1}{3!}}\left(A_{\alpha \beta \gamma \delta \cdots }+A_{\gamma \alpha \beta \delta \cdots }+A_{\beta \gamma \alpha \delta \cdots }+A_{\alpha \gamma \beta \delta \cdots }+A_{\gamma \beta \alpha \delta \cdots }+A_{\beta \alpha \gamma \delta \cdots }\right)$
The symmetrization is distributive over addition;
$A_{(\alpha }\left(B_{\beta )\gamma \cdots }+C_{\beta )\gamma \cdots }\right)=A_{(\alpha }B_{\beta )\gamma \cdots }+A_{(\alpha }C_{\beta )\gamma \cdots }$
Indices are not part of the symmetrization when they are:
• not on the same level, for example;
$A_{(\alpha }B^{\beta }{}_{\gamma )}={\dfrac {1}{2!}}\left(A_{\alpha }B^{\beta }{}_{\gamma }+A_{\gamma }B^{\beta }{}_{\alpha }\right)$
• within the parentheses and between vertical bars (i.e. |⋅⋅⋅|), modifying the previous example;
$A_{(\alpha }B_{|\beta |}{}_{\gamma )}={\dfrac {1}{2!}}\left(A_{\alpha }B_{\beta \gamma }+A_{\gamma }B_{\beta \alpha }\right)$
Here the α and γ indices are symmetrized, β is not.
Antisymmetric or alternating part of tensor
Square brackets, [ ], around multiple indices denotes the antisymmetrized part of the tensor. For p antisymmetrizing indices – the sum over the permutations of those indices ασ(i) multiplied by the signature of the permutation sgn(σ) is taken, then divided by the number of permutations:
${\begin{aligned}&A_{[\alpha _{1}\cdots \alpha _{p}]\alpha _{p+1}\cdots \alpha _{q}}\\[3pt]={}&{\dfrac {1}{p!}}\sum _{\sigma }\operatorname {sgn}(\sigma )A_{\alpha _{\sigma (1)}\cdots \alpha _{\sigma (p)}\alpha _{p+1}\cdots \alpha _{q}}\\={}&\delta _{\alpha _{1}\cdots \alpha _{p}}^{\beta _{1}\dots \beta _{p}}A_{\beta _{1}\cdots \beta _{p}\alpha _{p+1}\cdots \alpha _{q}}\\\end{aligned}}$
where δβ1⋅⋅⋅βp
α1⋅⋅⋅αp
is the generalized Kronecker delta of degree 2p, with scaling as defined below.
For example, two antisymmetrizing indices imply:
$A_{[\alpha \beta ]\gamma \cdots }={\dfrac {1}{2!}}\left(A_{\alpha \beta \gamma \cdots }-A_{\beta \alpha \gamma \cdots }\right)$
while three antisymmetrizing indices imply:
$A_{[\alpha \beta \gamma ]\delta \cdots }={\dfrac {1}{3!}}\left(A_{\alpha \beta \gamma \delta \cdots }+A_{\gamma \alpha \beta \delta \cdots }+A_{\beta \gamma \alpha \delta \cdots }-A_{\alpha \gamma \beta \delta \cdots }-A_{\gamma \beta \alpha \delta \cdots }-A_{\beta \alpha \gamma \delta \cdots }\right)$
as for a more specific example, if F represents the electromagnetic tensor, then the equation
$0=F_{[\alpha \beta ,\gamma ]}={\dfrac {1}{3!}}\left(F_{\alpha \beta ,\gamma }+F_{\gamma \alpha ,\beta }+F_{\beta \gamma ,\alpha }-F_{\beta \alpha ,\gamma }-F_{\alpha \gamma ,\beta }-F_{\gamma \beta ,\alpha }\right)\,$
represents Gauss's law for magnetism and Faraday's law of induction.
As before, the antisymmetrization is distributive over addition;
$A_{[\alpha }\left(B_{\beta ]\gamma \cdots }+C_{\beta ]\gamma \cdots }\right)=A_{[\alpha }B_{\beta ]\gamma \cdots }+A_{[\alpha }C_{\beta ]\gamma \cdots }$
As with symmetrization, indices are not antisymmetrized when they are:
• not on the same level, for example;
$A_{[\alpha }B^{\beta }{}_{\gamma ]}={\dfrac {1}{2!}}\left(A_{\alpha }B^{\beta }{}_{\gamma }-A_{\gamma }B^{\beta }{}_{\alpha }\right)$
• within the square brackets and between vertical bars (i.e. |⋅⋅⋅|), modifying the previous example;
$A_{[\alpha }B_{|\beta |}{}_{\gamma ]}={\dfrac {1}{2!}}\left(A_{\alpha }B_{\beta \gamma }-A_{\gamma }B_{\beta \alpha }\right)$
Here the α and γ indices are antisymmetrized, β is not.
Sum of symmetric and antisymmetric parts
Any tensor can be written as the sum of its symmetric and antisymmetric parts on two indices:
$A_{\alpha \beta \gamma \cdots }=A_{(\alpha \beta )\gamma \cdots }+A_{[\alpha \beta ]\gamma \cdots }$
as can be seen by adding the above expressions for A(αβ)γ⋅⋅⋅ and A[αβ]γ⋅⋅⋅. This does not hold for other than two indices.
Differentiation
See also: Four-gradient, d'Alembertian, and Intrinsic derivative
For compactness, derivatives may be indicated by adding indices after a comma or semicolon.[11][12]
Partial derivative
While most of the expressions of the Ricci calculus are valid for arbitrary bases, the expressions involving partial derivatives of tensor components with respect to coordinates apply only with a coordinate basis: a basis that is defined through differentiation with respect to the coordinates. Coordinates are typically denoted by xμ, but do not in general form the components of a vector. In flat spacetime with linear coordinatization, a tuple of differences in coordinates, Δxμ, can be treated as a contravariant vector. With the same constraints on the space and on the choice of coordinate system, the partial derivatives with respect to the coordinates yield a result that is effectively covariant. Aside from use in this special case, the partial derivatives of components of tensors do not in general transform covariantly, but are useful in building expressions that are covariant, albeit still with a coordinate basis if the partial derivatives are explicitly used, as with the covariant, exterior and Lie derivatives below.
To indicate partial differentiation of the components of a tensor field with respect to a coordinate variable xγ, a comma is placed before an appended lower index of the coordinate variable.
$A_{\alpha \beta \cdots ,\gamma }={\dfrac {\partial }{\partial x^{\gamma }}}A_{\alpha \beta \cdots }$
This may be repeated (without adding further commas):
$A_{\alpha _{1}\alpha _{2}\cdots \alpha _{p}\,,\,\alpha _{p+1}\cdots \alpha _{q}}={\dfrac {\partial }{\partial x^{\alpha _{q}}}}\cdots {\dfrac {\partial }{\partial x^{\alpha _{p+2}}}}{\dfrac {\partial }{\partial x^{\alpha _{p+1}}}}A_{\alpha _{1}\alpha _{2}\cdots \alpha _{p}}.$
These components do not transform covariantly, unless the expression being differentiated is a scalar. This derivative is characterized by the product rule and the derivatives of the coordinates
$x^{\alpha }{}_{,\gamma }=\delta _{\gamma }^{\alpha },$
where δ is the Kronecker delta.
Covariant derivative
The covariant derivative is only defined if a connection is defined. For any tensor field, a semicolon ( ; ) placed before an appended lower (covariant) index indicates covariant differentiation. Less common alternatives to the semicolon include a forward slash ( / )[13] or in three-dimensional curved space a single vertical bar ( | ).[14]
The covariant derivative of a scalar function, a contravariant vector and a covariant vector are:
$f_{;\beta }=f_{,\beta }$
$A^{\alpha }{}_{;\beta }=A^{\alpha }{}_{,\beta }+\Gamma ^{\alpha }{}_{\gamma \beta }A^{\gamma }$
$A_{\alpha ;\beta }=A_{\alpha ,\beta }-\Gamma ^{\gamma }{}_{\alpha \beta }A_{\gamma }\,,$ ;\beta }=A_{\alpha ,\beta }-\Gamma ^{\gamma }{}_{\alpha \beta }A_{\gamma }\,,}
where Γαγβ are the connection coefficients.
For an arbitrary tensor:[15]
${\begin{aligned}T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s};\gamma }&\\=T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s},\gamma }&+\,\Gamma ^{\alpha _{1}}{}_{\delta \gamma }T^{\delta \alpha _{2}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}+\cdots +\Gamma ^{\alpha _{r}}{}_{\delta \gamma }T^{\alpha _{1}\cdots \alpha _{r-1}\delta }{}_{\beta _{1}\cdots \beta _{s}}\\&-\,\Gamma ^{\delta }{}_{\beta _{1}\gamma }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\delta \beta _{2}\cdots \beta _{s}}-\cdots -\Gamma ^{\delta }{}_{\beta _{s}\gamma }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s-1}\delta }\,.\end{aligned}}$
An alternative notation for the covariant derivative of any tensor is the subscripted nabla symbol ∇β. For the case of a vector field Aα:[16]
$\nabla _{\beta }A^{\alpha }=A^{\alpha }{}_{;\beta }\,.$
The covariant formulation of the directional derivative of any tensor field along a vector vγ may be expressed as its contraction with the covariant derivative, e.g.:
$v^{\gamma }A_{\alpha ;\gamma }\,.$ ;\gamma }\,.}
The components of this derivative of a tensor field transform covariantly, and hence form another tensor field, despite subexpressions (the partial derivative and the connection coefficients) separately not transforming covariantly.
This derivative is characterized by the product rule:
$(A^{\alpha }{}_{\beta \cdots }B^{\gamma }{}_{\delta \cdots })_{;\epsilon }=A^{\alpha }{}_{\beta \cdots ;\epsilon }B^{\gamma }{}_{\delta \cdots }+A^{\alpha }{}_{\beta \cdots }B^{\gamma }{}_{\delta \cdots ;\epsilon }\,.$ ;\epsilon }B^{\gamma }{}_{\delta \cdots }+A^{\alpha }{}_{\beta \cdots }B^{\gamma }{}_{\delta \cdots ;\epsilon }\,.}
Connection types
A Koszul connection on the tangent bundle of a differentiable manifold is called an affine connection.
A connection is a metric connection when the covariant derivative of the metric tensor vanishes:
$g_{\mu \nu ;\xi }=0\,.$ ;\xi }=0\,.}
An affine connection that is also a metric connection is called a Riemannian connection. A Riemannian connection that is torsion-free (i.e., for which the torsion tensor vanishes: Tαβγ = 0) is a Levi-Civita connection.
The Γαβγ for a Levi-Civita connection in a coordinate basis are called Christoffel symbols of the second kind.
Exterior derivative
The exterior derivative of a totally antisymmetric type (0, s) tensor field with components Aα1⋅⋅⋅αs (also called a differential form) is a derivative that is covariant under basis transformations. It does not depend on either a metric tensor or a connection: it requires only the structure of a differentiable manifold. In a coordinate basis, it may be expressed as the antisymmetrization of the partial derivatives of the tensor components:[17]: 232–233
$(\mathrm {d} A)_{\gamma \alpha _{1}\cdots \alpha _{s}}={\frac {\partial }{\partial x^{[\gamma }}}A_{\alpha _{1}\cdots \alpha _{s}]}=A_{[\alpha _{1}\cdots \alpha _{s},\gamma ]}.$
This derivative is not defined on any tensor field with contravariant indices or that is not totally antisymmetric. It is characterized by a graded product rule.
Lie derivative
The Lie derivative is another derivative that is covariant under basis transformations. Like the exterior derivative, it does not depend on either a metric tensor or a connection. The Lie derivative of a type (r, s) tensor field T along (the flow of) a contravariant vector field Xρ may be expressed using a coordinate basis as[18]
${\begin{aligned}({\mathcal {L}}_{X}T)^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}&\\=X^{\gamma }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s},\gamma }&-\,X^{\alpha _{1}}{}_{,\gamma }T^{\gamma \alpha _{2}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}-\cdots -X^{\alpha _{r}}{}_{,\gamma }T^{\alpha _{1}\cdots \alpha _{r-1}\gamma }{}_{\beta _{1}\cdots \beta _{s}}\\&+\,X^{\gamma }{}_{,\beta _{1}}T^{\alpha _{1}\cdots \alpha _{r}}{}_{\gamma \beta _{2}\cdots \beta _{s}}+\cdots +X^{\gamma }{}_{,\beta _{s}}T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s-1}\gamma }\,.\end{aligned}}$
This derivative is characterized by the product rule and the fact that the Lie derivative of a contravariant vector field along itself is zero:
$({\mathcal {L}}_{X}X)^{\alpha }=X^{\gamma }X^{\alpha }{}_{,\gamma }-X^{\alpha }{}_{,\gamma }X^{\gamma }=0\,.$
Notable tensors
Kronecker delta
The Kronecker delta is like the identity matrix when multiplied and contracted:
${\begin{aligned}\delta _{\beta }^{\alpha }\,A^{\beta }&=A^{\alpha }\\\delta _{\nu }^{\mu }\,B_{\mu }&=B_{\nu }.\end{aligned}}$
The components δα
β
are the same in any basis and form an invariant tensor of type (1, 1), i.e. the identity of the tangent bundle over the identity mapping of the base manifold, and so its trace is an invariant.[19] Its trace is the dimensionality of the space; for example, in four-dimensional spacetime,
$\delta _{\rho }^{\rho }=\delta _{0}^{0}+\delta _{1}^{1}+\delta _{2}^{2}+\delta _{3}^{3}=4.$
The Kronecker delta is one of the family of generalized Kronecker deltas. The generalized Kronecker delta of degree 2p may be defined in terms of the Kronecker delta by (a common definition includes an additional multiplier of p! on the right):
$\delta _{\beta _{1}\cdots \beta _{p}}^{\alpha _{1}\cdots \alpha _{p}}=\delta _{\beta _{1}}^{[\alpha _{1}}\cdots \delta _{\beta _{p}}^{\alpha _{p}]},$
and acts as an antisymmetrizer on p indices:
$\delta _{\beta _{1}\cdots \beta _{p}}^{\alpha _{1}\cdots \alpha _{p}}\,A^{\beta _{1}\cdots \beta _{p}}=A^{[\alpha _{1}\cdots \alpha _{p}]}.$
Torsion tensor
An affine connection has a torsion tensor Tαβγ:
$T^{\alpha }{}_{\beta \gamma }=\Gamma ^{\alpha }{}_{\beta \gamma }-\Gamma ^{\alpha }{}_{\gamma \beta }-\gamma ^{\alpha }{}_{\beta \gamma },$
where γαβγ are given by the components of the Lie bracket of the local basis, which vanish when it is a coordinate basis.
For a Levi-Civita connection this tensor is defined to be zero, which for a coordinate basis gives the equations
$\Gamma ^{\alpha }{}_{\beta \gamma }=\Gamma ^{\alpha }{}_{\gamma \beta }.$
Riemann curvature tensor
If this tensor is defined as
$R^{\rho }{}_{\sigma \mu \nu }=\Gamma ^{\rho }{}_{\nu \sigma ,\mu }-\Gamma ^{\rho }{}_{\mu \sigma ,\nu }+\Gamma ^{\rho }{}_{\mu \lambda }\Gamma ^{\lambda }{}_{\nu \sigma }-\Gamma ^{\rho }{}_{\nu \lambda }\Gamma ^{\lambda }{}_{\mu \sigma }\,,$
then it is the commutator of the covariant derivative with itself:[20][21]
$A_{\nu ;\rho \sigma }-A_{\nu ;\sigma \rho }=A_{\beta }R^{\beta }{}_{\nu \rho \sigma }\,,$ ;\rho \sigma }-A_{\nu ;\sigma \rho }=A_{\beta }R^{\beta }{}_{\nu \rho \sigma }\,,}
since the connection is torsionless, which means that the torsion tensor vanishes.
This can be generalized to get the commutator for two covariant derivatives of an arbitrary tensor as follows:
${\begin{aligned}T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s};\gamma \delta }&-T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s};\delta \gamma }\\&\!\!\!\!\!\!\!\!\!\!=-R^{\alpha _{1}}{}_{\rho \gamma \delta }T^{\rho \alpha _{2}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}-\cdots -R^{\alpha _{r}}{}_{\rho \gamma \delta }T^{\alpha _{1}\cdots \alpha _{r-1}\rho }{}_{\beta _{1}\cdots \beta _{s}}\\&+R^{\sigma }{}_{\beta _{1}\gamma \delta }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\sigma \beta _{2}\cdots \beta _{s}}+\cdots +R^{\sigma }{}_{\beta _{s}\gamma \delta }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s-1}\sigma }\,\end{aligned}}$
which are often referred to as the Ricci identities.[22]
Metric tensor
The metric tensor gαβ is used for lowering indices and gives the length of any space-like curve
${\text{length}}=\int _{y_{1}}^{y_{2}}{\sqrt {g_{\alpha \beta }{\frac {dx^{\alpha }}{d\gamma }}{\frac {dx^{\beta }}{d\gamma }}}}\,d\gamma \,,$
where γ is any smooth strictly monotone parameterization of the path. It also gives the duration of any time-like curve
${\text{duration}}=\int _{t_{1}}^{t_{2}}{\sqrt {{\frac {-1}{c^{2}}}g_{\alpha \beta }{\frac {dx^{\alpha }}{d\gamma }}{\frac {dx^{\beta }}{d\gamma }}}}\,d\gamma \,,$
where γ is any smooth strictly monotone parameterization of the trajectory. See also Line element.
The inverse matrix gαβ of the metric tensor is another important tensor, used for raising indices:
$g^{\alpha \beta }g_{\beta \gamma }=\delta _{\gamma }^{\alpha }\,.$
Part of a series of articles about
Calculus
• Fundamental theorem
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Definitions
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• infinitesimal
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In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime).
Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita,[23] it was used by Albert Einstein to develop his general theory of relativity. Unlike the infinitesimal calculus, tensor calculus allows presentation of physics equations in a form that is independent of the choice of coordinates on the manifold.
Tensor calculus
Tensor calculus has many applications in physics, engineering and computer science including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory, and machine learning.
Working with a main proponent of the exterior calculus Elie Cartan, the influential geometer Shiing-Shen Chern summarizes the role of tensor calculus:[24]
In our subject of differential geometry, where you talk about manifolds, one difficulty is that the geometry is described by coordinates, but the coordinates do not have meaning. They are allowed to undergo transformation. And in order to handle this kind of situation, an important tool is the so-called tensor analysis, or Ricci calculus, which was new to mathematicians. In mathematics you have a function, you write down the function, you calculate, or you add, or you multiply, or you can differentiate. You have something very concrete. In geometry the geometric situation is described by numbers, but you can change your numbers arbitrarily. So to handle this, you need the Ricci calculus.
Syntax
Tensor notation makes use of upper and lower indexes on objects that are used to label a variable object as covariant (lower index), contravariant (upper index), or mixed covariant and contravariant (having both upper and lower indexes). In fact in conventional math syntax we make use of covariant indexes when dealing with Cartesian coordinate systems $(x_{1},x_{2},x_{3})$ frequently without realizing this is a limited use of tensor syntax as covariant indexed components.
Tensor notation allows upper index on an object that may be confused with normal power operations from conventional math syntax.
Key concepts
Vector decomposition
Tensors notation allows a vector (${\vec {V}}$) to be decomposed into an Einstein summation representing the tensor contraction of a basis vector (${\vec {Z}}_{i}$ or ${\vec {Z}}^{i}$) with a component vector ($V_{i}$ or $V^{i}$).
${\vec {V}}=V^{i}{\vec {Z}}_{i}=V_{i}{\vec {Z}}^{i}$
Every vector has two different representations, one referred to as contravariant component ($V^{i}$) with a covariant basis (${\vec {Z}}_{i}$), and the other as a covariant component ($V_{i}$) with a contravariant basis (${\vec {Z}}^{i}$). Tensor objects with all upper indexes are referred to as contravariant, and tensor objects with all lower indexes are referred to as covariant. The need to distinguish between contravariant and covariant arises from the fact that when we dot an arbitrary vector with its basis vector related to a particular coordinate system, there are two ways of interpreting this dot product, either we view it as the projection of the basis vector onto the arbitrary vector, or we view it as the projection of the arbitrary vector onto the basis vector, both views of the dot product are entirely equivalent, but have different component elements and different basis vectors:
${\vec {V}}\cdot {\vec {Z}}_{i}=V_{i}={\vec {V}}^{T}{\vec {Z}}_{i}={\vec {Z}}_{i}^{T}{\vec {V}}={\mathrm {proj} _{{\vec {Z}}^{i}}({\vec {V}})}\cdot {\vec {Z}}_{i}={\mathrm {proj} _{\vec {V}}({\vec {Z}}^{i})}\cdot {\vec {V}}$
${\vec {V}}\cdot {\vec {Z}}^{i}=V^{i}={\vec {V}}^{T}{\vec {Z}}^{i}={{\vec {Z}}^{i}}^{T}{\vec {V}}={\mathrm {proj} _{{\vec {Z}}_{i}}({\vec {V}})}\cdot {\vec {Z}}^{i}={\mathrm {proj} _{\vec {V}}({\vec {Z}}_{i})}\cdot {\vec {V}}$
For example, in physics you start with a vector field, you decompose it with respect to the covariant basis, and that's how you get the contravariant coordinates. For orthonormal cartesian coordinates, the covariant and contravariant basis are identical, since the basis set in this case is just the identity matrix, however, for non-affine coordinate system such as polar or spherical there is a need to distinguish between decomposition by use of contravariant or covariant basis set for generating the components of the coordinate system.
Covariant vector decomposition
${\vec {V}}=V^{i}{\vec {Z}}_{i}$
variable description Type
${\vec {V}}$ vector invariant
$V^{i}$ contravariant components (ordered set of scalars) variant
${\vec {Z}}_{i}$ covariant bases (ordered set of vectors) variant
Contravariant vector decomposition
${\vec {V}}=V_{i}{\vec {Z}}^{i}$
variable description type
${\vec {V}}$ vector invariant
$V_{i}$ covariant components (ordered set of scalars) variant
${\vec {Z}}^{i}$ contravariant bases (ordered set of covectors) variant
Metric tensor
The metric tensor represents a matrix with scalar elements ($Z_{ij}$ or $Z^{ij}$) and is a tensor object which is used to raise or lower the index on another tensor object by an operation called contraction, thus allowing a covariant tensor to be converted to a contravariant tensor, and vice versa.
Example of lowering index using metric tensor:
$T_{i}=Z_{ij}T^{j}$
Example of raising index using metric tensor:
$T^{i}=Z^{ij}T_{j}$
The metric tensor is defined as:
$Z_{ij}={\vec {Z}}_{i}\cdot {\vec {Z}}_{j}$
$Z^{ij}={\vec {Z}}^{i}\cdot {\vec {Z}}^{j}$
This means that if we take every permutation of a basis vector set and dotted them against each other, and then arrange them into a square matrix, we would have a metric tensor. The caveat here is which of the two vectors in the permutation is used for projection against the other vector, that is the distinguishing property of the covariant metric tensor in comparison with the contravariant metric tensor.
Two flavors of metric tensors exist: (1) the contravariant metric tensor ($Z^{ij}$), and (2) the covariant metric tensor ($Z_{ij}$). These two flavors of metric tensor are related by the identity:
$Z_{ik}Z^{jk}=\delta _{i}^{j}$
For an orthonormal Cartesian coordinate system, the metric tensor is just the kronecker delta $\delta _{ij}$ or $\delta ^{ij}$, which is just a tensor equivalent of the identity matrix, and $\delta _{ij}=\delta ^{ij}=\delta _{j}^{i}$.
Jacobian
In addition a tensor can be readily converted from an unbarred($x$) to a barred coordinate(${\bar {x}}$) system having different sets of basis vectors:
$f(x^{1},x^{2},\dots ,x^{n})=f{\bigg (}x^{1}({\bar {x}}),x^{2}({\bar {x}}),\dots ,x^{n}({\bar {x}}){\bigg )}={\bar {f}}({\bar {x}}^{1},{\bar {x}}^{2},\dots ,{\bar {x}}^{n})={\bar {f}}{\bigg (}{\bar {x}}^{1}(x),{\bar {x}}^{2}(x),\dots ,{\bar {x}}^{n}(x){\bigg )}$
by use of Jacobian matrix relationships between the barred and unbarred coordinate system (${\bar {J}}=J^{-1}$). The Jacobian between the barred and unbarred system is instrumental in defining the covariant and contravariant basis vectors, in that in order for these vectors to exist they need to satisfy the following relationship relative to the barred and unbarred system:
Contravariant vectors are required to obey the laws:
$v^{i}={\bar {v}}^{r}{\frac {\partial x^{i}({\bar {x}})}{\partial {\bar {x}}^{r}}}$
${\bar {v}}^{i}=v^{r}{\frac {\partial {\bar {x}}^{i}(x)}{\partial x^{r}}}$
Covariant vectors are required to obey the laws:
$v_{i}={\bar {v}}_{r}{\frac {\partial {\bar {x}}^{i}(x)}{\partial x^{r}}}$
${\bar {v}}_{i}=v_{r}{\frac {\partial x^{r}({\bar {x}})}{\partial {\bar {x}}^{i}}}$
There are two flavors of Jacobian matrix:
1. The J matrix representing the change from unbarred to barred coordinates. To find J, we take the "barred gradient", i.e. partial derive with respect to ${\bar {x}}^{i}$:
$J={\bar {\nabla }}f(x({\bar {x}}))$
2. The ${\bar {J}}$ matrix, representing the change from barred to unbarred coordinates. To find ${\bar {J}}$, we take the "unbarred gradient", i.e. partial derive with respect to $x^{i}$:
${\bar {J}}=\nabla {\bar {f}}({\bar {x}}(x))$
Gradient vector
Tensor calculus provides a generalization to the gradient vector formula from standard calculus that works in all coordinate systems:
$\nabla F=\nabla _{i}F{\vec {Z}}^{i}$
Where:
$\nabla _{i}F={\frac {\partial F}{\partial Z^{i}}}$
In contrast, for standard calculus, the gradient vector formula is dependent on the coordinate system in use (example: Cartesian gradient vector formula vs. the polar gradient vector formula vs. the spherical gradient vector formula, etc.). In standard calculus, each coordinate system has its own specific formula, unlike tensor calculus that has only one gradient formula that is equivalent for all coordinate systems. This is made possible by an understanding of the metric tensor that tensor calculus makes use of.
Tensor calculus
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In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime).
Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita,[25] it was used by Albert Einstein to develop his general theory of relativity. Unlike the infinitesimal calculus, tensor calculus allows presentation of physics equations in a form that is independent of the choice of coordinates on the manifold.
Tensor calculus has many applications in physics, engineering and computer science including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory, and machine learning.
Working with a main proponent of the exterior calculus Elie Cartan, the influential geometer Shiing-Shen Chern summarizes the role of tensor calculus:[26]
In our subject of differential geometry, where you talk about manifolds, one difficulty is that the geometry is described by coordinates, but the coordinates do not have meaning. They are allowed to undergo transformation. And in order to handle this kind of situation, an important tool is the so-called tensor analysis, or Ricci calculus, which was new to mathematicians. In mathematics you have a function, you write down the function, you calculate, or you add, or you multiply, or you can differentiate. You have something very concrete. In geometry the geometric situation is described by numbers, but you can change your numbers arbitrarily. So to handle this, you need the Ricci calculus.
Syntax
Tensor notation makes use of upper and lower indexes on objects that are used to label a variable object as covariant (lower index), contravariant (upper index), or mixed covariant and contravariant (having both upper and lower indexes). In fact in conventional math syntax we make use of covariant indexes when dealing with Cartesian coordinate systems $(x_{1},x_{2},x_{3})$ frequently without realizing this is a limited use of tensor syntax as covariant indexed components.
Tensor notation allows upper index on an object that may be confused with normal power operations from conventional math syntax.
Vector decomposition
Tensors notation allows a vector (${\vec {V}}$) to be decomposed into an Einstein summation representing the tensor contraction of a basis vector (${\vec {Z}}_{i}$ or ${\vec {Z}}^{i}$) with a component vector ($V_{i}$ or $V^{i}$).
${\vec {V}}=V^{i}{\vec {Z}}_{i}=V_{i}{\vec {Z}}^{i}$
Every vector has two different representations, one referred to as contravariant component ($V^{i}$) with a covariant basis (${\vec {Z}}_{i}$), and the other as a covariant component ($V_{i}$) with a contravariant basis (${\vec {Z}}^{i}$). Tensor objects with all upper indexes are referred to as contravariant, and tensor objects with all lower indexes are referred to as covariant. The need to distinguish between contravariant and covariant arises from the fact that when we dot an arbitrary vector with its basis vector related to a particular coordinate system, there are two ways of interpreting this dot product, either we view it as the projection of the basis vector onto the arbitrary vector, or we view it as the projection of the arbitrary vector onto the basis vector, both views of the dot product are entirely equivalent, but have different component elements and different basis vectors:
${\vec {V}}\cdot {\vec {Z}}_{i}=V_{i}={\vec {V}}^{T}{\vec {Z}}_{i}={\vec {Z}}_{i}^{T}{\vec {V}}={\mathrm {proj} _{{\vec {Z}}^{i}}({\vec {V}})}\cdot {\vec {Z}}_{i}={\mathrm {proj} _{\vec {V}}({\vec {Z}}^{i})}\cdot {\vec {V}}$
${\vec {V}}\cdot {\vec {Z}}^{i}=V^{i}={\vec {V}}^{T}{\vec {Z}}^{i}={{\vec {Z}}^{i}}^{T}{\vec {V}}={\mathrm {proj} _{{\vec {Z}}_{i}}({\vec {V}})}\cdot {\vec {Z}}^{i}={\mathrm {proj} _{\vec {V}}({\vec {Z}}_{i})}\cdot {\vec {V}}$
For example, in physics you start with a vector field, you decompose it with respect to the covariant basis, and that's how you get the contravariant coordinates. For orthonormal cartesian coordinates, the covariant and contravariant basis are identical, since the basis set in this case is just the identity matrix, however, for non-affine coordinate system such as polar or spherical there is a need to distinguish between decomposition by use of contravariant or covariant basis set for generating the components of the coordinate system.
Covariant vector decomposition
${\vec {V}}=V^{i}{\vec {Z}}_{i}$
variable description Type
${\vec {V}}$ vector invariant
$V^{i}$ contravariant components (ordered set of scalars) variant
${\vec {Z}}_{i}$ covariant bases (ordered set of vectors) variant
Contravariant vector decomposition
${\vec {V}}=V_{i}{\vec {Z}}^{i}$
variable description type
${\vec {V}}$ vector invariant
$V_{i}$ covariant components (ordered set of scalars) variant
${\vec {Z}}^{i}$ contravariant bases (ordered set of covectors) variant
Metric tensor
The metric tensor represents a matrix with scalar elements ($Z_{ij}$ or $Z^{ij}$) and is a tensor object which is used to raise or lower the index on another tensor object by an operation called contraction, thus allowing a covariant tensor to be converted to a contravariant tensor, and vice versa.
Example of lowering index using metric tensor:
$T_{i}=Z_{ij}T^{j}$
Example of raising index using metric tensor:
$T^{i}=Z^{ij}T_{j}$
The metric tensor is defined as:
$Z_{ij}={\vec {Z}}_{i}\cdot {\vec {Z}}_{j}$
$Z^{ij}={\vec {Z}}^{i}\cdot {\vec {Z}}^{j}$
This means that if we take every permutation of a basis vector set and dotted them against each other, and then arrange them into a square matrix, we would have a metric tensor. The caveat here is which of the two vectors in the permutation is used for projection against the other vector, that is the distinguishing property of the covariant metric tensor in comparison with the contravariant metric tensor.
Two flavors of metric tensors exist: (1) the contravariant metric tensor ($Z^{ij}$), and (2) the covariant metric tensor ($Z_{ij}$). These two flavors of metric tensor are related by the identity:
$Z_{ik}Z^{jk}=\delta _{i}^{j}$
For an orthonormal Cartesian coordinate system, the metric tensor is just the kronecker delta $\delta _{ij}$ or $\delta ^{ij}$, which is just a tensor equivalent of the identity matrix, and $\delta _{ij}=\delta ^{ij}=\delta _{j}^{i}$.
Jacobian
In addition a tensor can be readily converted from an unbarred($x$) to a barred coordinate(${\bar {x}}$) system having different sets of basis vectors:
$f(x^{1},x^{2},\dots ,x^{n})=f{\bigg (}x^{1}({\bar {x}}),x^{2}({\bar {x}}),\dots ,x^{n}({\bar {x}}){\bigg )}={\bar {f}}({\bar {x}}^{1},{\bar {x}}^{2},\dots ,{\bar {x}}^{n})={\bar {f}}{\bigg (}{\bar {x}}^{1}(x),{\bar {x}}^{2}(x),\dots ,{\bar {x}}^{n}(x){\bigg )}$
by use of Jacobian matrix relationships between the barred and unbarred coordinate system (${\bar {J}}=J^{-1}$). The Jacobian between the barred and unbarred system is instrumental in defining the covariant and contravariant basis vectors, in that in order for these vectors to exist they need to satisfy the following relationship relative to the barred and unbarred system:
Contravariant vectors are required to obey the laws:
$v^{i}={\bar {v}}^{r}{\frac {\partial x^{i}({\bar {x}})}{\partial {\bar {x}}^{r}}}$
${\bar {v}}^{i}=v^{r}{\frac {\partial {\bar {x}}^{i}(x)}{\partial x^{r}}}$
Covariant vectors are required to obey the laws:
$v_{i}={\bar {v}}_{r}{\frac {\partial {\bar {x}}^{i}(x)}{\partial x^{r}}}$
${\bar {v}}_{i}=v_{r}{\frac {\partial x^{r}({\bar {x}})}{\partial {\bar {x}}^{i}}}$
There are two flavors of Jacobian matrix:
1. The J matrix representing the change from unbarred to barred coordinates. To find J, we take the "barred gradient", i.e. partial derive with respect to ${\bar {x}}^{i}$:
$J={\bar {\nabla }}f(x({\bar {x}}))$
2. The ${\bar {J}}$ matrix, representing the change from barred to unbarred coordinates. To find ${\bar {J}}$, we take the "unbarred gradient", i.e. partial derive with respect to $x^{i}$:
${\bar {J}}=\nabla {\bar {f}}({\bar {x}}(x))$
Gradient vector
Tensor calculus provides a generalization to the gradient vector formula from standard calculus that works in all coordinate systems:
$\nabla F=\nabla _{i}F{\vec {Z}}^{i}$
Where:
$\nabla _{i}F={\frac {\partial F}{\partial Z^{i}}}$
In contrast, for standard calculus, the gradient vector formula is dependent on the coordinate system in use (example: Cartesian gradient vector formula vs. the polar gradient vector formula vs. the spherical gradient vector formula, etc.). In standard calculus, each coordinate system has its own specific formula, unlike tensor calculus that has only one gradient formula that is equivalent for all coordinate systems. This is made possible by an understanding of the metric tensor that tensor calculus makes use of.
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References
1. "Interview with Shiing Shen Chern" (PDF). Notices of the AMS. 45 (7): 860–5. August 1998.
2. J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. pp. 85–86, §3.5. ISBN 0-7167-0344-0.
3. R. Penrose (2007). The Road to Reality. Vintage books. ISBN 978-0-679-77631-4.
4. Ricci, Gregorio; Levi-Civita, Tullio (March 1900). "Méthodes de calcul différentiel absolu et leurs applications" [Methods of the absolute differential calculus and their applications]. Mathematische Annalen (in French). Springer. 54 (1–2): 125–201. doi:10.1007/BF01454201. S2CID 120009332. Retrieved 19 October 2019.
5. Schouten, Jan A. (1924). R. Courant (ed.). Der Ricci-Kalkül – Eine Einführung in die neueren Methoden und Probleme der mehrdimensionalen Differentialgeometrie (Ricci Calculus – An introduction in the latest methods and problems in multi-dimensional differential geometry). Grundlehren der mathematischen Wissenschaften (in German). Vol. 10. Berlin: Springer Verlag.
6. C. Møller (1952), The Theory of Relativity, p. 234 is an example of a variation: 'Greek indices run from 1 to 3, Latin indices from 1 to 4'
7. T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, p. 67, ISBN 978-1107-602601
8. J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 91. ISBN 0-7167-0344-0.
9. T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, p. 67, ISBN 978-1107-602601
10. J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. pp. 61, 202–203, 232. ISBN 0-7167-0344-0.
11. G. Woan (2010). The Cambridge Handbook of Physics Formulas. Cambridge University Press. ISBN 978-0-521-57507-2.
12. Covariant derivative – Mathworld, Wolfram
13. T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, p. 298, ISBN 978-1107-602601
14. J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. pp. 510, §21.5. ISBN 0-7167-0344-0.
15. T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, p. 299, ISBN 978-1107-602601
16. D. McMahon (2006). Relativity. Demystified. McGraw Hill. p. 67. ISBN 0-07-145545-0.
17. R. Penrose (2007). The Road to Reality. Vintage books. ISBN 978-0-679-77631-4.
18. Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds, p. 130
19. Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds, p. 85
20. Synge J.L.; Schild A. (1949). Tensor Calculus. first Dover Publications 1978 edition. pp. 83, p. 107.
21. P. A. M. Dirac. General Theory of Relativity. pp. 20–21.
22. Lovelock, David; Hanno Rund (1989). Tensors, Differential Forms, and Variational Principles. p. 84.
23. Ricci, Gregorio; Levi-Civita, Tullio (March 1900). "Méthodes de calcul différentiel absolu et leurs applications" [Methods of the absolute differential calculus and their applications]. Mathematische Annalen (in French). Springer. 54 (1–2): 125–201. doi:10.1007/BF01454201. S2CID 120009332.
24. "Interview with Shiing Shen Chern" (PDF). Notices of the AMS. 45 (7): 860–5. August 1998.
25. Ricci, Gregorio; Levi-Civita, Tullio (March 1900). "Méthodes de calcul différentiel absolu et leurs applications" [Methods of the absolute differential calculus and their applications]. Mathematische Annalen (in French). Springer. 54 (1–2): 125–201. doi:10.1007/BF01454201. S2CID 120009332.
26. "Interview with Shiing Shen Chern" (PDF). Notices of the AMS. 45 (7): 860–5. August 1998.
Further reading
• Dimitrienko, Yuriy (2002). Tensor Analysis and Nonlinear Tensor Functions. Springer. ISBN 1-4020-1015-X.
• Sokolnikoff, Ivan S (1951). Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua. Wiley. ISBN 0471810525.
• Borisenko, A.I.; Tarapov, I.E. (1979). Vector and Tensor Analysis with Applications (2nd ed.). Dover. ISBN 0486638332.
• Itskov, Mikhail (2015). Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics (2nd ed.). Springer. ISBN 9783319163420.
• Tyldesley, J. R. (1973). An introduction to Tensor Analysis: For Engineers and Applied Scientists. Longman. ISBN 0-582-44355-5.
• Kay, D. C. (1988). Tensor Calculus. Schaum’s Outlines. McGraw Hill. ISBN 0-07-033484-6.
• Grinfeld, P. (2014). Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer. ISBN 978-1-4614-7866-9.
• Dimitrienko, Yuriy (2002). Tensor Analysis and Nonlinear Tensor Functions. Springer. ISBN 1-4020-1015-X.
• Sokolnikoff, Ivan S (1951). Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua. Wiley. ISBN 0471810525.
• Borisenko, A.I.; Tarapov, I.E. (1979). Vector and Tensor Analysis with Applications (2nd ed.). Dover. ISBN 0486638332.
• Itskov, Mikhail (2015). Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics (2nd ed.). Springer. ISBN 9783319163420.
• Tyldesley, J. R. (1973). An introduction to Tensor Analysis: For Engineers and Applied Scientists. Longman. ISBN 0-582-44355-5.
• Kay, D. C. (1988). Tensor Calculus. Schaum’s Outlines. McGraw Hill. ISBN 0-07-033484-6.
• Grinfeld, P. (2014). Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer. ISBN 978-1-4614-7866-9.
External links
• Dullemond, Kees; Peeters, Kasper (1991–2010). "Introduction to Tensor Calculus" (PDF). Retrieved 17 May 2018.
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• Dullemond, Kees; Peeters, Kasper (1991–2010). "Introduction to Tensor Calculus" (PDF). Retrieved 17 May 2018.
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Notes
1. While the raising and lowering of indices is dependent on a metric tensor, the covariant derivative is only dependent on the connection while the exterior derivative and the Lie derivative are dependent on neither.
References
Sources
• Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds (First Dover 1980 ed.), The Macmillan Company, ISBN 0-486-64039-6
• Danielson, Donald A. (2003). Vectors and Tensors in Engineering and Physics (2/e ed.). Westview (Perseus). ISBN 978-0-8133-4080-7.
• Dimitrienko, Yuriy (2002). Tensor Analysis and Nonlinear Tensor Functions. Kluwer Academic Publishers (Springer). ISBN 1-4020-1015-X.
• Lovelock, David; Hanno Rund (1989) [1975]. Tensors, Differential Forms, and Variational Principles. Dover. ISBN 978-0-486-65840-7.
• C. Møller (1952), The Theory of Relativity (3rd ed.), Oxford University Press
• Synge J.L.; Schild A. (1949). Tensor Calculus. first Dover Publications 1978 edition. ISBN 978-0-486-63612-2.
• J.R. Tyldesley (1975), An introduction to Tensor Analysis: For Engineers and Applied Scientists, Longman, ISBN 0-582-44355-5
• D.C. Kay (1988), Tensor Calculus, Schaum’s Outlines, McGraw Hill (USA), ISBN 0-07-033484-6
• T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, ISBN 978-1107-602601
Tensors
Glossary of tensor theory
Scope
Mathematics
• Coordinate system
• Differential geometry
• Dyadic algebra
• Euclidean geometry
• Exterior calculus
• Multilinear algebra
• Tensor algebra
• Tensor calculus
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Notation
• Abstract index notation
• Einstein notation
• Index notation
• Multi-index notation
• Penrose graphical notation
• Ricci calculus
• Tetrad (index notation)
• Van der Waerden notation
• Voigt notation
Tensor
definitions
• Tensor (intrinsic definition)
• Tensor field
• Tensor density
• Tensors in curvilinear coordinates
• Mixed tensor
• Antisymmetric tensor
• Symmetric tensor
• Tensor operator
• Tensor bundle
• Two-point tensor
Operations
• Covariant derivative
• Exterior covariant derivative
• Exterior derivative
• Exterior product
• Hodge star operator
• Lie derivative
• Raising and lowering indices
• Symmetrization
• Tensor contraction
• Tensor product
• Transpose (2nd-order tensors)
Related
abstractions
• Affine connection
• Basis
• Cartan formalism (physics)
• Connection form
• Covariance and contravariance of vectors
• Differential form
• Dimension
• Exterior form
• Fiber bundle
• Geodesic
• Levi-Civita connection
• Linear map
• Manifold
• Matrix
• Multivector
• Pseudotensor
• Spinor
• Vector
• Vector space
Notable tensors
Mathematics
• Kronecker delta
• Levi-Civita symbol
• Metric tensor
• Nonmetricity tensor
• Ricci curvature
• Riemann curvature tensor
• Torsion tensor
• Weyl tensor
Physics
• Moment of inertia
• Angular momentum tensor
• Spin tensor
• Cauchy stress tensor
• stress–energy tensor
• Einstein tensor
• EM tensor
• Gluon field strength tensor
• Metric tensor (GR)
Mathematicians
• Élie Cartan
• Augustin-Louis Cauchy
• Elwin Bruno Christoffel
• Albert Einstein
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• Carl Friedrich Gauss
• Hermann Grassmann
• Tullio Levi-Civita
• Gregorio Ricci-Curbastro
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• Jan Arnoldus Schouten
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• Hermann Weyl
| Wikipedia |
Invariants of tensors
In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor $\mathbf {A} $ are the coefficients of the characteristic polynomial[1]
$\ p(\lambda )=\det(\mathbf {A} -\lambda \mathbf {I} )$,
where $\mathbf {I} $ is the identity operator and $\lambda _{i}\in \mathbb {C} $ represent the polynomial's eigenvalues.
More broadly, any scalar-valued function $f(\mathbf {A} )$ is an invariant of $\mathbf {A} $ if and only if $f(\mathbf {Q} \mathbf {A} \mathbf {Q} ^{T})=f(\mathbf {A} )$ for all orthogonal $\mathbf {Q} $. This means that a formula expressing an invariant in terms of components, $A_{ij}$, will give the same result for all Cartesian bases. For example, even though individual diagonal components of $\mathbf {A} $ will change with a change in basis, the sum of diagonal components will not change.
Properties
The principal invariants do not change with rotations of the coordinate system (they are objective, or in more modern terminology, satisfy the principle of material frame-indifference) and any function of the principal invariants is also objective.
Calculation of the invariants of rank two tensors
In a majority of engineering applications, the principal invariants of (rank two) tensors of dimension three are sought, such as those for the right Cauchy-Green deformation tensor.
Principal invariants
For such tensors, the principal invariants are given by:
${\begin{aligned}I_{1}&=\mathrm {tr} (\mathbf {A} )=A_{11}+A_{22}+A_{33}=\lambda _{1}+\lambda _{2}+\lambda _{3}\\I_{2}&={\frac {1}{2}}\left((\mathrm {tr} (\mathbf {A} ))^{2}-\mathrm {tr} \left(\mathbf {A} ^{2}\right)\right)=A_{11}A_{22}+A_{22}A_{33}+A_{11}A_{33}-A_{12}A_{21}-A_{23}A_{32}-A_{13}A_{31}=\lambda _{1}\lambda _{2}+\lambda _{1}\lambda _{3}+\lambda _{2}\lambda _{3}\\I_{3}&=\det(\mathbf {A} )=-A_{13}A_{22}A_{31}+A_{12}A_{23}A_{31}+A_{13}A_{21}A_{32}-A_{11}A_{23}A_{32}-A_{12}A_{21}A_{33}+A_{11}A_{22}A_{33}=\lambda _{1}\lambda _{2}\lambda _{3}\end{aligned}}$
For symmetric tensors, these definitions are reduced.[2]
The correspondence between the principal invariants and the characteristic polynomial of a tensor, in tandem with the Cayley–Hamilton theorem reveals that
$\ \mathbf {A} ^{3}-I_{1}\mathbf {A} ^{2}+I_{2}\mathbf {A} -I_{3}\mathbf {I} =0$
where $\mathbf {I} $ is the second-order identity tensor.
Main invariants
In addition to the principal invariants listed above, it is also possible to introduce the notion of main invariants[3][4]
${\begin{aligned}J_{1}&=\lambda _{1}+\lambda _{2}+\lambda _{3}=I_{1}\\J_{2}&=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}=I_{1}^{2}-2I_{2}\\J_{3}&=\lambda _{1}^{3}+\lambda _{2}^{3}+\lambda _{3}^{3}=I_{1}^{3}-3I_{1}I_{2}+3I_{3}\end{aligned}}$
which are functions of the principal invariants above. These are the coefficients of the characteristic polynomial of the deviator $\mathbf {A} -(\mathrm {tr} (\mathbf {A} )/3)\mathbf {I} $, such that it is traceless. The separation of a tensor into a component that is a multiple of the identity and a traceless component is standard in hydrodynamics, where the former is called isotropic, providing the modified pressure, and the latter is called deviatoric, providing shear effects.
Mixed invariants
Furthermore, mixed invariants between pairs of rank two tensors may also be defined.[4]
Calculation of the invariants of order two tensors of higher dimension
These may be extracted by evaluating the characteristic polynomial directly, using the Faddeev-LeVerrier algorithm for example.
Calculation of the invariants of higher order tensors
The invariants of rank three, four, and higher order tensors may also be determined.[5]
Engineering applications
A scalar function $f$ that depends entirely on the principal invariants of a tensor is objective, i.e., independent of rotations of the coordinate system. This property is commonly used in formulating closed-form expressions for the strain energy density, or Helmholtz free energy, of a nonlinear material possessing isotropic symmetry.[6]
This technique was first introduced into isotropic turbulence by Howard P. Robertson in 1940 where he was able to derive Kármán–Howarth equation from the invariant principle.[7] George Batchelor and Subrahmanyan Chandrasekhar exploited this technique and developed an extended treatment for axisymmetric turbulence.[8][9][10]
Invariants of non-symmetric tensors
A real tensor $\mathbf {A} $ in 3D (i.e., one with a 3x3 component matrix) has as many as six independent invariants, three being the invariants of its symmetric part and three characterizing the orientation of the axial vector of the skew-symmetric part relative to the principal directions of the symmetric part. For example, if the Cartesian components of $\mathbf {A} $ are
$[A]={\begin{bmatrix}931&5480&-717\\-5120&1650&1090\\1533&-610&1169\end{bmatrix}},$
the first step would be to evaluate the axial vector $\mathbf {w} $ associated with the skew-symmetric part. Specifically, the axial vector has components
${\begin{aligned}w_{1}&={\frac {A_{32}-A_{23}}{2}}=-850\\w_{2}&={\frac {A_{13}-A_{31}}{2}}=-1125\\w_{3}&={\frac {A_{21}-A_{12}}{2}}=-5300\end{aligned}}$
The next step finds the principal values of the symmetric part of $\mathbf {A} $. Even though the eigenvalues of a real non-symmetric tensor might be complex, the eigenvalues of its symmetric part will always be real and therefore can be ordered from largest to smallest. The corresponding orthonormal principal basis directions can be assigned senses to ensure that the axial vector $\mathbf {w} $ points within the first octant. With respect to that special basis, the components of $\mathbf {A} $ are
$[A']={\begin{bmatrix}1875&-2500&3125\\2500&1250&-3750\\-3125&3750&625\end{bmatrix}},$
The first three invariants of $\mathbf {A} $ are the diagonal components of this matrix: $a_{1}=A'_{11}=1875,a_{2}=A'_{22}=1250,a_{3}=A'_{33}=625$ (equal to the ordered principal values of the tensor's symmetric part). The remaining three invariants are the axial vector's components in this basis: $w'_{1}=A'_{32}=3750,w'_{2}=A'_{13}=3125,w'_{3}=A'_{21}=2500$. Note: the magnitude of the axial vector, ${\sqrt {\mathbf {w} \cdot \mathbf {w} }}$, is the sole invariant of the skew part of $\mathbf {A} $, whereas these distinct three invariants characterize (in a sense) "alignment" between the symmetric and skew parts of $\mathbf {A} $. Incidentally, it is a myth that a tensor is positive definite if its eigenvalues are positive. Instead, it is positive definite if and only if the eigenvalues of its symmetric part are positive.
See also
• Symmetric polynomial
• Elementary symmetric polynomial
• Newton's identities
• Invariant theory
References
1. Spencer, A. J. M. (1980). Continuum Mechanics. Longman. ISBN 0-582-44282-6.
2. Kelly, PA. "Lecture Notes: An introduction to Solid Mechanics" (PDF). Retrieved 27 May 2018.
3. Kindlmann, G. "Tensor Invariants and their Gradients" (PDF). Retrieved 24 Jan 2019.
4. Schröder, Jörg; Neff, Patrizio (2010). Poly-, Quasi- and Rank-One Convexity in Applied Mechanics. Springer.
5. Betten, J. (1987). "Irreducible Invariants of Fourth-Order Tensors". Mathematical Modelling. 8: 29–33. doi:10.1016/0270-0255(87)90535-5.
6. Ogden, R. W. (1984). Non-Linear Elastic Deformations. Dover.
7. Robertson, H. P. (1940). "The Invariant Theory of Isotropic Turbulence". Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press. 36 (2): 209–223. Bibcode:1940PCPS...36..209R. doi:10.1017/S0305004100017199. S2CID 122767772.
8. Batchelor, G. K. (1946). "The Theory of Axisymmetric Turbulence". Proc. R. Soc. Lond. A. 186 (1007): 480–502. Bibcode:1946RSPSA.186..480B. doi:10.1098/rspa.1946.0060.
9. Chandrasekhar, S. (1950). "The Theory of Axisymmetric Turbulence". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 242 (855): 557–577. Bibcode:1950RSPTA.242..557C. doi:10.1098/rsta.1950.0010. S2CID 123358727.
10. Chandrasekhar, S. (1950). "The Decay of Axisymmetric Turbulence". Proc. R. Soc. A. 203 (1074): 358–364. Bibcode:1950RSPSA.203..358C. doi:10.1098/rspa.1950.0143. S2CID 121178989.
| Wikipedia |
Multilinear multiplication
In multilinear algebra, applying a map that is the tensor product of linear maps to a tensor is called a multilinear multiplication.
Abstract definition
Let $F$ be a field of characteristic zero, such as $\mathbb {R} $ or $\mathbb {C} $. Let $V_{k}$ be a finite-dimensional vector space over $F$, and let ${\mathcal {A}}\in V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}$ be an order-d simple tensor, i.e., there exist some vectors $\mathbf {v} _{k}\in V_{k}$ such that ${\mathcal {A}}=\mathbf {v} _{1}\otimes \mathbf {v} _{2}\otimes \cdots \otimes \mathbf {v} _{d}$. If we are given a collection of linear maps $A_{k}:V_{k}\to W_{k}$, then the multilinear multiplication of ${\mathcal {A}}$ with $(A_{1},A_{2},\ldots ,A_{d})$ is defined[1] as the action on ${\mathcal {A}}$ of the tensor product of these linear maps,[2] namely
${\begin{aligned}A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d}:V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}&\to W_{1}\otimes W_{2}\otimes \cdots \otimes W_{d},\\\mathbf {v} _{1}\otimes \mathbf {v} _{2}\otimes \cdots \otimes \mathbf {v} _{d}&\mapsto A_{1}(\mathbf {v} _{1})\otimes A_{2}(\mathbf {v} _{2})\otimes \cdots \otimes A_{d}(\mathbf {v} _{d})\end{aligned}}$
Since the tensor product of linear maps is itself a linear map,[2] and because every tensor admits a tensor rank decomposition,[1] the above expression extends linearly to all tensors. That is, for a general tensor ${\mathcal {A}}\in V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}$, the multilinear multiplication is
${\begin{aligned}&{\mathcal {B}}:=(A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d})({\mathcal {A}})\\[4pt]={}&(A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d})\left(\sum _{i=1}^{r}\mathbf {a} _{i}^{1}\otimes \mathbf {a} _{i}^{2}\otimes \cdots \otimes \mathbf {a} _{i}^{d}\right)\\[5pt]={}&\sum _{i=1}^{r}A_{1}(\mathbf {a} _{i}^{1})\otimes A_{2}(\mathbf {a} _{i}^{2})\otimes \cdots \otimes A_{d}(\mathbf {a} _{i}^{d})\end{aligned}}$
where $ {\mathcal {A}}=\sum _{i=1}^{r}\mathbf {a} _{i}^{1}\otimes \mathbf {a} _{i}^{2}\otimes \cdots \otimes \mathbf {a} _{i}^{d}$ with $\mathbf {a} _{i}^{k}\in V_{k}$ is one of ${\mathcal {A}}$'s tensor rank decompositions. The validity of the above expression is not limited to a tensor rank decomposition; in fact, it is valid for any expression of ${\mathcal {A}}$ as a linear combination of pure tensors, which follows from the universal property of the tensor product.
It is standard to use the following shorthand notations in the literature for multilinear multiplications:
$(A_{1},A_{2},\ldots ,A_{d})\cdot {\mathcal {A}}:=(A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d})({\mathcal {A}})$
and
$A_{k}\cdot _{k}{\mathcal {A}}:=(\operatorname {Id} _{V_{1}},\ldots ,\operatorname {Id} _{V_{k-1}},A_{k},\operatorname {Id} _{V_{k+1}},\ldots ,\operatorname {Id} _{V_{d}})\cdot {\mathcal {A}},$
where $\operatorname {Id} _{V_{k}}:V_{k}\to V_{k}$ is the identity operator.
Definition in coordinates
In computational multilinear algebra it is conventional to work in coordinates. Assume that an inner product is fixed on $V_{k}$ and let $V_{k}^{*}$ denote the dual vector space of $V_{k}$. Let $\{e_{1}^{k},\ldots ,e_{n_{k}}^{k}\}$ be a basis for $V_{k}$, let $\{(e_{1}^{k})^{*},\ldots ,(e_{n_{k}}^{k})^{*}\}$ be the dual basis, and let $\{f_{1}^{k},\ldots ,f_{m_{k}}^{k}\}$ be a basis for $W_{k}$. The linear map $ M_{k}=\sum _{i=1}^{m_{k}}\sum _{j=1}^{n_{k}}m_{i,j}^{(k)}f_{i}^{k}\otimes (e_{j}^{k})^{*}$ is then represented by the matrix ${\widehat {M}}_{k}=[m_{i,j}^{(k)}]\in F^{m_{k}\times n_{k}}$. Likewise, with respect to the standard tensor product basis $\{e_{j_{1}}^{1}\otimes e_{j_{2}}^{2}\otimes \cdots \otimes e_{j_{d}}^{d}\}_{j_{1},j_{2},\ldots ,j_{d}}$, the abstract tensor
${\mathcal {A}}=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}e_{j_{1}}^{1}\otimes e_{j_{2}}^{2}\otimes \cdots \otimes e_{j_{d}}^{d}$
is represented by the multidimensional array ${\widehat {\mathcal {A}}}=[a_{j_{1},j_{2},\ldots ,j_{d}}]\in F^{n_{1}\times n_{2}\times \cdots \times n_{d}}$ . Observe that
${\widehat {\mathcal {A}}}=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d},$
where $\mathbf {e} _{j}^{k}\in F^{n_{k}}$ is the jth standard basis vector of $F^{n_{k}}$ and the tensor product of vectors is the affine Segre map $\otimes :(\mathbf {v} ^{(1)},\mathbf {v} ^{(2)},\ldots ,\mathbf {v} ^{(d)})\mapsto [v_{i_{1}}^{(1)}v_{i_{2}}^{(2)}\cdots v_{i_{d}}^{(d)}]_{i_{1},i_{2},\ldots ,i_{d}}$ :(\mathbf {v} ^{(1)},\mathbf {v} ^{(2)},\ldots ,\mathbf {v} ^{(d)})\mapsto [v_{i_{1}}^{(1)}v_{i_{2}}^{(2)}\cdots v_{i_{d}}^{(d)}]_{i_{1},i_{2},\ldots ,i_{d}}} . It follows from the above choices of bases that the multilinear multiplication ${\mathcal {B}}=(M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {A}}$ becomes
${\begin{aligned}{\widehat {\mathcal {B}}}&=({\widehat {M}}_{1},{\widehat {M}}_{2},\ldots ,{\widehat {M}}_{d})\cdot \sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d}\\&=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}({\widehat {M}}_{1},{\widehat {M}}_{2},\ldots ,{\widehat {M}}_{d})\cdot (\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d})\\&=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}({\widehat {M}}_{1}\mathbf {e} _{j_{1}}^{1})\otimes ({\widehat {M}}_{2}\mathbf {e} _{j_{2}}^{2})\otimes \cdots \otimes ({\widehat {M}}_{d}\mathbf {e} _{j_{d}}^{d}).\end{aligned}}$
The resulting tensor ${\widehat {\mathcal {B}}}$ lives in $F^{m_{1}\times m_{2}\times \cdots \times m_{d}}$.
Element-wise definition
From the above expression, an element-wise definition of the multilinear multiplication is obtained. Indeed, since ${\widehat {\mathcal {B}}}$ is a multidimensional array, it may be expressed as
${\widehat {\mathcal {B}}}=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}b_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d},$
where $b_{j_{1},j_{2},\ldots ,j_{d}}\in F$ are the coefficients. Then it follows from the above formulae that
${\begin{aligned}&\left((\mathbf {e} _{i_{1}}^{1})^{T},(\mathbf {e} _{i_{2}}^{2})^{T},\ldots ,(\mathbf {e} _{i_{d}}^{d})^{T}\right)\cdot {\widehat {\mathcal {B}}}\\={}&\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}b_{j_{1},j_{2},\ldots ,j_{d}}\left((\mathbf {e} _{i_{1}}^{1})^{T}\mathbf {e} _{j_{1}}^{1}\right)\otimes \left((\mathbf {e} _{i_{2}}^{2})^{T}\mathbf {e} _{j_{2}}^{2}\right)\otimes \cdots \otimes \left((\mathbf {e} _{i_{d}}^{d})^{T}\mathbf {e} _{j_{d}}^{d}\right)\\={}&\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}b_{j_{1},j_{2},\ldots ,j_{d}}\delta _{i_{1},j_{1}}\cdot \delta _{i_{2},j_{2}}\cdots \delta _{i_{d},j_{d}}\\={}&b_{i_{1},i_{2},\ldots ,i_{d}},\end{aligned}}$
where $\delta _{i,j}$ is the Kronecker delta. Hence, if ${\mathcal {B}}=(M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {A}}$, then
${\begin{aligned}&b_{i_{1},i_{2},\ldots ,i_{d}}=\left((\mathbf {e} _{i_{1}}^{1})^{T},(\mathbf {e} _{i_{2}}^{2})^{T},\ldots ,(\mathbf {e} _{i_{d}}^{d})^{T}\right)\cdot {\widehat {\mathcal {B}}}\\={}&\left((\mathbf {e} _{i_{1}}^{1})^{T},(\mathbf {e} _{i_{2}}^{2})^{T},\ldots ,(\mathbf {e} _{i_{d}}^{d})^{T}\right)\cdot ({\widehat {M}}_{1},{\widehat {M}}_{2},\ldots ,{\widehat {M}}_{d})\cdot \sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d}\\={}&\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}((\mathbf {e} _{i_{1}}^{1})^{T}{\widehat {M}}_{1}\mathbf {e} _{j_{1}}^{1})\otimes ((\mathbf {e} _{i_{2}}^{2})^{T}{\widehat {M}}_{2}\mathbf {e} _{j_{2}}^{2})\otimes \cdots \otimes ((\mathbf {e} _{i_{d}}^{d})^{T}{\widehat {M}}_{d}\mathbf {e} _{j_{d}}^{d})\\={}&\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}m_{i_{1},j_{1}}^{(1)}\cdot m_{i_{2},j_{2}}^{(2)}\cdots m_{i_{d},j_{d}}^{(d)},\end{aligned}}$
where the $m_{i,j}^{(k)}$ are the elements of ${\widehat {M}}_{k}$ as defined above.
Properties
Let ${\mathcal {A}}\in V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}$ be an order-d tensor over the tensor product of $F$-vector spaces.
Since a multilinear multiplication is the tensor product of linear maps, we have the following multilinearity property (in the construction of the map):[1][2]
$A_{1}\otimes \cdots \otimes A_{k-1}\otimes (\alpha A_{k}+\beta B)\otimes A_{k+1}\otimes \cdots \otimes A_{d}=\alpha A_{1}\otimes \cdots \otimes A_{d}+\beta A_{1}\otimes \cdots \otimes A_{k-1}\otimes B\otimes A_{k+1}\otimes \cdots \otimes A_{d}$
Multilinear multiplication is a linear map:[1][2]
$(M_{1},M_{2},\ldots ,M_{d})\cdot (\alpha {\mathcal {A}}+\beta {\mathcal {B}})=\alpha \;(M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {A}}+\beta \;(M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {B}}$
It follows from the definition that the composition of two multilinear multiplications is also a multilinear multiplication:[1][2]
$(M_{1},M_{2},\ldots ,M_{d})\cdot \left((K_{1},K_{2},\ldots ,K_{d})\cdot {\mathcal {A}}\right)=(M_{1}\circ K_{1},M_{2}\circ K_{2},\ldots ,M_{d}\circ K_{d})\cdot {\mathcal {A}},$
where $M_{k}:U_{k}\to W_{k}$ and $K_{k}:V_{k}\to U_{k}$ are linear maps.
Observe specifically that multilinear multiplications in different factors commute,
$M_{k}\cdot _{k}\left(M_{\ell }\cdot _{\ell }{\mathcal {A}}\right)=M_{\ell }\cdot _{\ell }\left(M_{k}\cdot _{k}{\mathcal {A}}\right)=M_{k}\cdot _{k}M_{\ell }\cdot _{\ell }{\mathcal {A}},$
if $k\neq \ell .$
Computation
The factor-k multilinear multiplication $M_{k}\cdot _{k}{\mathcal {A}}$ can be computed in coordinates as follows. Observe first that
${\begin{aligned}M_{k}\cdot _{k}{\mathcal {A}}&=M_{k}\cdot _{k}\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d}\\&=\sum _{j_{1}=1}^{n_{1}}\cdots \sum _{j_{k-1}=1}^{n_{k-1}}\sum _{j_{k+1}=1}^{n_{k+1}}\cdots \sum _{j_{d}=1}^{n_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \cdots \otimes \mathbf {e} _{j_{k-1}}^{k-1}\otimes M_{k}\left(\sum _{j_{k}=1}^{n_{k}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{k}}^{k}\right)\otimes \mathbf {e} _{j_{k+1}}^{k+1}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d}.\end{aligned}}$
Next, since
$F^{n_{1}}\otimes F^{n_{2}}\otimes \cdots \otimes F^{n_{d}}\simeq F^{n_{k}}\otimes (F^{n_{1}}\otimes \cdots \otimes F^{n_{k-1}}\otimes F^{n_{k+1}}\otimes \cdots \otimes F^{n_{d}})\simeq F^{n_{k}}\otimes F^{n_{1}\cdots n_{k-1}n_{k+1}\cdots n_{d}},$
there is a bijective map, called the factor-k standard flattening,[1] denoted by $(\cdot )_{(k)}$, that identifies $M_{k}\cdot _{k}{\mathcal {A}}$ with an element from the latter space, namely
$\left(M_{k}\cdot _{k}{\mathcal {A}}\right)_{(k)}:=\sum _{j_{1}=1}^{n_{1}}\cdots \sum _{j_{k-1}=1}^{n_{k-1}}\sum _{j_{k+1}=1}^{n_{k+1}}\cdots \sum _{j_{d}=1}^{n_{d}}M_{k}\left(\sum _{j_{k}=1}^{n_{k}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{k}}^{k}\right)\otimes \mathbf {e} _{\mu _{k}(j_{1},\ldots ,j_{k-1},j_{k+1},\ldots ,j_{d})}:=M_{k}{\mathcal {A}}_{(k)},$
where $\mathbf {e} _{j}$is the jth standard basis vector of $F^{N_{k}}$, $N_{k}=n_{1}\cdots n_{k-1}n_{k+1}\cdots n_{d}$, and ${\mathcal {A}}_{(k)}\in F^{n_{k}}\otimes F^{N_{k}}\simeq F^{n_{k}\times N_{k}}$ is the factor-k flattening matrix of ${\mathcal {A}}$ whose columns are the factor-k vectors $[a_{j_{1},\ldots ,j_{k-1},i,j_{k+1},\ldots ,j_{d}}]_{i=1}^{n_{k}}$ in some order, determined by the particular choice of the bijective map
$\mu _{k}:[1,n_{1}]\times \cdots \times [1,n_{k-1}]\times [1,n_{k+1}]\times \cdots \times [1,n_{d}]\to [1,N_{k}].$
In other words, the multilinear multiplication $(M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {A}}$ can be computed as a sequence of d factor-k multilinear multiplications, which themselves can be implemented efficiently as classic matrix multiplications.
Applications
The higher-order singular value decomposition (HOSVD) factorizes a tensor given in coordinates ${\mathcal {A}}\in F^{n_{1}\times n_{2}\times \cdots \times n_{d}}$ as the multilinear multiplication ${\mathcal {A}}=(U_{1},U_{2},\ldots ,U_{d})\cdot {\mathcal {S}}$, where $U_{k}\in F^{n_{k}\times n_{k}}$ are orthogonal matrices and ${\mathcal {S}}\in F^{n_{1}\times n_{2}\times \cdots \times n_{d}}$.
Further reading
1. M., Landsberg, J. (2012). Tensors : geometry and applications. Providence, R.I.: American Mathematical Society. ISBN 9780821869079. OCLC 733546583.{{cite book}}: CS1 maint: multiple names: authors list (link)
2. Multilinear Algebra | Werner Greub | Springer. Universitext. Springer. 1978. ISBN 9780387902845.
| Wikipedia |
Tensor network
Tensor networks or tensor network states are a class of variational wave functions used in the study of many-body quantum systems.[1] Tensor networks extend one-dimensional matrix product states to higher dimensions while preserving some of their useful mathematical properties.[2]
The wave function is encoded as a tensor contraction of a network of individual tensors.[3] The structure of the individual tensors can impose global symmetries on the wave function (such as antisymmetry under exchange of fermions) or restrict the wave function to specific quantum numbers, like total charge, angular momentum, or spin. It is also possible to derive strict bounds on quantities like entanglement and correlation length using the mathematical structure of the tensor network.[4] This has made tensor networks useful in theoretical studies of quantum information in many-body systems. They have also proved useful in variational studies of ground states, excited states, and dynamics of strongly correlated many-body systems.[5]
Diagrammatic notation
In general, a tensor network diagram (Penrose diagram) can be viewed as a graph where nodes (or vertices) represent individual tensors, while edges represent summation over an index. Free indices are depicted as edges (or legs) attached to a single vertex only.[6] Sometimes, there is also additional meaning to a node's shape. For instance, one can use trapezoids for unitary matrices or tensors with similar behaviour. This way, flipped trapezoids would be interpreted as complex conjugates to them.
Connection to machine learning
Tensor networks have been adapted for supervised learning,[7] taking advantage of similar mathematical structure in variational studies in quantum mechanics and large-scale machine learning. This crossover has spurred collaboration between researchers in artificial intelligence and quantum information science. In June 2019, Google, the Perimeter Institute for Theoretical Physics, and X (company), released TensorNetwork,[8] an open-source library for efficient tensor calculations.[9]
The main interest in tensor networks and their study from the perspective of machine learning is to reduce the number of trainable parameters (in a layer) by approximating a high-order tensor with a network of lower-order ones. Using the so-called tensor train technique (TT),[10] one can reduce an N-order tensor (containing exponentially many trainable parameters) to a chain of N tensors of order 2 or 3, which gives us a polynomial number of parameters.
See also
• Tensor
• Tensor diagrams
• Tensor contraction
• Tensor Processing Unit (TPU)
• Tensor rank decomposition
• Einstein Notation
References
1. Orús, Román (5 August 2019). "Tensor networks for complex quantum systems". Nature Reviews Physics. 1 (9): 538–550. arXiv:1812.04011. Bibcode:2019NatRP...1..538O. doi:10.1038/s42254-019-0086-7. ISSN 2522-5820. S2CID 118989751.
2. Orús, Román (2014-10-01). "A practical introduction to tensor networks: Matrix product states and projected entangled pair states". Annals of Physics. 349: 117–158. arXiv:1306.2164. Bibcode:2014AnPhy.349..117O. doi:10.1016/j.aop.2014.06.013. ISSN 0003-4916. S2CID 118349602.
3. Biamonte, Jacob; Bergholm, Ville (2017-07-31). "Tensor Networks in a Nutshell". arXiv:1708.00006 [quant-ph].
4. Verstraete, F.; Wolf, M. M.; Perez-Garcia, D.; Cirac, J. I. (2006-06-06). "Criticality, the Area Law, and the Computational Power of Projected Entangled Pair States". Physical Review Letters. 96 (22): 220601. arXiv:quant-ph/0601075. Bibcode:2006PhRvL..96v0601V. doi:10.1103/PhysRevLett.96.220601. hdl:1854/LU-8590963. PMID 16803296. S2CID 119396305.
5. Montangero, Simone (28 November 2018). Introduction to tensor network methods : numerical simulations of low-dimensional many-body quantum systems. Cham, Switzerland. ISBN 978-3-030-01409-4. OCLC 1076573498.{{cite book}}: CS1 maint: location missing publisher (link)
6. "The Tensor Network". Tensor Network. Retrieved 2022-07-30.
7. Stoudenmire, E. Miles; Schwab, David J. (2017-05-18). "Supervised Learning with Quantum-Inspired Tensor Networks". Advances in Neural Information Processing Systems. 29: 4799. arXiv:1605.05775.
8. google/TensorNetwork, 2021-01-30, retrieved 2021-02-02
9. "Introducing TensorNetwork, an Open Source Library for Efficient Tensor Calculations". Google AI Blog. Retrieved 2021-02-02.
10. Oseledets, I. V. (2011-01-01). "Tensor-Train Decomposition". SIAM Journal on Scientific Computing. 33 (5): 2295–2317. Bibcode:2011SJSC...33.2295O. doi:10.1137/090752286. ISSN 1064-8275. S2CID 207059098.
| Wikipedia |
Topological tensor product
In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products (see Tensor product of Hilbert spaces), but for general Banach spaces or locally convex topological vector spaces the theory is notoriously subtle.
Motivation
One of the original motivations for topological tensor products ${\hat {\otimes }}$ is the fact that tensor products of the spaces of smooth functions on $\mathbb {R} ^{n}$ do not behave as expected. There is an injection
$C^{\infty }(\mathbb {R} ^{n})\otimes C^{\infty }(\mathbb {R} ^{m})\hookrightarrow C^{\infty }(\mathbb {R} ^{n+m})$
but this is not an isomorphism. For example, the function $f(x,y)=e^{xy}$ cannot be expressed as a finite linear combination of smooth functions in $C^{\infty }(\mathbb {R} _{x})\otimes C^{\infty }(\mathbb {R} _{y}).$[1] We only get an isomorphism after constructing the topological tensor product; i.e.,
$C^{\infty }(\mathbb {R} ^{n})\mathop {\hat {\otimes }} C^{\infty }(\mathbb {R} ^{m})\cong C^{\infty }(\mathbb {R} ^{n+m}).$
This article first details the construction in the Banach space case. $C^{\infty }(\mathbb {R} ^{n})$ is not a Banach space and further cases are discussed at the end.
Tensor products of Hilbert spaces
Main article: Tensor product of Hilbert spaces
The algebraic tensor product of two Hilbert spaces A and B has a natural positive definite sesquilinear form (scalar product) induced by the sesquilinear forms of A and B. So in particular it has a natural positive definite quadratic form, and the corresponding completion is a Hilbert space A ⊗ B, called the (Hilbert space) tensor product of A and B.
If the vectors ai and bj run through orthonormal bases of A and B, then the vectors ai⊗bj form an orthonormal basis of A ⊗ B.
Cross norms and tensor products of Banach spaces
We shall use the notation from (Ryan 2002) in this section. The obvious way to define the tensor product of two Banach spaces $A$ and $B$ is to copy the method for Hilbert spaces: define a norm on the algebraic tensor product, then take the completion in this norm. The problem is that there is more than one natural way to define a norm on the tensor product.
If $A$ and $B$ are Banach spaces the algebraic tensor product of $A$ and $B$ means the tensor product of $A$ and $B$ as vector spaces and is denoted by $A\otimes B.$ The algebraic tensor product $A\otimes B$ consists of all finite sums
$x=\sum _{i=1}^{n}a_{i}\otimes b_{i},$
where $n$ is a natural number depending on $x$ and $a_{i}\in A$ and $b_{i}\in B$ for $i=1,\ldots ,n.$
When $A$ and $B$ are Banach spaces, a crossnorm (or cross norm) $p$ on the algebraic tensor product $A\otimes B$ is a norm satisfying the conditions
$p(a\otimes b)=\|a\|\|b\|,$
$p'(a'\otimes b')=\|a'\|\|b'\|.$
Here $a^{\prime }$ and $b^{\prime }$ are elements of the topological dual spaces of $A$ and $B,$ respectively, and $p^{\prime }$ is the dual norm of $p.$ The term reasonable crossnorm is also used for the definition above.
There is a cross norm $\pi $ called the projective cross norm, given by
$\pi (x)=\inf \left\{\sum _{i=1}^{n}\|a_{i}\|\|b_{i}\|:x=\sum _{i=1}^{n}a_{i}\otimes b_{i}\right\},$
where $x\in A\otimes B.$
It turns out that the projective cross norm agrees with the largest cross norm ((Ryan 2002), proposition 2.1).
There is a cross norm $\varepsilon $ called the injective cross norm, given by
$\varepsilon (x)=\sup \left\{\left|(a'\otimes b')(x)\right|:a'\in A',b'\in B',\|a'\|=\|b'\|=1\right\}$
where $x\in A\otimes B.$ Here $A^{\prime }$ and $B^{\prime }$ denote the topological duals of $A$ and $B,$ respectively.
Note hereby that the injective cross norm is only in some reasonable sense the "smallest".
The completions of the algebraic tensor product in these two norms are called the projective and injective tensor products, and are denoted by $A\operatorname {\hat {\otimes }} _{\pi }B$ and $A\operatorname {\hat {\otimes }} _{\varepsilon }B.$
When $A$ and $B$ are Hilbert spaces, the norm used for their Hilbert space tensor product is not equal to either of these norms in general. Some authors denote it by $\sigma ,$ so the Hilbert space tensor product in the section above would be $A\operatorname {\hat {\otimes }} _{\sigma }B.$
A uniform crossnorm $\alpha $ is an assignment to each pair $(X,Y)$ of Banach spaces of a reasonable crossnorm on $X\otimes Y$ so that if $X,W,Y,Z$ are arbitrary Banach spaces then for all (continuous linear) operators $S:X\to W$ and $T:Y\to Z$ the operator $S\otimes T:X\otimes _{\alpha }Y\to W\otimes _{\alpha }Z$ is continuous and $\|S\otimes T\|\leq \|S\|\|T\|.$ If $A$ and $B$ are two Banach spaces and $\alpha $ is a uniform cross norm then $\alpha $ defines a reasonable cross norm on the algebraic tensor product $A\otimes B.$ The normed linear space obtained by equipping $A\otimes B$ with that norm is denoted by $A\otimes _{\alpha }B.$ The completion of $A\otimes _{\alpha }B,$ which is a Banach space, is denoted by $A\operatorname {\hat {\otimes }} _{\alpha }B.$ The value of the norm given by $\alpha $ on $A\otimes B$ and on the completed tensor product $A\operatorname {\hat {\otimes }} _{\alpha }B$ for an element $x$ in $A\operatorname {\hat {\otimes }} _{\alpha }B$ (or $A\otimes _{\alpha }B$) is denoted by $\alpha _{A,B}(x){\text{ or }}\alpha (x).$
A uniform crossnorm $\alpha $ is said to be finitely generated if, for every pair $(X,Y)$ of Banach spaces and every $u\in X\otimes Y,$
$\alpha (u;X\otimes Y)=\inf\{\alpha (u;M\otimes N):\dim M,\dim N<\infty \}.$
A uniform crossnorm $\alpha $ is cofinitely generated if, for every pair $(X,Y)$ of Banach spaces and every $u\in X\otimes Y,$
$\alpha (u)=\sup\{\alpha ((Q_{E}\otimes Q_{F})u;(X/E)\otimes (Y/F)):\dim X/E,\dim Y/F<\infty \}.$
A tensor norm is defined to be a finitely generated uniform crossnorm. The projective cross norm $\pi $ and the injective cross norm $\varepsilon $ defined above are tensor norms and they are called the projective tensor norm and the injective tensor norm, respectively.
If $A$ and $B$ are arbitrary Banach spaces and $\alpha $ is an arbitrary uniform cross norm then
$\varepsilon _{A,B}(x)\leq \alpha _{A,B}(x)\leq \pi _{A,B}(x).$
Tensor products of locally convex topological vector spaces
See also: Injective tensor product and Projective tensor product
The topologies of locally convex topological vector spaces $A$ and $B$ are given by families of seminorms. For each choice of seminorm on $A$ and on $B$ we can define the corresponding family of cross norms on the algebraic tensor product $A\otimes B,$ and by choosing one cross norm from each family we get some cross norms on $A\otimes B,$ defining a topology. There are in general an enormous number of ways to do this. The two most important ways are to take all the projective cross norms, or all the injective cross norms. The completions of the resulting topologies on $A\otimes B$ are called the projective and injective tensor products, and denoted by $A\otimes _{\gamma }B$ and $A\otimes _{\lambda }B.$ There is a natural map from $A\otimes _{\gamma }B$ to $A\otimes _{\lambda }B.$
If $A$ or $B$ is a nuclear space then the natural map from $A\otimes _{\gamma }B$ to $A\otimes _{\lambda }B$ is an isomorphism. Roughly speaking, this means that if $A$ or $B$ is nuclear, then there is only one sensible tensor product of $A$ and $B$. This property characterizes nuclear spaces.
See also
• Fréchet space – A locally convex topological vector space that is also a complete metric space
• Fredholm kernel – type of a kernel on a Banach spacePages displaying wikidata descriptions as a fallback
• Inductive tensor product – binary operation on topological vector spacesPages displaying wikidata descriptions as a fallback
• Injective tensor product
• Projective tensor product – tensor product defined on two topological vector spacesPages displaying wikidata descriptions as a fallback
• Projective topology – Coarsest topology making certain functions continuousPages displaying short descriptions of redirect targets
• Tensor product of Hilbert spaces – Tensor product space endowed with a special inner product
References
1. "What is an example of a smooth function in C∞(R2) which is not contained in C∞(R)⊗C∞(R)".
• Ryan, R.A. (2002), Introduction to Tensor Products of Banach Spaces, New York: Springer.
• Grothendieck, A. (1955), "Produits tensoriels topologiques et espaces nucléaires", Memoirs of the American Mathematical Society, 16.
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Glossary of tensor theory
This is a glossary of tensor theory. For expositions of tensor theory from different points of view, see:
• Tensor
• Tensor (intrinsic definition)
• Application of tensor theory in engineering science
For some history of the abstract theory see also multilinear algebra.
Classical notation
Ricci calculus
The earliest foundation of tensor theory – tensor index notation.[1]
Order of a tensor
The components of a tensor with respect to a basis is an indexed array. The order of a tensor is the number of indices needed. Some texts may refer to the tensor order using the term degree or rank.
Rank of a tensor
The rank of a tensor is the minimum number of rank-one tensor that must be summed to obtain the tensor. A rank-one tensor may be defined as expressible as the outer product of the number of nonzero vectors needed to obtain the correct order.
Dyadic tensor
A dyadic tensor is a tensor of order two, and may be represented as a square matrix. In contrast, a dyad is specifically a dyadic tensor of rank one.
Einstein notation
This notation is based on the understanding that whenever a multidimensional array contains a repeated index letter, the default interpretation is that the product is summed over all permitted values of the index. For example, if aij is a matrix, then under this convention aii is its trace. The Einstein convention is widely used in physics and engineering texts, to the extent that if summation is not to be applied, it is normal to note that explicitly.
Kronecker delta
Levi-Civita symbol
Covariant tensor
Contravariant tensor
The classical interpretation is by components. For example, in the differential form aidxi the components ai are a covariant vector. That means all indices are lower; contravariant means all indices are upper.
Mixed tensor
This refers to any tensor that has both lower and upper indices.
Cartesian tensor
Cartesian tensors are widely used in various branches of continuum mechanics, such as fluid mechanics and elasticity. In classical continuum mechanics, the space of interest is usually 3-dimensional Euclidean space, as is the tangent space at each point. If we restrict the local coordinates to be Cartesian coordinates with the same scale centered at the point of interest, the metric tensor is the Kronecker delta. This means that there is no need to distinguish covariant and contravariant components, and furthermore there is no need to distinguish tensors and tensor densities. All Cartesian-tensor indices are written as subscripts. Cartesian tensors achieve considerable computational simplification at the cost of generality and of some theoretical insight.
Contraction of a tensor
Raising and lowering indices
Symmetric tensor
Antisymmetric tensor
Multiple cross products
Algebraic notation
This avoids the initial use of components, and is distinguished by the explicit use of the tensor product symbol.
Tensor product
If v and w are vectors in vector spaces V and W respectively, then
$v\otimes w$
is a tensor in
$V\otimes W.$
That is, the ⊗ operation is a binary operation, but it takes values into a fresh space (it is in a strong sense external). The ⊗ operation is a bilinear map; but no other conditions are applied to it.
Pure tensor
A pure tensor of V ⊗ W is one that is of the form v ⊗ w.
It could be written dyadically aibj, or more accurately aibj ei ⊗ fj, where the ei are a basis for V and the fj a basis for W. Therefore, unless V and W have the same dimension, the array of components need not be square. Such pure tensors are not generic: if both V and W have dimension greater than 1, there will be tensors that are not pure, and there will be non-linear conditions for a tensor to satisfy, to be pure. For more see Segre embedding.
Tensor algebra
In the tensor algebra T(V) of a vector space V, the operation $\otimes $ becomes a normal (internal) binary operation. A consequence is that T(V) has infinite dimension unless V has dimension 0. The free algebra on a set X is for practical purposes the same as the tensor algebra on the vector space with X as basis.
Hodge star operator
Exterior power
The wedge product is the anti-symmetric form of the ⊗ operation. The quotient space of T(V) on which it becomes an internal operation is the exterior algebra of V; it is a graded algebra, with the graded piece of weight k being called the k-th exterior power of V.
Symmetric power, symmetric algebra
This is the invariant way of constructing polynomial algebras.
Applications
Metric tensor
Strain tensor
Stress–energy tensor
Tensor field theory
Jacobian matrix
Tensor field
Tensor density
Lie derivative
Tensor derivative
Differential geometry
Abstract algebra
Tensor product of fields
This is an operation on fields, that does not always produce a field.
Tensor product of R-algebras
Clifford module
A representation of a Clifford algebra which gives a realisation of a Clifford algebra as a matrix algebra.
Tor functors
These are the derived functors of the tensor product, and feature strongly in homological algebra. The name comes from the torsion subgroup in abelian group theory.
Symbolic method of invariant theory
Derived category
Grothendieck's six operations
These are highly abstract approaches used in some parts of geometry.
Spinors
See:
Spin group
Spin-c group
Spinor
Pin group
Pinors
Spinor field
Killing spinor
Spin manifold
References
1. Ricci, Gregorio; Levi-Civita, Tullio (March 1900), "Méthodes de calcul différentiel absolu et leurs applications" [Absolute differential calculation methods & their applications], Mathematische Annalen (in French), Springer, 54 (1–2): 125–201, doi:10.1007/BF01454201, S2CID 120009332
Books
• Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds (First Dover 1980 ed.), The Macmillan Company, ISBN 0-486-64039-6
• Danielson, Donald A. (2003). Vectors and Tensors in Engineering and Physics (2/e ed.). Westview (Perseus). ISBN 978-0-8133-4080-7.
• Dimitrienko, Yuriy (2002). Tensor Analysis and Nonlinear Tensor Functions. Kluwer Academic Publishers (Springer). ISBN 1-4020-1015-X.
• Lovelock, David; Hanno Rund (1989) [1975]. Tensors, Differential Forms, and Variational Principles. Dover. ISBN 978-0-486-65840-7.
• Synge, John L; Schild, Alfred (1949). Tensor Calculus. Dover Publications 1978 edition. ISBN 978-0-486-63612-2.
Tensors
Glossary of tensor theory
Scope
Mathematics
• Coordinate system
• Differential geometry
• Dyadic algebra
• Euclidean geometry
• Exterior calculus
• Multilinear algebra
• Tensor algebra
• Tensor calculus
• Physics
• Engineering
• Computer vision
• Continuum mechanics
• Electromagnetism
• General relativity
• Transport phenomena
Notation
• Abstract index notation
• Einstein notation
• Index notation
• Multi-index notation
• Penrose graphical notation
• Ricci calculus
• Tetrad (index notation)
• Van der Waerden notation
• Voigt notation
Tensor
definitions
• Tensor (intrinsic definition)
• Tensor field
• Tensor density
• Tensors in curvilinear coordinates
• Mixed tensor
• Antisymmetric tensor
• Symmetric tensor
• Tensor operator
• Tensor bundle
• Two-point tensor
Operations
• Covariant derivative
• Exterior covariant derivative
• Exterior derivative
• Exterior product
• Hodge star operator
• Lie derivative
• Raising and lowering indices
• Symmetrization
• Tensor contraction
• Tensor product
• Transpose (2nd-order tensors)
Related
abstractions
• Affine connection
• Basis
• Cartan formalism (physics)
• Connection form
• Covariance and contravariance of vectors
• Differential form
• Dimension
• Exterior form
• Fiber bundle
• Geodesic
• Levi-Civita connection
• Linear map
• Manifold
• Matrix
• Multivector
• Pseudotensor
• Spinor
• Vector
• Vector space
Notable tensors
Mathematics
• Kronecker delta
• Levi-Civita symbol
• Metric tensor
• Nonmetricity tensor
• Ricci curvature
• Riemann curvature tensor
• Torsion tensor
• Weyl tensor
Physics
• Moment of inertia
• Angular momentum tensor
• Spin tensor
• Cauchy stress tensor
• stress–energy tensor
• Einstein tensor
• EM tensor
• Gluon field strength tensor
• Metric tensor (GR)
Mathematicians
• Élie Cartan
• Augustin-Louis Cauchy
• Elwin Bruno Christoffel
• Albert Einstein
• Leonhard Euler
• Carl Friedrich Gauss
• Hermann Grassmann
• Tullio Levi-Civita
• Gregorio Ricci-Curbastro
• Bernhard Riemann
• Jan Arnoldus Schouten
• Woldemar Voigt
• Hermann Weyl
| Wikipedia |
Classical Hamiltonian quaternions
William Rowan Hamilton invented quaternions, a mathematical entity in 1843. This article describes Hamilton's original treatment of quaternions, using his notation and terms. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties. Mathematically, quaternions discussed differ from the modern definition only by the terminology which is used.
For the history of quaternions, see history of quaternions. For a more general treatment of quaternions, see quaternion.
Classical elements of a quaternion
Hamilton defined a quaternion as the quotient of two directed lines in tridimensional space;[1] or, more generally, as the quotient of two vectors.[2]
A quaternion can be represented as the sum of a scalar and a vector. It can also be represented as the product of its tensor and its versor.
Scalar
Main article: Scalar (mathematics)
Hamilton invented the term scalars for the real numbers, because they span the "scale of progression from positive to negative infinity"[3] or because they represent the "comparison of positions upon one common scale".[4] Hamilton regarded ordinary scalar algebra as the science of pure time.[5]
Vector
See also: Vector space
Hamilton defined a vector as "a right line ... having not only length but also direction".[6] Hamilton derived the word vector from the Latin vehere, to carry.[7]
Hamilton conceived a vector as the "difference of its two extreme points."[6] For Hamilton, a vector was always a three-dimensional entity, having three co-ordinates relative to any given co-ordinate system, including but not limited to both polar and rectangular systems.[8] He therefore referred to vectors as "triplets".
Hamilton defined addition of vectors in geometric terms, by placing the origin of the second vector at the end of the first.[9] He went on to define vector subtraction.
By adding a vector to itself multiple times, he defined multiplication of a vector by an integer, then extended this to division by an integer, and multiplication (and division) of a vector by a rational number. Finally, by taking limits, he defined the result of multiplying a vector α by any scalar x as a vector β with the same direction as α if x is positive; the opposite direction to α if x is negative; and a length that is |x| times the length of α.[10]
The quotient of two parallel or anti-parallel vectors is therefore a scalar with absolute value equal to the ratio of the lengths of the two vectors; the scalar is positive if the vectors are parallel and negative if they are anti-parallel.[11]
Unit vector
A unit vector is a vector of length one. Examples of unit vectors include i, j and k.
Tensor
Note: The use of the word tensor by Hamilton does not coincide with modern terminology. Hamilton's tensor is actually the absolute value on the quaternion algebra, which makes it a normed vector space.
Hamilton defined tensor as a positive numerical quantity, or, more properly, signless number.[12][13][14] A tensor can be thought of as a positive scalar.[15] The "tensor" can be thought of as representing a "stretching factor."[16]
Hamilton introduced the term tensor in his first book, Lectures on Quaternions, based on lectures he gave shortly after his invention of the quaternions:
• it seems convenient to enlarge by definition the signification of the new word tensor, so as to render it capable of including also those other cases in which we operate on a line by diminishing instead of increasing its length ; and generally by altering that length in any definite ratio. We shall thus (as was hinted at the end of the article in question) have fractional and even incommensurable tensors, which will simply be numerical multipliers, and will all be positive or (to speak more properly) SignLess Numbers, that is, unclothed with the algebraic signs of positive and negative ; because, in the operation here considered, we abstract from the directions (as well as from the situations) of the lines which are compared or operated on.
Each quaternion has a tensor, which is a measure of its magnitude (in the same way as the length of a vector is a measure of a vectors' magnitude). When a quaternion is defined as the quotient of two vectors, its tensor is the ratio of the lengths of these vectors.
Versor
Main article: versor
A versor is a quaternion with a tensor of 1. Alternatively, a versor can be defined as the quotient of two equal-length vectors.[17][18]
In general a versor defines all of the following: a directional axis; the plane normal to that axis; and an angle of rotation.[19]
When a versor and a vector which lies in the plane of the versor are multiplied, the result is a new vector of the same length but turned by the angle of the versor.
Vector arc
Since every unit vector can be thought of as a point on a unit sphere, and since a versor can be thought of as the quotient of two vectors, a versor has a representative great circle arc, called a vector arc, connecting these two points, drawn from the divisor or lower part of quotient, to the dividend or upper part of the quotient.[20][21]
Right versor
When the arc of a versor has the magnitude of a right angle, then it is called a right versor, a right radial or quadrantal versor.
Degenerate forms
There are two special degenerate versor cases, called the unit-scalars.[22] These two scalars (negative and positive unity) can be thought of as scalar quaternions. These two scalars are special limiting cases, corresponding to versors with angles of either zero or π.
Unlike other versors, these two cannot be represented by a unique arc. The arc of 1 is a single point, and –1 can be represented by an infinite number of arcs, because there are an infinite number of shortest lines between antipodal points of a sphere.
Quaternion
Every quaternion can be decomposed into a scalar and a vector.
$q=\mathbf {S} (q)+\mathbf {V} (q)$
These two operations S and V are called "take the Scalar of" and "take the vector of" a quaternion. The vector part of a quaternion is also called the right part.[23]
Every quaternion is equal to a versor multiplied by the tensor of the quaternion. Denoting the versor of a quaternion by
$\mathbf {U} q$
and the tensor of a quaternion by
$\mathbf {T} q$
we have
$q=\mathbf {T} q\mathbf {U} q$
Right quaternion
A right quaternion is a quaternion whose scalar component is zero,
$S(q)=0$
The angle of a right quaternion is 90 degrees. A right quaternion can also be thought of as a vector plus a zero scalar. Right quaternions may be put in what was called the standard trinomial form. For example, if Q is a right quaternion, it may be written as:
$Q=xi+yj+zk$[24]
Four operations
Four operations are of fundamental importance in quaternion notation.[25]
+ − ÷ ×
In particular it is important to understand that there is a single operation of multiplication, a single operation of division, and a single operation each of addition and subtraction. This single multiplication operator can operate on any of the types of mathematical entities. Likewise every kind of entity can be divided, added or subtracted from any other type of entity. Understanding the meaning of the subtraction symbol is critical in quaternion theory, because it leads to an understanding of the concept of a vector.
Ordinal operators
The two ordinal operations in classical quaternion notation were addition and subtraction or + and −.
These marks are:
"...characteristics of synthesis and analysis of a state of progression, according as this state is considered as being derived from, or compared with, some other state of that progression."[26]
Subtraction
Subtraction is a type of analysis called ordinal analysis[27]
...let space be now regarded as the field of progression which is to be studied, and POINTS as states of that progression. ...I am led to regard the word "Minus," or the mark −, in geometry, as the sign or characteristic of analysis of one geometric position (in space), as compared with another (such) position. The comparison of one mathematical point with another with a view to the determination of what may be called their ordinal relation, or their relative position in space...[28]
The first example of subtraction is to take the point A to represent the earth, and the point B to represent the sun, then an arrow drawn from A to B represents the act of moving or vection from A to B.
B − A
this represents the first example in Hamilton's lectures of a vector. In this case the act of traveling from the earth to the sun.[29][30]
Addition
Addition is a type of analysis called ordinal synthesis.[31]
Addition of vectors and scalars
Vectors and scalars can be added. When a vector is added to a scalar, a completely different entity, a quaternion is created.
A vector plus a scalar is always a quaternion even if the scalar is zero. If the scalar added to the vector is zero then the new quaternion produced is called a right quaternion. It has an angle characteristic of 90 degrees.
Cardinal operations
The two Cardinal operations[32] in quaternion notation are geometric multiplication and geometric division and can be written:
÷, ×
It is not required to learn the following more advanced terms in order to use division and multiplication.
Division is a kind of analysis called cardinal analysis.[33] Multiplication is a kind of synthesis called cardinal synthesis[34]
Division
Classically, the quaternion was viewed as the ratio of two vectors, sometimes called a geometric fraction.
If OA and OB represent two vectors drawn from the origin O to two other points A and B, then the geometric fraction was written as
$OA:OB$
Alternately if the two vectors are represented by α and β the quotient was written as
$\alpha \div \beta $
or
${\frac {\alpha }{\beta }}$
Hamilton asserts: "The quotient of two vectors is generally a quaternion".[35] Lectures on Quaternions also first introduces the concept of a quaternion as the quotient of two vectors:
Logically and by definition,[36][37]
if ${\frac {\alpha }{\beta }}=q$
then ${q}\times {\beta }=\alpha .$.
In Hamilton's calculus the product is not commutative, i.e., the order of the variables is of great importance. If the order of q and β were to be reversed the result would not in general be α. The quaternion q can be thought of as an operator that changes β into α, by first rotating it, formerly an act of version and then changing the length of it, formerly called an act of tension.
Also by definition the quotient of two vectors is equal to the numerator times the reciprocal of the denominator. Since multiplication of vectors is not commutative, the order cannot be changed in the following expression.
${\frac {\alpha }{\beta }}=\,{\alpha }\times {\frac {1}{\beta }}$
Again the order of the two quantities on the right hand side is significant.
Hardy presents the definition of division in terms of mnemonic cancellation rules. "Canceling being performed by an upward right hand stroke".[38]
If alpha and beta are vectors and q is a quaternion such that
${\frac {\alpha }{\beta }}=q$
then $\alpha \beta ^{-1}=q$
and ${\frac {\alpha }{\beta }}.\beta =\alpha \beta ^{-1}.\beta =\alpha $[39]
$\times $ and $\div $ are inverse operations, such that:
$\beta \div \alpha \times \alpha =\beta $ and $q\times \alpha \div \alpha =q$[40]
and
$\gamma =(\gamma \div \beta )\times (\beta \div \alpha )\times \alpha $[41]
An important way to think of q is as an operator that changes β into α, by first rotating it (version) and then changing its length (tension).
$\gamma \div \alpha =(\gamma \div \beta )\times (\beta \div \alpha )$[42]
Division of the unit vectors i, j, k
The results of using the division operator on i, j, and k was as follows.[43]
$ij=k$${\frac {k}{j}}=i$
$jk=i$${\frac {i}{k}}=j$
$ki=j$${\frac {j}{i}}=k$
$ji=-k$${\frac {-k}{i}}=j$
$kj=-i$${\frac {-i}{j}}=k$
$ik=-j$${\frac {-j}{k}}=i$
$i(-j)=-k$${\frac {-k}{-j}}=i$
$i(-k)=j$${\frac {j}{-k}}=i$
$k(-i)=-j$${\frac {-j}{-i}}=k$
$k(-j)=i$${\frac {i}{-j}}=k$
$j(-k)=-i$${\frac {-i}{-k}}=j$
$j(-i)=k$${\frac {k}{-i}}=j$
The reciprocal of a unit vector is the vector reversed.[44]
${\frac {1}{i}}=i^{-1}=-i$
Because a unit vector and its reciprocal are parallel to each other but point in opposite directions, the product of a unit vector and its reciprocal have a special case commutative property, for example if a is any unit vector then:[45]
${\frac {1}{a}}a=(-a)a=1=a(-a)=a{\frac {1}{a}}.$
However, in the more general case involving more than one vector (whether or not it is a unit vector) the commutative property does not hold.[46] For example:
$i{\frac {k}{i}}$ ≠ ${\frac {k}{i}}i.$
This is because k/i is carefully defined as:
${\frac {k}{i}}=k{\frac {1}{i}}=ki^{-1}=k(-i)=-(ki)=-(j)=-j$.
So that:
$i{\frac {k}{i}}=i(-j)=-k$,
however
${\frac {k}{i}}i=(-j)i=-(ji)=-(-k)=k$
Division of two parallel vectors
While in general the quotient of two vectors is a quaternion, If α and β are two parallel vectors then the quotient of these two vectors is a scalar. For example, if
$\alpha =ai$,
and $\beta =bi$ then
$\alpha \div \beta ={\frac {\alpha }{\beta }}={\frac {ai}{bi}}={\frac {a}{b}}$
Where a/b is a scalar.[47]
Division of two non-parallel vectors
The quotient of two vectors is in general the quaternion:
$q={\frac {\alpha }{\beta }}$$={\frac {T\alpha }{T\beta }}(\cos \phi +\epsilon \sin \phi )$
Where α and β are two non-parallel vectors, φ is that angle between them, and ε is a unit vector perpendicular to the plane of the vectors α and β, with its direction given by the standard right hand rule.[48]
Multiplication
Classical quaternion notation had only one concept of multiplication. Multiplication of two real numbers, two imaginary numbers or a real number by an imaginary number in the classical notation system was the same operation.
Multiplication of a scalar and a vector was accomplished with the same single multiplication operator; multiplication of two vectors of quaternions used this same operation as did multiplication of a quaternion and a vector or of two quaternions.
Factor, Faciend and Factum
Factor × Faciend = Factum[49]
When two quantities are multiplied the first quantity is called the factor,[50] the second quantity is called the faciend and the result is called the factum.
Distributive
In classical notation, multiplication was distributive. Understanding this makes it simple to see why the product of two vectors in classical notation produced a quaternion.
$q=(ai+bj+ck)\times (ei+fj+gk)$
$q=ae({i}\times {i})+af({i}\times {j})+ag({i}\times {k})+be({j}\times {i})+bf({j}\times {j})+bg({j}\times {k})+ce({k}\times {i})+cf({k}\times {j})+cg({k}\times {k})$
Using the quaternion multiplication table we have:
$q=ae(-1)+af(+k)+ag(-j)+be(-k)+bf(-1)+bg(+i)+ce(+j)+cf(-i)+cg(-1)$
Then collecting terms:
$q=-ae-bf-cg+(bg-cf)i+(ce-ag)j+(af-be)k$
The first three terms are a scalar.
Letting
$w=-ae-bf-cg$
$x=(bg-cf)$
$y=(ce-ag)$
$z=(af-be)$
So that the product of two vectors is a quaternion, and can be written in the form:
$q=w+xi+yj+zk$
Product of two right quaternions
The product of two right quaternions is generally a quaternion.
Let α and β be the right quaternions that result from taking the vectors of two quaternions:
$\alpha =\mathbf {V} p$
$\beta =\mathbf {V} q$
Their product in general is a new quaternion represented here by r. This product is not ambiguous because classical notation has only one product.
$r=\,\alpha \beta ;$ ;}
Like all quaternions r may now be decomposed into its vector and scalar parts.
$r=\mathbf {S} r+\mathbf {V} r$
The terms on the right are called scalar of the product, and the vector of the product[51] of two right quaternions.
Note: "Scalar of the product" corresponds to Euclidean scalar product of two vectors up to the change of sign (multiplication to −1).
Other operators in detail
Scalar and vector
Two important operations in two the classical quaternion notation system were S(q) and V(q) which meant take the scalar part of, and take the imaginary part, what Hamilton called the vector part of the quaternion. Here S and V are operators acting on q. Parenthesis can be omitted in these kinds of expressions without ambiguity. Classical notation:
$q=\,\mathbf {S} q+\mathbf {V} q$
Here, q is a quaternion. Sq is the scalar of the quaternion while Vq is the vector of the quaternion.
Conjugate
K is the conjugate operator. The conjugate of a quaternion is a quaternion obtained by multiplying the vector part of the first quaternion by minus one.
If
$q=\,\mathbf {S} q+\mathbf {V} q$
then
$\mathbf {K} q=\mathbf {S} \,q-\mathbf {V} q$.
The expression
$r=\,\mathbf {K} q$,
means, assign the quaternion r the value of the conjugate of the quaternion q.
Tensor
T is the tensor operator. It returns a kind of number called a tensor.
The tensor of a positive scalar is that scalar. The tensor of a negative scalar is the absolute value of the scalar (i.e., without the negative sign). For example:
$\mathbf {T} (5)=5$
$\mathbf {T} (-5)=5$
The tensor of a vector is by definition the length of the vector. For example, if:
$\alpha =xi+yj+zk$
Then
$\mathbf {T} \alpha ={\sqrt {x^{2}+y^{2}+z^{2}}}$
The tensor of a unit vector is one. Since the versor of a vector is a unit vector, the tensor of the versor of any vector is always equal to unity. Symbolically:
$\mathbf {TU} \alpha =1$[52]
A quaternion is by definition the quotient of two vectors and the tensor of a quaternion is by definition the quotient of the tensors of these two vectors. In symbols:
$q={\frac {\alpha }{\beta }}.$
$\mathbf {T} q={\frac {\mathbf {T} \alpha }{\mathbf {T} \beta }}.$[53]
From this definition it can be shown that a useful formula for the tensor of a quaternion is:[54]
$\mathbf {T} q={\sqrt {w^{2}+x^{2}+y^{2}+z^{2}}}$
It can also be proven from this definition that another formula to obtain the tensor of a quaternion is from the common norm, defined as the product of a quaternion and its conjugate. The square root of the common norm of a quaternion is equal to its tensor.
$\mathbf {T} q={\sqrt {qKq}}$
A useful identity is that the square of the tensor of a quaternion is equal to the tensor of the square of a quaternion, so that the parentheses may be omitted.[55]
$(\mathbf {T} q)^{2}=\mathbf {T} (q^{2})=\mathbf {T} q^{2}$
Also, the tensors of conjugate quaternions are equal.[56]
$\mathbf {TK} q=\mathbf {T} q$
The tensor of a quaternion is now called its norm.
Axis and angle
Taking the angle of a non-scalar quaternion, resulted in a value greater than zero and less than π.[57][58]
When a non-scalar quaternion is viewed as the quotient of two vectors, then the axis of the quaternion is a unit vector perpendicular to the plane of the two vectors in this original quotient, in a direction specified by the right hand rule.[59] The angle is the angle between the two vectors.
In symbols,
$u=Ax.q$
$\theta =\angle q$
Reciprocal
If
$q={\frac {\alpha }{\beta }}$
then its reciprocal is defined as
${\frac {1}{q}}=q^{-1}={\frac {\beta }{\alpha }}$
The expression:
${q}\times {\alpha }\times {\frac {1}{q}}$
Reciprocals have many important applications,[60][61] for example rotations, particularly when q is a versor. A versor has an easy formula for its reciprocal.[62]
${\frac {1}{(\mathbf {U} q)}}=\mathbf {S.U} q-\mathbf {V.U} q=\mathbf {K.U} q$
In words the reciprocal of a versor is equal to its conjugate. The dots between operators show the order of the operations, and also help to indicate that S and U for example, are two different operations rather than a single operation named SU.
Common norm
The product of a quaternion with its conjugate is its common norm.[63]
The operation of taking the common norm of a quaternion is represented with the letter N. By definition the common norm is the product of a quaternion with its conjugate. It can be proven[64][65] that common norm is equal to the square of the tensor of a quaternion. However this proof does not constitute a definition. Hamilton gives exact, independent definitions of both the common norm and the tensor. This norm was adopted as suggested from the theory of numbers, however to quote Hamilton "they will not often be wanted". The tensor is generally of greater utility. The word norm does not appear in Lectures on Quaternions, and only twice in the table of contents of Elements of Quaternions.
In symbols:
$\mathbf {N} q=\,q\mathbf {K} q=\,(\mathbf {T} q)^{2}$
The common norm of a versor is always equal to positive unity.[66]
$\mathbf {NU} q=\mathbf {U} q.\mathbf {KU} q=1$
Biquaternions
Geometrically real and geometrically imaginary numbers
In classical quaternion literature the equation
$q^{2}=-1$
was thought to have infinitely many solutions that were called geometrically real. These solutions are the unit vectors that form the surface of a unit sphere.
A geometrically real quaternion is one that can be written as a linear combination of i, j and k, such that the squares of the coefficients add up to one. Hamilton demonstrated that there had to be additional roots of this equation in addition to the geometrically real roots. Given the existence of the imaginary scalar, a number of expressions can be written and given proper names. All of these were part of Hamilton's original quaternion calculus. In symbols:
$q+q'{\sqrt {-1}}$
where q and q′ are real quaternions, and the square root of minus one is the imaginary of ordinary algebra, and are called an imaginary or symbolical roots[67] and not a geometrically real vector quantity.
Imaginary scalar
Geometrically Imaginary quantities are additional roots of the above equation of a purely symbolic nature. In article 214 of Elements Hamilton proves that if there is an i, j and k there also has to be another quantity h which is an imaginary scalar, which he observes should have already occurred to anyone who had read the preceding articles with attention.[68] Article 149 of Elements is about Geometrically Imaginary numbers and includes a footnote introducing the term biquaternion.[69] The terms imaginary of ordinary algebra and scalar imaginary are sometimes used for these geometrically imaginary quantities.
Geometrically Imaginary roots to an equation were interpreted in classical thinking as geometrically impossible situations. Article 214 of Elements of Quaternions explores the example of the equation of a line and a circle that do not intersect, as being indicated by the equation having only a geometrically imaginary root.[70]
In Hamilton's later writings he proposed using the letter h to denote the imaginary scalar[71][72][73]
Biquaternion
Main article: Biquaternion
On page 665 of Elements of Quaternions Hamilton defines a biquaternion to be a quaternion with complex number coefficients. The scalar part of a biquaternion is then a complex number called a biscalar. The vector part of a biquaternion is a bivector consisting of three complex components. The biquaternions are then the complexification of the original (real) quaternions.
Other double quaternions
Hamilton invented the term associative to distinguish between the imaginary scalar (known by now as a complex number) which is both commutative and associative, and four other possible roots of negative unity which he designated L, M, N and O, mentioning them briefly in appendix B of Lectures on Quaternions and in private letters. However, non-associative roots of minus one do not appear in Elements of Quaternions. Hamilton died before he worked on these strange entities. His son claimed them to be "bows reserved for the hands of another Ulysses".[74]
See also
• Cayley–Dickson construction
• Octonions
• Frobenius theorem
Footnotes
1. Hamilton 1853 pg. 60 at Google Books
2. Hardy 1881 pg. 32 at Google Books
3. Hamilton, in the Philosophical magazine, as cited in the OED.
4. Hamilton (1866) Book I Chapter II Article 17 at Google Books
5. Hamilton 1853, pg 2 paragraph 3 of introduction. Refers to his early article "Algebra as the Science of pure time". at Google Books
6. Hamilton (1866) Book I Chapter I Article 1 at Google Books
7. Hamilton (1853) Lecture I Article 15, introduction of term vector, from vehere at Google Books
8. Hamilton (1853) Lecture I Article 17 vector is natural triplet at Google Books
9. aHamilton (1866) Book I Chapter I Article 6 at Google Books
10. Hamilton (1866) Book I Chapter I Article 15 at Google Books
11. Hamilton (1866) Book I Chapter II Article 19 at Google Books
12. Hamilton 1853 pg 57 at Google Books
13. Hardy 1881 pg 5 at Google Books
14. Tait 1890 pg.31 explains Hamilton's older definition of a tensor as a positive number at Google Books
15. Hamilton 1989 pg 165, refers to a tensor as a positive scalar. at Google Books
16. (1890), pg 32 31 at Google Books
17. Hamilton 1898 section 8 pg 133 art 151 On the versor of a quaternion or a vector and some general formula of transformation at Google Books
18. Hamilton (1899), art 156 pg 135, introduction of term versor at Google Books
19. Hamilton (1899), Section 8 article 151 pg 133 at Google Books
20. Hamilton 1898 section 9 art 162 pg 142 Vector Arcs considered as representative of versors of quaternions at Google Books
21. (1881), art. 49 pg 71-72 71 at Google Books
22. Elements of Quaternions Article 147 pg 130 130 at Google Books
23. See Elements of Quaternions Section 13 starting on page 190 at Google Books
24. Hamilton (1899), Section 14 article 221 on page 233 at Google Books
25. Hamilton 1853 pg 4 at Google Books
26. Hamilton 1853 art 5 pg 4 -5 at Google Books
27. Hamilton pg 33 at Google Books
28. Hamilton 1853 pg 5-6 at Google Books
29. see Hamilton 1853 pg 8-15 at Google Books
30. Hamilton 1853 pg 15 introduction of the term vector as the difference between two points. at Google Books
31. Hamilton 1853 pg.19 Hamilton associates plus sign with ordinal synthesis at Google Books
32. Hamilton (1853), pg 35, Hamilton first introduces cardinal operations at Google Books
33. Hamilton 1953 pg.36 Division defined as cardinal analysis at Google Books
34. Hamilton 1853 pg 37 at Google Books
35. Hamilton (1899), Article 112 page 110 at Google Books
36. Hardy (1881), pg 32 at Google Books
37. Hamilton Lectures on Quaternions page 37 at Google Books
38. Elements of quaternions at Google Books
39. Tait Treaties on Quaternions at Google Books
40. Hamilton Lectures On Quaternions pg 38 at Google Books
41. Hamilton Lectures on quaternions page 41 at Google Books
42. Hamilton Lectures on quaternions pg 42 at Google Books
43. Hardy (1881), page 40-41 at Google Books
44. Hardy 1887 pg 45 formula 29 at Google Books
45. Hardy 1887 pg 45 formula 30 at Google Books
46. Hardy 1887 pg 46 at Google Books
47. Elements of Quaternions, book one. at Google Books
48. Hardy (1881), pg 39 article 25 at Google Books
49. Hamilton 1853 pg. 27 explains Factor Faciend and Factum at Google Books
50. Hamilton 1898 section 103 at Google Books
51. (1887) scalar of the product vector of the product defined, pg 57 at Google Books
52. Hamilton 1898 pg164 Tensor of the versor of a vector is unity. at Google Books
53. Elements of Quaternions, Ch. 11 at Google Books
54. Hardy (1881), pg 65 at Google Books
55. Hamilton 1898 pg 169 art 190 Tensor of the square is the square of the tensor at Google Books
56. Hamilton 1898 pg 167 art. 187 equation 12 Tensors of conjugate quaternions are equal at Google Books
57. "Hamilton (1853), pg 164, art 148".
58. Hamilton (1899), pg 118 at Google Books
59. Hamilton (1899), pg 118 at Google Books
60. See Goldstein (1980) Chapter 7 for the same function written in matrix notation
61. "Lorentz Transforms Hamilton (1853), pg 268 1853".
62. Hardy (1881), pg 71 at Google Books
63. Hamilton (1899), pg 128 -129 at Google Books
64. See foot note at bottom of page, were word proven is highlighted. at Google Books
65. See Hamilton 1898 pg. 169 art. 190 for proof of relationship between tensor and common norm at Google Books
66. Hamilton 1899 pg 138 at Google Books
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69. Elements of Quaternions Article 149 at Google Books
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72. Hamilton Elements Article 274 pg 300 Example of use of h notation at Google Books
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74. Hamilton, William Rowan (1899). Elements of Quaternions. London, New York, and Bombay: Longmans, Green, and Co. p. v. ISBN 9780828402194.
References
• W.R. Hamilton (1853), Lectures on Quaternions at Google Books Dublin: Hodges and Smith
• W.R. Hamilton (1866), Elements of Quaternions at Google Books, 2nd edition, edited by Charles Jasper Joly, Longmans Green & Company.
• A.S. Hardy (1887), Elements of Quaternions
• P.G. Tait (1890), An Elementary Treatise on Quaternions, Cambridge: C.J. Clay and Sons
• Herbert Goldstein(1980), Classical Mechanics, 2nd edition, Library of congress catalog number QA805.G6 1980
| Wikipedia |
Tensor operator
In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator.[1]
"Spherical tensor operator" redirects here. For the closely related concept, see spherical basis.
The general notion of scalar, vector, and tensor operators
In quantum mechanics, physical observables that are scalars, vectors, and tensors, must be represented by scalar, vector, and tensor operators, respectively. Whether something is a scalar, vector, or tensor depends on how it is viewed by two observers whose coordinate frames are related to each other by a rotation. Alternatively, one may ask how, for a single observer, a physical quantity transforms if the state of the system is rotated. Consider, for example, a system consisting of a molecule of mass $M$, traveling with a definite center of mass momentum, $p{\mathbf {\hat {z}} }$, in the $z$ direction. If we rotate the system by $90^{\circ }$ about the $y$ axis, the momentum will change to $p{\mathbf {\hat {x}} }$, which is in the $x$ direction. The center-of-mass kinetic energy of the molecule will, however, be unchanged at $p^{2}/2M$. The kinetic energy is a scalar and the momentum is a vector, and these two quantities must be represented by a scalar and a vector operator, respectively. By the latter in particular, we mean an operator whose expected values in the initial and the rotated states are $p{\mathbf {\hat {z}} }$ and $p{\mathbf {\hat {x}} }$. The kinetic energy on the other hand must be represented by a scalar operator, whose expected value must be the same in the initial and the rotated states.
In the same way, tensor quantities must be represented by tensor operators. An example of a tensor quantity (of rank two) is the electrical quadrupole moment of the above molecule. Likewise, the octupole and hexadecapole moments would be tensors of rank three and four, respectively.
Other examples of scalar operators are the total energy operator (more commonly called the Hamiltonian), the potential energy, and the dipole-dipole interaction energy of two atoms. Examples of vector operators are the momentum, the position, the orbital angular momentum, ${\mathbf {L} }$, and the spin angular momentum, ${\mathbf {S} }$. (Fine print: Angular momentum is a vector as far as rotations are concerned, but unlike position or momentum it does not change sign under space inversion, and when one wishes to provide this information, it is said to be a pseudovector.)
Scalar, vector and tensor operators can also be formed by products of operators. For example, the scalar product ${\mathbf {L} }\cdot {\mathbf {S} }$ of the two vector operators, ${\mathbf {L} }$ and ${\mathbf {S} }$, is a scalar operator, which figures prominently in discussions of the spin–orbit interaction. Similarly, the quadrupole moment tensor of our example molecule has the nine components
$Q_{ij}=\sum _{\alpha }q_{\alpha }\left(3r_{\alpha ,i}r_{\alpha ,j}-r_{\alpha }^{2}\delta _{ij}\right).$
Here, the indices $i$ and $j$ can independently take on the values 1, 2, and 3 (or $x$, $y$, and $z$) corresponding to the three Cartesian axes, the index $\alpha $ runs over all particles (electrons and nuclei) in the molecule, $q_{\alpha }$ is the charge on particle $\alpha $, and $r_{\alpha ,i}$ is the $i$-th component of the position of this particle. Each term in the sum is a tensor operator. In particular, the nine products $r_{\alpha ,i}r_{\alpha ,j}$ together form a second rank tensor, formed by taking the direct product of the vector operator ${\mathbf {r} }_{\alpha }$ with itself.
Rotations of quantum states
Quantum rotation operator
The rotation operator about the unit vector n (defining the axis of rotation) through angle θ is
$U[R(\theta ,{\hat {\mathbf {n} }})]=\exp \left(-{\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right)$
where J = (Jx, Jy, Jz) are the rotation generators (also the angular momentum matrices):
$J_{x}={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}}\,\quad J_{y}={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&i&0\\-i&0&i\\0&-i&0\end{pmatrix}}\,\quad J_{z}=\hbar {\begin{pmatrix}-1&0&0\\0&0&0\\0&0&1\end{pmatrix}}$
and let ${\widehat {R}}={\widehat {R}}(\theta ,{\hat {\mathbf {n} }})$ be a rotation matrix. According to the Rodrigues' rotation formula, the rotation operator then amounts to
$U[R(\theta ,{\hat {\mathbf {n} }})]=1\!\!1-{\frac {i\sin \theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} -{\frac {1-\cos \theta }{\hbar ^{2}}}({\hat {\mathbf {n} }}\cdot \mathbf {J} )^{2}.$
An operator ${\widehat {\Omega }}$ is invariant under a unitary transformation U if
${\widehat {\Omega }}={U}^{\dagger }{\widehat {\Omega }}U;$
in this case for the rotation ${\widehat {U}}(R)$,
${\widehat {\Omega }}={U(R)}^{\dagger }{\widehat {\Omega }}U(R)=\exp \left({\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right){\widehat {\Omega }}\exp \left(-{\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right).$
Angular momentum eigenkets
The orthonormal basis set for total angular momentum is $|j,m\rangle $, where j is the total angular momentum quantum number and m is the magnetic angular momentum quantum number, which takes values −j, −j + 1, ..., j − 1, j. A general state within the j subspace
$|\psi \rangle =\sum _{m}c_{jm}|j,m\rangle $
rotates to a new state by:
$|{\bar {\psi }}\rangle =U(R)|\psi \rangle =\sum _{m}c_{jm}U(R)|j,m\rangle $
Using the completeness condition:
$I=\sum _{m'}|j,m'\rangle \langle j,m'|$
we have
$|{\bar {\psi }}\rangle =IU(R)|\psi \rangle =\sum _{mm'}c_{jm}|j,m'\rangle \langle j,m'|U(R)|j,m\rangle $
Introducing the Wigner D matrix elements:
${D(R)}_{m'm}^{(j)}=\langle j,m'|U(R)|j,m\rangle $
gives the matrix multiplication:
$|{\bar {\psi }}\rangle =\sum _{mm'}c_{jm}D_{m'm}^{(j)}|j,m'\rangle \quad \Rightarrow \quad |{\bar {\psi }}\rangle =D^{(j)}|\psi \rangle $
For one basis ket:
$|{\overline {j,m}}\rangle =\sum _{m'}{D(R)}_{m'm}^{(j)}|j,m'\rangle $
For the case of orbital angular momentum, the eigenstates $|\ell ,m\rangle $ of the orbital angular momentum operator L and solutions of Laplace's equation on a 3d sphere are spherical harmonics:
$Y_{\ell }^{m}(\theta ,\phi )=\langle \theta ,\phi |\ell ,m\rangle ={\sqrt {{(2\ell +1) \over 4\pi }{(\ell -m)! \over (\ell +m)!}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\phi }$
where Pℓm is an associated Legendre polynomial, ℓ is the orbital angular momentum quantum number, and m is the orbital magnetic quantum number which takes the values −ℓ, −ℓ + 1, ... ℓ − 1, ℓ The formalism of spherical harmonics have wide applications in applied mathematics, and are closely related to the formalism of spherical tensors, as shown below.
Spherical harmonics are functions of the polar and azimuthal angles, ϕ and θ respectively, which can be conveniently collected into a unit vector n(θ, ϕ) pointing in the direction of those angles, in the Cartesian basis it is:
${\hat {\mathbf {n} }}(\theta ,\phi )=\cos \phi \sin \theta \mathbf {e} _{x}+\sin \phi \sin \theta \mathbf {e} _{y}+\cos \theta \mathbf {e} _{z}$
So a spherical harmonic can also be written $Y_{\ell }^{m}=\langle \mathbf {n} |\ell m\rangle $. Spherical harmonic states $|m,\ell \rangle $ rotate according to the inverse rotation matrix $U(R^{-1})$, while $|\ell ,m\rangle $ rotates by the initial rotation matrix ${\widehat {U}}(R)$.
$|{\overline {\ell ,m}}\rangle =\sum _{m'}D_{m'm}^{(\ell )}[U(R^{-1})]|\ell ,m'\rangle \,,\quad |{\overline {\hat {\mathbf {n} }}}\rangle =U(R)|{\hat {\mathbf {n} }}\rangle $
Rotation of tensor operators
We define the Rotation of an operator by requiring that the expectation value of the original operator ${\widehat {\mathbf {A} }}$ with respect to the initial state be equal to the expectation value of the rotated operator with respect to the rotated state,
$\langle \psi '|{\widehat {A'}}|\psi '\rangle =\langle \psi |{\widehat {A}}|\psi \rangle $
Now as,
$|\psi \rangle ~\rightarrow ~|\psi '\rangle =U(R)|\psi \rangle \,,\quad \langle \psi |~\rightarrow ~\langle \psi '|=\langle \psi |U^{\dagger }(R)$
we have,
$\langle \psi |U^{\dagger }(R){\widehat {A}}'U(R)|\psi \rangle =\langle \psi |{\widehat {A}}|\psi \rangle $
since, $|\psi \rangle $ is arbitrary,
$U^{\dagger }(R){\widehat {A}}'U(R)={\widehat {A}}$
Scalar operators
A scalar operator is invariant under rotations:[2]
$U(R)^{\dagger }{\widehat {S}}U(R)={\widehat {S}}$
This is equivalent to saying a scalar operator commutes with the rotation generators:
$\left[{\widehat {S}},{\widehat {\mathbf {J} }}\right]=0$
Examples of scalar operators include
• the energy operator:
${\widehat {E}}\psi =i\hbar {\frac {\partial }{\partial t}}\psi $
• potential energy V (in the case of a central potential only)
${\widehat {V}}(r,t)\psi (\mathbf {r} ,t)=V(r,t)\psi (\mathbf {r} ,t)$
• kinetic energy T:
${\widehat {T}}\psi (\mathbf {r} ,t)=-{\frac {\hbar ^{2}}{2m}}(\nabla ^{2}\psi )(\mathbf {r} ,t)$
• the spin–orbit coupling:
${\widehat {\mathbf {L} }}\cdot {\widehat {\mathbf {S} }}={\widehat {L}}_{x}{\widehat {S}}_{x}+{\widehat {L}}_{y}{\widehat {S}}_{y}+{\widehat {L}}_{z}{\widehat {S}}_{z}\,.$
Vector operators
Vector operators (as well as pseudovector operators) are a set of 3 operators that can be rotated according to:[2]
${U(R)}^{\dagger }{\widehat {V}}_{i}U(R)=\sum _{j}R_{ij}{\widehat {V}}_{j}$
from this and the infinitesimal rotation operator and its Hermitian conjugate, and ignoring second order term in $(\delta \theta )^{2}$, one can derive the commutation relation with the rotation generator:
$\left[{\widehat {V}}_{a},{\widehat {J}}_{b}\right]\approx i\hbar \varepsilon _{abc}{\widehat {V}}_{c}$
where εijk is the Levi-Civita symbol, which all vector operators must satisfy, by construction. As the symbol εijk is a pseudotensor, pseudovector operators are invariant up to a sign: +1 for proper rotations and −1 for improper rotations.
Vector operators include
• the position operator:
${\widehat {\mathbf {r} }}\psi =\mathbf {r} \psi $
• the momentum operator:
${\widehat {\mathbf {p} }}\psi =-i\hbar \nabla \psi $
and peusodovector operators include
• the orbital angular momentum operator:
${\widehat {\mathbf {L} }}\psi =-i\hbar \mathbf {r} \times \nabla \psi $
• as well the spin operator S, and hence the total angular momentum
${\widehat {\mathbf {J} }}={\widehat {\mathbf {L} }}+{\widehat {\mathbf {S} }}\,.$
In Dirac notation:
$\langle {\bar {\psi }}|{\widehat {V}}_{a}|{\bar {\psi }}\rangle =\langle \psi |{U(R)}^{\dagger }{\widehat {V}}_{a}U(R)|\psi \rangle =\sum _{b}R_{ab}\langle \psi |{\widehat {V}}_{b}|\psi \rangle $
and since |Ψ⟩ is any quantum state, the same result follows:
${U(R)}^{\dagger }{\widehat {V}}_{a}U(R)=\sum _{b}R_{ab}{\widehat {V}}_{b}$
Note that here, the term "vector" is used two different ways: kets such as |ψ⟩ are elements of abstract Hilbert spaces, while the vector operator is defined as a quantity whose components transform in a certain way under rotations.
Spherical vector operators
A vector operator in the spherical basis is V = (V+1, V0, V−1) where the components are:[2]
$V_{+1}=-{\frac {1}{\sqrt {2}}}(V_{x}+iV_{y})\,\quad V_{-1}={\frac {1}{\sqrt {2}}}(V_{x}-iV_{y})\,,\quad V_{0}=V_{z}\,,$
and the commutators with the rotation generators are:
${\begin{aligned}\left[J_{z},V_{q}\right]&=qV_{q}\\[1ex]\left[J_{\pm },V_{0}\right]&={\sqrt {2}}V_{\pm }\\[1ex]\left[J_{\pm },V_{\mp }\right]&={\sqrt {2}}V_{0}\\[1ex]\left[J_{\pm },V_{\pm }\right]&=0\end{aligned}}$
where q is a placeholder for the spherical basis labels (+1, 0, −1), and:
$J_{\pm }=J_{x}\pm iJ_{y}\,,$
(some authors may place a factor of 1/2 on the left hand side of the equation) and raise (J+) or lower (J−) the total magnetic quantum number m by one unit. In the spherical basis the generators are:
$J_{\pm 1}=\mp {\frac {1}{\sqrt {2}}}J_{\pm }\,,\quad J_{0}=J_{z}$
The rotation transformation in the spherical basis (originally written in the Cartesian basis) is then:
${U(R)}^{\dagger }{\widehat {V}}_{q}U(R)=\sum _{q'}{{D(R)}_{qq'}^{(1)}}^{*}{\widehat {V}}_{q'}$
One can generalize the vector operator concept easily to tensorial operators, shown next.
Tensor operators and their reducible and irreducible representations
A tensor operator can be rotated according to:[2]
$U(R)^{\dagger }{\widehat {T}}_{pqr\cdots }U(R)=R_{pi}R_{qj}R_{rk}\cdots {\widehat {T}}_{ijk\cdots }$
Consider a dyadic tensor with components $T_{ij}=a_{i}b_{j}$. This rotates infinitesimally according to:
$U(R)^{\dagger }{\widehat {T}}_{pq}U(R)=R_{pi}R_{qj}{\widehat {T}}_{ij}=R_{pi}{\widehat {a}}_{i}R_{qj}{\widehat {b}}_{j}$
Cartesian dyadic tensors of the form
${\hat {\mathbf {T} }}=\mathbf {e} _{i}{\widehat {a}}_{i}\otimes \mathbf {e} _{j}{\widehat {b}}_{j}=\mathbf {e} _{i}\otimes \mathbf {e} _{j}{\widehat {a}}_{i}{\widehat {b}}_{j}$
where a and b are two vector operators:
${\hat {\mathbf {a} }}=\mathbf {e} _{i}{\widehat {a}}_{i}\,,\quad {\hat {\mathbf {b} }}=\mathbf {e} _{j}{\widehat {b}}_{j}$
are reducible, which means they can be re-expressed in terms of a and b as a rank 0 tensor (scalar), plus a rank 1 tensor (an antisymmetric tensor), plus a rank 2 tensor (a symmetric tensor with zero trace):
$\mathbf {T} =\mathbf {T} ^{(0)}+\mathbf {T} ^{(1)}+\mathbf {T} ^{(2)}$
where the first term
${\widehat {T}}_{ij}^{(0)}={\frac {{\widehat {a}}_{k}{\widehat {b}}_{k}}{3}}\delta _{ij}$
includes just one component, a scalar equivalently written (a·b)/3, the second
${\widehat {T}}_{ij}^{(1)}={\frac {1}{2}}\left[{\widehat {a}}_{i}{\widehat {b}}_{j}-{\widehat {a}}_{j}{\widehat {b}}_{i}\right]={\widehat {a}}_{[i}{\widehat {b}}_{j]}$
includes three independent components, equivalently the components of (a×b)/2, and the third
${\widehat {T}}_{ij}^{(2)}={\tfrac {1}{2}}\left({\widehat {a}}_{i}{\widehat {b}}_{j}+{\widehat {a}}_{j}{\widehat {b}}_{i}\right)-{\tfrac {1}{3}}{\widehat {a}}_{k}{\widehat {b}}_{k}\delta _{ij}={\widehat {a}}_{(i}{\widehat {b}}_{j)}-T_{ij}^{(0)}$
includes five independent components. Throughout, δij is the Kronecker delta, the components of the identity matrix. The number in the superscripted brackets denotes the tensor rank. These three terms are irreducible, which means they cannot be decomposed further and still be tensors satisfying the defining transformation laws under which they must be invariant. These also correspond to the number of spherical harmonic functions 2ℓ + 1 for ℓ = 0, 1, 2, the same as the ranks for each tensor. Each of the irreducible representations T(1), T(2) ... transform like angular momentum eigenstates according to the number of independent components.
Example of a Tensor operator,
• The Quadrupole moment operator,
$Q_{ij}=\sum _{\alpha }q_{\alpha }(3r_{\alpha i}r_{\alpha j}-r_{\alpha }^{2}\delta _{ij})$
• Two Tensor operators can be multiplied to give another Tensor operator.
$T_{ij}=V_{i}W_{j}$
In general,
$T_{i_{1}i_{2}\cdots j_{1}j_{2}\cdots }=V_{i_{1}i_{2}\cdots }W_{j_{1}j_{2}\cdots }$
Note: This is just an example, in general, a tensor operator cannot be written as the product of two Tensor operators as given in the above example.
Spherical tensor operators
Continuing the previous example of the second order dyadic tensor T = a ⊗ b, casting each of a and b into the spherical basis and substituting into T gives the spherical tensor operators of the second order, which are:
${\begin{aligned}{\widehat {T}}_{\pm 2}^{(2)}&={\widehat {a}}_{\pm 1}{\widehat {b}}_{\pm 1}\\[1ex]{\widehat {T}}_{\pm 1}^{(2)}&={\tfrac {1}{\sqrt {2}}}\left({\widehat {a}}_{\pm 1}{\widehat {b}}_{0}+{\widehat {a}}_{0}{\widehat {b}}_{\pm 1}\right)\\[1ex]{\widehat {T}}_{0}^{(2)}&={\tfrac {1}{\sqrt {6}}}\left({\widehat {a}}_{+1}{\widehat {b}}_{-1}+{\widehat {a}}_{-1}{\widehat {b}}_{+1}+2{\widehat {a}}_{0}{\widehat {b}}_{0}\right)\end{aligned}}$
Using the infinitesimal rotation operator and its Hermitian conjugate, one can derive the commutation relation in the spherical basis:
$\left[J_{a},{\widehat {T}}_{q}^{(2)}\right]=\sum _{q'}{D(J_{a})}_{qq'}^{(2)}{\widehat {T}}_{q'}^{(2)}=\sum _{q'}\langle j{=}2,m{=}q|J_{a}|j{=}2,m{=}q'\rangle {\widehat {T}}_{q'}^{(2)}$
and the finite rotation transformation in the spherical basis is:
${U(R)}^{\dagger }{\widehat {T}}_{q}^{(2)}U(R)=\sum _{q'}{{D(R)}_{qq'}^{(2)}}^{*}{\widehat {T}}_{q'}^{(2)}$
In general, tensor operators can be constructed from two perspectives.[3]
One way is to specify how spherical tensors transform under a physical rotation - a group theoretical definition. A rotated angular momentum eigenstate can be decomposed into a linear combination of the initial eigenstates: the coefficients in the linear combination consist of Wigner rotation matrix entries. Spherical tensor operators are sometimes defined as the set of operators that transform just like the eigenkets under a rotation.
A spherical tensor $T_{q}^{(k)}$ of rank $k$ is defined to rotate into $T_{q'}^{(k)}$according to:
${U(R)}^{\dagger }{\widehat {T}}_{q}^{(k)}U(R)=\sum _{q'}{D(R)_{qq'}^{(k)}}^{*}{\widehat {T}}_{q'}^{(k)}$
where q = k, k − 1, ..., −k + 1, −k. For spherical tensors, k and q are analogous labels to ℓ and m respectively, for spherical harmonics. Some authors write Tkq instead of Tq(k), with or without the parentheses enclosing the rank number k.
Another related procedure requires that the spherical tensors satisfy certain commutation relations with respect to the rotation generators Jx, Jy, Jz - an algebraic definition.
The commutation relations of the angular momentum components with the tensor operators are:
${\begin{aligned}\left[J_{\pm },{\widehat {T}}_{q}^{(k)}\right]&=\hbar {\sqrt {(k\mp q)(k\pm q+1)}}{\widehat {T}}_{q\pm 1}^{(k)}\\[1ex]\left[J_{z},{\widehat {T}}_{q}^{(k)}\right]&=\hbar q{\widehat {T}}_{q}^{(k)}\end{aligned}}$
For any 3d vector, not just a unit vector, and not just the position vector:
$\mathbf {a} =a_{x}\mathbf {e} _{x}+a_{y}\mathbf {e} _{y}+a_{z}\mathbf {e} _{z}$
a spherical tensor is a spherical harmonic as a function of this vector a, and in Dirac notation:
$T_{q}^{(k)}=Y_{\ell =k}^{m=q}(\mathbf {a} )=\langle \mathbf {a} |k,q\rangle $
(the super and subscripts switch places for the corresponding labels ℓ ↔ k and m ↔ q which spherical tensors and spherical harmonics use).
Spherical harmonic states and spherical tensors can also be constructed out of the Clebsch–Gordan coefficients. Irreducible spherical tensors can build higher rank spherical tensors; if Aq1(k1) and Bq2(k2) are two spherical tensors of ranks k1 and k2 respectively, then:
$T_{q}^{(k)}=\sum _{q_{1},q_{2}}\langle k_{1},k_{2};q_{1},q_{2}|k_{1},k_{2};k,q\rangle A_{q_{1}}^{(k_{1})}B_{q_{2}}^{(k_{2})}$
is a spherical tensor of rank k.
The Hermitian adjoint of a spherical tensor may be defined as
$(T^{\dagger })_{q}^{(k)}=(-1)^{k-q}(T_{-q}^{(k)})^{\dagger }.$
There is some arbitrariness in the choice of the phase factor: any factor containing (−1)±q will satisfy the commutation relations.[4] The above choice of phase has the advantages of being real and that the tensor product of two commuting Hermitian operators is still Hermitian.[5] Some authors define it with a different sign on q, without the k, or use only the floor of k.[6]
Angular momentum and spherical harmonics
Orbital angular momentum and spherical harmonics
Orbital angular momentum operators have the ladder operators:
$L_{\pm }=L_{x}\pm iL_{y}$
which raise or lower the orbital magnetic quantum number mℓ by one unit. This has almost exactly the same form as the spherical basis, aside from constant multiplicative factors.
Spherical tensor operators and quantum spin
See also: Spin-weighted spherical harmonics and spinor spherical harmonics
Spherical tensors can also be formed from algebraic combinations of the spin operators Sx, Sy, Sz, as matrices, for a spin system with total quantum number j = ℓ + s (and ℓ = 0). Spin operators have the ladder operators:
$S_{\pm }=S_{x}\pm iS_{y}$
which raise or lower the spin magnetic quantum number ms by one unit.
Applications
Spherical bases have broad applications in pure and applied mathematics and physical sciences where spherical geometries occur.
Dipole radiative transitions in a single-electron atom (alkali)
The transition amplitude is proportional to matrix elements of the dipole operator between the initial and final states. We use an electrostatic, spinless model for the atom and we consider the transition from the initial energy level Enℓ to final level En′ℓ′. These levels are degenerate, since the energy does not depend on the magnetic quantum number m or m′. The wave functions have the form,
$\psi _{n\ell m}(r,\theta ,\phi )=R_{n\ell }(r)Y_{\ell m}(\theta ,\phi )$
The dipole operator is proportional to the position operator of the electron, so we must evaluate matrix elements of the form,
$\langle n'\ell 'm'|\mathbf {r} |n\ell m\rangle $
where, the initial state is on the right and the final one on the left. The position operator r has three components, and the initial and final levels consist of 2ℓ + 1 and 2ℓ′ + 1 degenerate states, respectively. Therefore if we wish to evaluate the intensity of a spectral line as it would be observed, we really have to evaluate 3(2ℓ′+ 1)(2ℓ+ 1) matrix elements, for example, 3×3×5 = 45 in a 3d → 2p transition. This is actually an exaggeration, as we shall see, because many of the matrix elements vanish, but there are still many non-vanishing matrix elements to be calculated.
A great simplification can be achieved by expressing the components of r, not with respect to the Cartesian basis, but with respect to the spherical basis. First we define,
$r_{q}={\hat {\mathbf {e} }}_{q}\cdot \mathbf {r} $
Next, by inspecting a table of the Yℓm′s, we find that for ℓ = 1 we have,
${\begin{aligned}rY_{11}(\theta ,\phi )&=&&-r{\sqrt {\frac {3}{8\pi }}}\sin(\theta )e^{i\phi }&=&{\sqrt {\frac {3}{4\pi }}}\left(-{\frac {x+iy}{\sqrt {2}}}\right)\\rY_{10}(\theta ,\phi )&=&&r{\sqrt {\frac {3}{4\pi }}}\cos(\theta )&=&{\sqrt {\frac {3}{4\pi }}}z\\rY_{1-1}(\theta ,\phi )&=&&r{\sqrt {\frac {3}{8\pi }}}\sin(\theta )e^{-i\phi }&=&{\sqrt {\frac {3}{4\pi }}}\left({\frac {x-iy}{\sqrt {2}}}\right)\end{aligned}}$
where, we have multiplied each Y1m by the radius r. On the right hand side we see the spherical components rq of the position vector r. The results can be summarized by,
$rY_{1q}(\theta ,\phi )={\sqrt {\frac {3}{4\pi }}}r_{q}$
for q = 1, 0, −1, where q appears explicitly as a magnetic quantum number. This equation reveals a relationship between vector operators and the angular momentum value ℓ = 1, something we will have more to say about presently. Now the matrix elements become a product of a radial integral times an angular integral,
$\langle n'\ell 'm'|r_{q}|n\ell m\rangle =\left(\int _{0}^{\infty }r^{2}drR_{n'\ell '}^{*}(r)rR_{n\ell }(r)\right)\left({\sqrt {\frac {4\pi }{3}}}\int \sin {(\theta )}d\Omega Y_{\ell 'm'}^{*}(\theta ,\phi )Y_{1q}(\theta ,\phi )Y_{\ell m}(\theta ,\phi )\right)$
We see that all the dependence on the three magnetic quantum numbers (m′,q,m) is contained in the angular part of the integral. Moreover, the angular integral can be evaluated by the three-Yℓm formula, whereupon it becomes proportional to the Clebsch-Gordan coefficient,
$\langle \ell 'm'|\ell 1mq\rangle $
The radial integral is independent of the three magnetic quantum numbers (m′, q, m), and the trick we have just used does not help us to evaluate it. But it is only one integral, and after it has been done, all the other integrals can be evaluated just by computing or looking up Clebsch–Gordan coefficients.
The selection rule m′ = q + m in the Clebsch–Gordan coefficient means that many of the integrals vanish, so we have exaggerated the total number of integrals that need to be done. But had we worked with the Cartesian components ri of r, this selection rule might not have been obvious. In any case, even with the selection rule, there may still be many nonzero integrals to be done (nine, in the case 3d → 2p). The example we have just given of simplifying the calculation of matrix elements for a dipole transition is really an application of the Wigner–Eckart theorem, which we take up later in these notes.
Magnetic resonance
The spherical tensor formalism provides a common platform for treating coherence and relaxation in nuclear magnetic resonance. In NMR and EPR, spherical tensor operators are employed to express the quantum dynamics of particle spin, by means of an equation of motion for the density matrix entries, or to formulate dynamics in terms of an equation of motion in Liouville space. The Liouville space equation of motion governs the observable averages of spin variables. When relaxation is formulated using a spherical tensor basis in Liouville space, insight is gained because the relaxation matrix exhibits the cross-relaxation of spin observables directly.[3]
See also
• Wigner–Eckart theorem
• Structure tensor
• Clebsch–Gordan coefficients for SU(3)
References
Notes
1. Jeevanjee, Nadir (2015). An Introduction to Tensors and Group Theory for Physicists (2nd ed.). Birkhauser. ISBN 978-0-8176-4714-8.
2. E. Abers (2004). "5". Quantum Mechanics. Addison Wesley. ISBN 978-0-13-146100-0.
3. R.D. Nielsen; B.H. Robinson (2006). "The Spherical Tensor Formalism Applied to Relaxation in Magnetic Resonance". Concepts in Magnetic Resonance Part A. 28A (4): 270–271. doi:10.1002/cmr.a.20055. Retrieved 2023-04-06.
4. McCarthy, Ian E.; Weigold, Erich (2005). Electron-Atom Collisions (Volume 5 of Cambridge Monographs on Atomic, Molecular and Chemical Physics). Cambridge University Press. p. 68. ISBN 9780521019682.
5. Edmonds, A. R. (1957). Angular Momentum in Quantum Mechanics. Princeton University Press. p. 78. ISBN 9780691025896.
6. Degl'Innocenti, M. Landi; Landolfi, M. (2006). Polarization in Spectral Lines. Springer Science & Business Media. p. 65. ISBN 9781402024153.
Sources
• P. T. Callaghan (2011). Translational Dynamics and Magnetic Resonance:Principles of Pulsed Gradient Spin Echo NMR. Oxford University Press. ISBN 978-0-191-621-048.
• V. V. Balashov; A. N. Grum-Grzhimailo; N.M. Kabachnik (2000). Polarization and Correlation Phenomena in Atomic Collisions: A Practical Theory Course. Springer. ISBN 9780306462665.
• J. A. Tuszynski (1990). Spherical Tensor Operators: Tables of Matrix Elements and Symmetries. World Scientific. ISBN 978-981-0202-835.
• L. Castellani; J. Wess (1996). Quantum Groups and Their Applications in Physics: Varenna on Lake Como, Villa Monastero, 28 June-8 July 1994. Società Italiana di Fisica, IOS. ISBN 978-905-199-24-72.
• Introduction to the Graphical Theory of Angular Momentum. Springer. 2009. ISBN 978-364-203-11-99.
• A. R. Edmonds (1996). Angular Momentum in Quantum Mechanics (2nd ed.). Princeton University Press. ISBN 978-0-691-025-896.
• L.J. Mueller (2011). "Tensors and rotations in NMR". Concepts in Magnetic Resonance Part A. 38A (5): 221–235. doi:10.1002/cmr.a.20224. S2CID 8889942.
• M.S. Anwar (2004). "Spherical Tensor Operators in NMR" (PDF).
• P. Callaghan (1993). Principles of nuclear magnetic resonance microscopy. Oxford University Press. pp. 56–57. ISBN 978-0-198-539-971.
Spherical harmonics
• G.W.F. Drake (2006). Springer Handbook of Atomic, Molecular, and Optical Physics (2nd ed.). Springer. p. 57. ISBN 978-0-3872-6308-3.
• F.A. Dahlen; J. Tromp (1998). Theoretical global seismology (2nd ed.). Princeton University Press. p. appendix C. ISBN 978-0-69100-1241.
• D.O. Thompson; D.E. Chimenti (1997). Review of Progress in Quantitative Nondestructive Evaluation. Review Of Progress In Quantitative Nondestructive Evaluation. Vol. 16. Springer. p. 1708. ISBN 978-0-3064-55971.
• H. Paetz; G. Schieck (2011). Nuclear Physics with Polarized Particles. Lecture Notes in Physics. Vol. 842. Springer. p. 31. ISBN 978-364-224-225-0.
• V. Devanathan (1999). Angular Momentum Techniques in Quantum Mechanics. Fundamental Theories of Physics. Vol. 108. Springer. pp. 34, 61. ISBN 978-0-7923-5866-4.
• V.D. Kleiman; R.N. Zare (1998). "5". A Companion to Angular Momentum. John Wiley & Sons. p. 112. ISBN 978-0-4711-9249-7.
Angular momentum and spin
• Devanathan, V (2002). "Vectors and Tensors in Spherical Basis". Angular Momentum Techniques in Quantum Mechanics. Fundamental Theories of Physics. Vol. 108. pp. 24–33. doi:10.1007/0-306-47123-X_3. ISBN 978-0-306-47123-0.
• K.T. Hecht (2000). Quantum mechanics. Graduate texts in contemporary physics. Springer. ISBN 978-0-387-989-198.
Condensed matter physics
• J.A. Mettes; J.B. Keith; R.B. McClurg (2002). "Molecular Crystal Global Phase Diagrams:I Method of Construction" (PDF).
• B.Henderson, R.H. Bartram (2005). Crystal-Field Engineering of Solid-State Laser Materials. Cambridge Studies in Modern Optics. Vol. 25. Cambridge University Press. p. 49. ISBN 978-0-52101-8012.
• Edward U. Condon, and Halis Odabaşı (1980). Atomic Structure. CUP Archive. ISBN 978-0-5212-98933.
• Melinda J. Duer, ed. (2008). "3". Solid State NMR Spectroscopy: Principles and Applications. John Wiley & Sons. p. 113. ISBN 978-0-4709-9938-7.
• K.D. Bonin; V.V. Kresin (1997). "2". Electric - Dipole Polarizabilities of Atoms, Molecules and Clusters. World Scientific. pp. 14–15. ISBN 978-981-022-493-6.
• A.E. McDermott, T.Polenova (2012). Solid State NMR Studies of Biopolymers. EMR handbooks. John Wiley & Sons. p. 42. ISBN 978-111-858-889-5.
Magnetic resonance
• L.J. Mueller (2011). "Tensors and rotations in NMR". Concepts in Magnetic Resonance Part A. 38A (5): 221–235. doi:10.1002/cmr.a.20224. S2CID 8889942.
• M.S. Anwar (2004). "Spherical Tensor Operators in NMR" (PDF).
• P. Callaghan (1993). Principles of nuclear magnetic resonance microscopy. Oxford University Press. pp. 56–57. ISBN 978-0-198-539-971.
Image processing
• M. Reisert; H. Burkhardt (2009). S. Aja-Fernández (ed.). Tensors in Image Processing and Computer Vision. Springer. ISBN 978-184-8822-993.
• D.H. Laidlaw; J. Weickert (2009). Visualization and Processing of Tensor Fields: Advances and Perspectives. Mathematics and Visualization. Springer. ISBN 978-354-088-378-4.
• M. Felsberg; E. Jonsson (2005). Energy Tensors: Quadratic, Phase Invariant Image Operators. Lecture Notes in Computer Science. Vol. 3663. Springer. pp. 493–500.
• E. König; S. Kremer (1979). "Tensor Operator Algebra for Point Groups". Magnetism Diagrams for Transition Metal Ions. Lecture Notes in Computer Science. Vol. 3663. Springer. pp. 13–20. doi:10.1007/978-1-4613-3003-5_3. ISBN 978-1-4613-3005-9.
External links
• (2012) Clebsch-Gordon (sic) coefficients and the tensor spherical harmonics
• The tensor spherical harmonics
• (2010) Irreducible Tensor Operators and the Wigner-Eckart Theorem Archived 2014-07-20 at the Wayback Machine
• Tensor operators
• M. Fowler (2008), Tensor Operators
• Tensor_Operators
• (2009) Tensor Operators and the Wigner Eckart Theorem
• The Wigner-Eckart theorem
• (2004) Rotational Transformations and Spherical Tensor Operators
• Tensor operators
• Evaluation of the matrix elements for radiative transitions
• D.K. Ghosh, (2013) Angular Momentum - III : Wigner- Eckart Theorem
• B. Baragiola (2002) Tensor Operators
• Spherical Tensors
Tensors
Glossary of tensor theory
Scope
Mathematics
• Coordinate system
• Differential geometry
• Dyadic algebra
• Euclidean geometry
• Exterior calculus
• Multilinear algebra
• Tensor algebra
• Tensor calculus
• Physics
• Engineering
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Notation
• Abstract index notation
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• Voigt notation
Tensor
definitions
• Tensor (intrinsic definition)
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Operations
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Related
abstractions
• Affine connection
• Basis
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Notable tensors
Mathematics
• Kronecker delta
• Levi-Civita symbol
• Metric tensor
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• Ricci curvature
• Riemann curvature tensor
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• Weyl tensor
Physics
• Moment of inertia
• Angular momentum tensor
• Spin tensor
• Cauchy stress tensor
• stress–energy tensor
• Einstein tensor
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• Gluon field strength tensor
• Metric tensor (GR)
Mathematicians
• Élie Cartan
• Augustin-Louis Cauchy
• Elwin Bruno Christoffel
• Albert Einstein
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• Carl Friedrich Gauss
• Hermann Grassmann
• Tullio Levi-Civita
• Gregorio Ricci-Curbastro
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• Jan Arnoldus Schouten
• Woldemar Voigt
• Hermann Weyl
| Wikipedia |
Tensor product (disambiguation)
Tensor product refers to one of several related binary operations, typically denoted $-\otimes -$. Usually, they are associative, unital, and symmetric (up to some appropriate kind of structural equivalence).
It may refer to:
• Tensor product of vector spaces, an operation on vector spaces (the original tensor product)
• Tensor product of modules, the same operation slightly generalized to modules over arbitrary rings
• Kronecker product, the tensor product of matrices (or vectors), which satisfies all the properties for vector spaces and allows a concrete representation
• Tensor product of Hilbert spaces, endowed with a special inner product as to remain a Hilbert space
• Other topological tensor products
• Tensor product of graphs, an operation on graphs, whose adjacency matrices are the Kronecker product of the component adjacency matrices
• Tensor product of algebras (or rings), on algebras over a field (or other commutative ring)
• Tensor product of representations, a special case in representation theory
• Tensor product of fields, an operation on fields—unlike most tensor products, the category of fields is not closed with respect to this operation (i.e., sometimes the product is not a field)
• "Categorified" concepts, applied "pointwise" on objects and morphisms:
• Tensor product of vector bundles
• Tensor product of sheaves of modules, essentially the same thing
• Tensor product of functors
Categories closed under a suitable tensor product are called "monoidal categories". Special types of monoidal categories exist with interesting properties.
| Wikipedia |
Tensor product of Hilbert spaces
In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly speaking, the tensor product is the metric space completion of the ordinary tensor product. This is an example of a topological tensor product. The tensor product allows Hilbert spaces to be collected into a symmetric monoidal category.[1]
Definition
Since Hilbert spaces have inner products, one would like to introduce an inner product, and therefore a topology, on the tensor product that arises naturally from those of the factors. Let $H_{1}$ and $H_{2}$ be two Hilbert spaces with inner products $\langle \cdot ,\cdot \rangle _{1}$ and $\langle \cdot ,\cdot \rangle _{2},$ respectively. Construct the tensor product of $H_{1}$ and $H_{2}$ as vector spaces as explained in the article on tensor products. We can turn this vector space tensor product into an inner product space by defining
$\left\langle \phi _{1}\otimes \phi _{2},\psi _{1}\otimes \psi _{2}\right\rangle =\left\langle \phi _{1},\psi _{1}\right\rangle _{1}\,\left\langle \phi _{2},\psi _{2}\right\rangle _{2}\quad {\mbox{for all }}\phi _{1},\psi _{1}\in H_{1}{\mbox{ and }}\phi _{2},\psi _{2}\in H_{2}$
and extending by linearity. That this inner product is the natural one is justified by the identification of scalar-valued bilinear maps on $H_{1}\times H_{2}$ and linear functionals on their vector space tensor product. Finally, take the completion under this inner product. The resulting Hilbert space is the tensor product of $H_{1}$ and $H_{2}.$
Explicit construction
The tensor product can also be defined without appealing to the metric space completion. If $H_{1}$ and $H_{2}$ are two Hilbert spaces, one associates to every simple tensor product $x_{1}\otimes x_{2}$ the rank one operator from $H_{1}^{*}$ to $H_{2}$ that maps a given $x^{*}\in H_{1}^{*}$ as
$x^{*}\mapsto x^{*}(x_{1})\,x_{2}.$
This extends to a linear identification between $H_{1}\otimes H_{2}$ and the space of finite rank operators from $H_{1}^{*}$ to $H_{2}.$ The finite rank operators are embedded in the Hilbert space $HS(H_{1}^{*},H_{2})$ of Hilbert–Schmidt operators from $H_{1}^{*}$ to $H_{2}.$ The scalar product in $HS(H_{1}^{*},H_{2})$ is given by
$\langle T_{1},T_{2}\rangle =\sum _{n}\left\langle T_{1}e_{n}^{*},T_{2}e_{n}^{*}\right\rangle ,$
where $\left(e_{n}^{*}\right)$ is an arbitrary orthonormal basis of $H_{1}^{*}.$
Under the preceding identification, one can define the Hilbertian tensor product of $H_{1}$ and $H_{2},$ that is isometrically and linearly isomorphic to $HS(H_{1}^{*},H_{2}).$
Universal property
The Hilbert tensor product $H_{1}\otimes H_{2}$ is characterized by the following universal property (Kadison & Ringrose 1997, Theorem 2.6.4):
Theorem — There is a weakly Hilbert–Schmidt mapping $p:H_{1}\times H_{2}\to H_{1}\otimes H_{2}$ such that, given any weakly Hilbert–Schmidt mapping $L:H_{1}\times H_{2}\to K$ to a Hilbert space $K,$ there is a unique bounded operator $T:H_{1}\otimes H_{2}\to K$ such that $L=Tp.$
A weakly Hilbert-Schmidt mapping $L:H_{1}\times H_{2}\to K$ is defined as a bilinear map for which a real number $d$ exists, such that
$\sum _{i,j=1}^{\infty }{\bigl |}\left\langle L(e_{i},f_{j}),u\right\rangle {\bigr |}^{2}\leq d^{2}\,\|u\|^{2}$
for all $u\in K$ and one (hence all) orthonormal bases $e_{1},e_{2},\ldots $ of $H_{1}$ and $f_{1},f_{2},\ldots $ of $H_{2}.$
As with any universal property, this characterizes the tensor product H uniquely, up to isomorphism. The same universal property, with obvious modifications, also applies for the tensor product of any finite number of Hilbert spaces. It is essentially the same universal property shared by all definitions of tensor products, irrespective of the spaces being tensored: this implies that any space with a tensor product is a symmetric monoidal category, and Hilbert spaces are a particular example thereof.
Infinite tensor products
If $H_{n}$ is a collection of Hilbert spaces and $\xi _{n}$ is a collection of unit vectors in these Hilbert spaces then the incomplete tensor product (or Guichardet tensor product) is the $L^{2}$ completion of the set of all finite linear combinations of simple tensor vectors $ \bigotimes _{n=1}^{\infty }\psi _{n}$ where all but finitely many of the $\psi _{n}$'s equal the corresponding $\xi _{n}.$[2]
Operator algebras
Let ${\mathfrak {A}}_{i}$ be the von Neumann algebra of bounded operators on $H_{i}$ for $i=1,2.$ Then the von Neumann tensor product of the von Neumann algebras is the strong completion of the set of all finite linear combinations of simple tensor products $A_{1}\otimes A_{2}$ where $A_{i}\in {\mathfrak {A}}_{i}$ for $i=1,2.$ This is exactly equal to the von Neumann algebra of bounded operators of $H_{1}\otimes H_{2}.$ Unlike for Hilbert spaces, one may take infinite tensor products of von Neumann algebras, and for that matter C*-algebras of operators, without defining reference states.[2] This is one advantage of the "algebraic" method in quantum statistical mechanics.
Properties
If $H_{1}$ and $H_{2}$ have orthonormal bases $\left\{\phi _{k}\right\}$ and $\left\{\psi _{l}\right\},$ respectively, then $\left\{\phi _{k}\otimes \psi _{l}\right\}$ is an orthonormal basis for $H_{1}\otimes H_{2}.$ In particular, the Hilbert dimension of the tensor product is the product (as cardinal numbers) of the Hilbert dimensions.
Examples and applications
The following examples show how tensor products arise naturally.
Given two measure spaces $X$ and $Y$, with measures $\mu $ and $\nu $ respectively, one may look at $L^{2}(X\times Y),$ the space of functions on $X\times Y$ that are square integrable with respect to the product measure $\mu \times \nu .$ If $f$ is a square integrable function on $X,$ and $g$ is a square integrable function on $Y,$ then we can define a function $h$ on $X\times Y$ by $h(x,y)=f(x)g(y).$ The definition of the product measure ensures that all functions of this form are square integrable, so this defines a bilinear mapping $L^{2}(X)\times L^{2}(Y)\to L^{2}(X\times Y).$ Linear combinations of functions of the form $f(x)g(y)$ are also in $L^{2}(X\times Y).$ It turns out that the set of linear combinations is in fact dense in $L^{2}(X\times Y),$ if $L^{2}(X)$ and $L^{2}(Y)$ are separable. This shows that $L^{2}(X)\otimes L^{2}(Y)$ is isomorphic to $L^{2}(X\times Y),$ and it also explains why we need to take the completion in the construction of the Hilbert space tensor product.
Similarly, we can show that $L^{2}(X;H)$, denoting the space of square integrable functions $X\to H,$ is isomorphic to $L^{2}(X)\otimes H$ if this space is separable. The isomorphism maps $f(x)\otimes \phi \in L^{2}(X)\otimes H$ to $f(x)\phi \in L^{2}(X;H)$ We can combine this with the previous example and conclude that $L^{2}(X)\otimes L^{2}(Y)$ and $L^{2}(X\times Y)$ are both isomorphic to $L^{2}\left(X;L^{2}(Y)\right).$
Tensor products of Hilbert spaces arise often in quantum mechanics. If some particle is described by the Hilbert space $H_{1},$ and another particle is described by $H_{2},$ then the system consisting of both particles is described by the tensor product of $H_{1}$ and $H_{2}.$ For example, the state space of a quantum harmonic oscillator is $L^{2}(\mathbb {R} ),$ so the state space of two oscillators is $L^{2}(\mathbb {R} )\otimes L^{2}(\mathbb {R} ),$ which is isomorphic to $L^{2}\left(\mathbb {R} ^{2}\right).$ Therefore, the two-particle system is described by wave functions of the form $\psi \left(x_{1},x_{2}\right).$ A more intricate example is provided by the Fock spaces, which describe a variable number of particles.
References
1. B. Coecke and E. O. Paquette, Categories for the practising physicist, in: New Structures for Physics, B. Coecke (ed.), Springer Lecture Notes in Physics, 2009. arXiv:0905.3010
2. Bratteli, O. and Robinson, D: Operator Algebras and Quantum Statistical Mechanics v.1, 2nd ed., page 144. Springer-Verlag, 2002.
Bibliography
• Kadison, Richard V.; Ringrose, John R. (1997). Fundamentals of the theory of operator algebras. Vol. I. Graduate Studies in Mathematics. Vol. 15. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-0819-1. MR 1468229..
• Weidmann, Joachim (1980). Linear operators in Hilbert spaces. Graduate Texts in Mathematics. Vol. 68. Berlin, New York: Springer-Verlag. ISBN 978-0-387-90427-6. MR 0566954..
Hilbert spaces
Basic concepts
• Adjoint
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Main results
• Bessel's inequality
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Examples
• Cn(K) with K compact & n<∞
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Topological tensor products and nuclear spaces
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| Wikipedia |
Tensor product of algebras
In mathematics, the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring is a field, the most common application of such products is to describe the product of algebra representations.
Algebraic structure → Ring theory
Ring theory
Basic concepts
Rings
• Subrings
• Ideal
• Quotient ring
• Fractional ideal
• Total ring of fractions
• Product of rings
• Free product of associative algebras
• Tensor product of algebras
Ring homomorphisms
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• Inner automorphism
• Frobenius endomorphism
Algebraic structures
• Module
• Associative algebra
• Graded ring
• Involutive ring
• Category of rings
• Initial ring $\mathbb {Z} $
• Terminal ring $0=\mathbb {Z} _{1}$
Related structures
• Field
• Finite field
• Non-associative ring
• Lie ring
• Jordan ring
• Semiring
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Commutative algebra
Commutative rings
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p-adic number theory and decimals
• Direct limit/Inverse limit
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• p-adic rationals $\mathbb {Z} [1/p]$
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• p-adic integers $\mathbb {Z} _{p}$
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Operator algebra
Definition
Let R be a commutative ring and let A and B be R-algebras. Since A and B may both be regarded as R-modules, their tensor product
$A\otimes _{R}B$
is also an R-module. The tensor product can be given the structure of a ring by defining the product on elements of the form a ⊗ b by[1][2]
$(a_{1}\otimes b_{1})(a_{2}\otimes b_{2})=a_{1}a_{2}\otimes b_{1}b_{2}$
and then extending by linearity to all of A ⊗R B. This ring is an R-algebra, associative and unital with identity element given by 1A ⊗ 1B.[3] where 1A and 1B are the identity elements of A and B. If A and B are commutative, then the tensor product is commutative as well.
The tensor product turns the category of R-algebras into a symmetric monoidal category.
Further properties
There are natural homomorphisms from A and B to A ⊗R B given by[4]
$a\mapsto a\otimes 1_{B}$
$b\mapsto 1_{A}\otimes b$
These maps make the tensor product the coproduct in the category of commutative R-algebras. The tensor product is not the coproduct in the category of all R-algebras. There the coproduct is given by a more general free product of algebras. Nevertheless, the tensor product of non-commutative algebras can be described by a universal property similar to that of the coproduct:
${\text{Hom}}(A\otimes B,X)\cong \lbrace (f,g)\in {\text{Hom}}(A,X)\times {\text{Hom}}(B,X)\mid \forall a\in A,b\in B:[f(a),g(b)]=0\rbrace ,$
where [-, -] denotes the commutator. The natural isomorphism is given by identifying a morphism $\phi :A\otimes B\to X$ on the left hand side with the pair of morphisms $(f,g)$ on the right hand side where $f(a):=\phi (a\otimes 1)$ and similarly $g(b):=\phi (1\otimes b)$.
Applications
The tensor product of commutative algebras is of frequent use in algebraic geometry. For affine schemes X, Y, Z with morphisms from X and Z to Y, so X = Spec(A), Y = Spec(R), and Z = Spec(B) for some commutative rings A, R, B, the fiber product scheme is the affine scheme corresponding to the tensor product of algebras:
$X\times _{Y}Z=\operatorname {Spec} (A\otimes _{R}B).$
More generally, the fiber product of schemes is defined by gluing together affine fiber products of this form.
Examples
See also: tensor product of modules § Examples
• The tensor product can be used as a means of taking intersections of two subschemes in a scheme: consider the $\mathbb {C} [x,y]$-algebras $\mathbb {C} [x,y]/f$, $\mathbb {C} [x,y]/g$, then their tensor product is $\mathbb {C} [x,y]/(f)\otimes _{\mathbb {C} [x,y]}\mathbb {C} [x,y]/(g)\cong \mathbb {C} [x,y]/(f,g)$, which describes the intersection of the algebraic curves f = 0 and g = 0 in the affine plane over C.
• More generally, if $A$ is a commutative ring and $I,J\subseteq A$ are ideals, then ${\frac {A}{I}}\otimes _{A}{\frac {A}{J}}\cong {\frac {A}{I+J}}$, with a unique isomorphism sending $(a+I)\otimes (b+J)$ to $(ab+I+J)$.
• Tensor products can be used as a means of changing coefficients. For example, $\mathbb {Z} [x,y]/(x^{3}+5x^{2}+x-1)\otimes _{\mathbb {Z} }\mathbb {Z} /5\cong \mathbb {Z} /5[x,y]/(x^{3}+x-1)$ and $\mathbb {Z} [x,y]/(f)\otimes _{\mathbb {Z} }\mathbb {C} \cong \mathbb {C} [x,y]/(f)$.
• Tensor products also can be used for taking products of affine schemes over a field. For example, $\mathbb {C} [x_{1},x_{2}]/(f(x))\otimes _{\mathbb {C} }\mathbb {C} [y_{1},y_{2}]/(g(y))$ is isomorphic to the algebra $\mathbb {C} [x_{1},x_{2},y_{1},y_{2}]/(f(x),g(y))$ which corresponds to an affine surface in $\mathbb {A} _{\mathbb {C} }^{4}$ if f and g are not zero.
See also
• Extension of scalars
• Tensor product of modules
• Tensor product of fields
• Linearly disjoint
• Multilinear subspace learning
Notes
1. Kassel (1995), p. 32.
2. Lang 2002, pp. 629–630.
3. Kassel (1995), p. 32.
4. Kassel (1995), p. 32.
References
• Kassel, Christian (1995), Quantum groups, Graduate texts in mathematics, vol. 155, Springer, ISBN 978-0-387-94370-1.
• Lang, Serge (2002) [first published in 1993]. Algebra. Graduate Texts in Mathematics. Vol. 21. Springer. ISBN 0-387-95385-X.
| Wikipedia |
Tensor product bundle
In differential geometry, the tensor product of vector bundles E, F (over same space $X$) is a vector bundle, denoted by E ⊗ F, whose fiber over a point $x\in X$ is the tensor product of vector spaces Ex ⊗ Fx.[1]
Not to be confused with a tensor bundle, a vector bundle whose section is a tensor field.
Example: If O is a trivial line bundle, then E ⊗ O = E for any E.
Example: E ⊗ E ∗ is canonically isomorphic to the endomorphism bundle End(E), where E ∗ is the dual bundle of E.
Example: A line bundle L has tensor inverse: in fact, L ⊗ L ∗ is (isomorphic to) a trivial bundle by the previous example, as End(L) is trivial. Thus, the set of the isomorphism classes of all line bundles on some topological space X forms an abelian group called the Picard group of X.
Variants
One can also define a symmetric power and an exterior power of a vector bundle in a similar way. For example, a section of $\Lambda ^{p}T^{*}M$ is a differential p-form and a section of $\Lambda ^{p}T^{*}M\otimes E$ is a differential p-form with values in a vector bundle E.
See also
• Tensor product of modules
Notes
1. To construct a tensor-product bundle over a paracompact base, first note the construction is clear for trivial bundles. For the general case, if the base is compact, choose E' such that E ⊕ E' is trivial. Choose F' in the same way. Then let E ⊗ F be the subbundle of (E ⊕ E') ⊗ (F ⊕ F') with the desired fibers. Finally, use the approximation argument to handle a non-compact base. See Hatcher for a general direct approach.
References
• Hatcher, Vector Bundles and K-Theory
| Wikipedia |
Tensor product of quadratic forms
In mathematics, the tensor product of quadratic forms is most easily understood when one views the quadratic forms as quadratic spaces. If R is a commutative ring where 2 is invertible (that is, R has characteristic ${\text{char}}(R)\neq 2$), and if $(V_{1},q_{1})$ and $(V_{2},q_{2})$ are two quadratic spaces over R, then their tensor product $(V_{1}\otimes V_{2},q_{1}\otimes q_{2})$ is the quadratic space whose underlying R-module is the tensor product $V_{1}\otimes V_{2}$ of R-modules and whose quadratic form is the quadratic form associated to the tensor product of the bilinear forms associated to $q_{1}$ and $q_{2}$.
In particular, the form $q_{1}\otimes q_{2}$ satisfies
$(q_{1}\otimes q_{2})(v_{1}\otimes v_{2})=q_{1}(v_{1})q_{2}(v_{2})\quad \forall v_{1}\in V_{1},\ v_{2}\in V_{2}$
(which does uniquely characterize it however). It follows from this that if the quadratic forms are diagonalizable (which is always possible if 2 is invertible in R), i.e.,
$q_{1}\cong \langle a_{1},...,a_{n}\rangle $
$q_{2}\cong \langle b_{1},...,b_{m}\rangle $
then the tensor product has diagonalization
$q_{1}\otimes q_{2}\cong \langle a_{1}b_{1},a_{1}b_{2},...a_{1}b_{m},a_{2}b_{1},...,a_{2}b_{m},...,a_{n}b_{1},...a_{n}b_{m}\rangle .$
| Wikipedia |
Sheaf of modules
In mathematics, a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf F such that, for any open subset U of X, F(U) is an O(U)-module and the restriction maps F(U) → F(V) are compatible with the restriction maps O(U) → O(V): the restriction of fs is the restriction of f times that of s for any f in O(U) and s in F(U).
The standard case is when X is a scheme and O its structure sheaf. If O is the constant sheaf ${\underline {\mathbf {Z} }}$, then a sheaf of O-modules is the same as a sheaf of abelian groups (i.e., an abelian sheaf).
If X is the prime spectrum of a ring R, then any R-module defines an OX-module (called an associated sheaf) in a natural way. Similarly, if R is a graded ring and X is the Proj of R, then any graded module defines an OX-module in a natural way. O-modules arising in such a fashion are examples of quasi-coherent sheaves, and in fact, on affine or projective schemes, all quasi-coherent sheaves are obtained this way.
Sheaves of modules over a ringed space form an abelian category.[1] Moreover, this category has enough injectives,[2] and consequently one can and does define the sheaf cohomology $\operatorname {H} ^{i}(X,-)$ as the i-th right derived functor of the global section functor $\Gamma (X,-)$.[3]
Examples
• Given a ringed space (X, O), if F is an O-submodule of O, then it is called the sheaf of ideals or ideal sheaf of O, since for each open subset U of X, F(U) is an ideal of the ring O(U).
• Let X be a smooth variety of dimension n. Then the tangent sheaf of X is the dual of the cotangent sheaf $\Omega _{X}$ and the canonical sheaf $\omega _{X}$ is the n-th exterior power (determinant) of $\Omega _{X}$.
• A sheaf of algebras is a sheaf of module that is also a sheaf of rings.
Operations
Let (X, O) be a ringed space. If F and G are O-modules, then their tensor product, denoted by
$F\otimes _{O}G$ or $F\otimes G$,
is the O-module that is the sheaf associated to the presheaf $U\mapsto F(U)\otimes _{O(U)}G(U).$ (To see that sheafification cannot be avoided, compute the global sections of $O(1)\otimes O(-1)=O$ where O(1) is Serre's twisting sheaf on a projective space.)
Similarly, if F and G are O-modules, then
${\mathcal {H}}om_{O}(F,G)$
denotes the O-module that is the sheaf $U\mapsto \operatorname {Hom} _{O|_{U}}(F|_{U},G|_{U})$.[4] In particular, the O-module
${\mathcal {H}}om_{O}(F,O)$
is called the dual module of F and is denoted by ${\check {F}}$. Note: for any O-modules E, F, there is a canonical homomorphism
${\check {E}}\otimes F\to {\mathcal {H}}om_{O}(E,F)$,
which is an isomorphism if E is a locally free sheaf of finite rank. In particular, if L is locally free of rank one (such L is called an invertible sheaf or a line bundle),[5] then this reads:
${\check {L}}\otimes L\simeq O,$
implying the isomorphism classes of invertible sheaves form a group. This group is called the Picard group of X and is canonically identified with the first cohomology group $\operatorname {H} ^{1}(X,{\mathcal {O}}^{*})$ (by the standard argument with Čech cohomology).
If E is a locally free sheaf of finite rank, then there is an O-linear map ${\check {E}}\otimes E\simeq \operatorname {End} _{O}(E)\to O$ given by the pairing; it is called the trace map of E.
For any O-module F, the tensor algebra, exterior algebra and symmetric algebra of F are defined in the same way. For example, the k-th exterior power
$\bigwedge ^{k}F$
is the sheaf associated to the presheaf $ U\mapsto \bigwedge _{O(U)}^{k}F(U)$. If F is locally free of rank n, then $ \bigwedge ^{n}F$ is called the determinant line bundle (though technically invertible sheaf) of F, denoted by det(F). There is a natural perfect pairing:
$\bigwedge ^{r}F\otimes \bigwedge ^{n-r}F\to \det(F).$
Let f: (X, O) →(X', O') be a morphism of ringed spaces. If F is an O-module, then the direct image sheaf $f_{*}F$ is an O'-module through the natural map O' →f*O (such a natural map is part of the data of a morphism of ringed spaces.)
If G is an O'-module, then the module inverse image $f^{*}G$ of G is the O-module given as the tensor product of modules:
$f^{-1}G\otimes _{f^{-1}O'}O$
where $f^{-1}G$ is the inverse image sheaf of G and $f^{-1}O'\to O$ is obtained from $O'\to f_{*}O$ by adjuction.
There is an adjoint relation between $f_{*}$ and $f^{*}$: for any O-module F and O'-module G,
$\operatorname {Hom} _{O}(f^{*}G,F)\simeq \operatorname {Hom} _{O'}(G,f_{*}F)$
as abelian group. There is also the projection formula: for an O-module F and a locally free O'-module E of finite rank,
$f_{*}(F\otimes f^{*}E)\simeq f_{*}F\otimes E.$
Properties
Let (X, O) be a ringed space. An O-module F is said to be generated by global sections if there is a surjection of O-modules:
$\bigoplus _{i\in I}O\to F\to 0.$
Explicitly, this means that there are global sections si of F such that the images of si in each stalk Fx generates Fx as Ox-module.
An example of such a sheaf is that associated in algebraic geometry to an R-module M, R being any commutative ring, on the spectrum of a ring Spec(R). Another example: according to Cartan's theorem A, any coherent sheaf on a Stein manifold is spanned by global sections. (cf. Serre's theorem A below.) In the theory of schemes, a related notion is ample line bundle. (For example, if L is an ample line bundle, some power of it is generated by global sections.)
An injective O-module is flasque (i.e., all restrictions maps F(U) → F(V) are surjective.)[6] Since a flasque sheaf is acyclic in the category of abelian sheaves, this implies that the i-th right derived functor of the global section functor $\Gamma (X,-)$ in the category of O-modules coincides with the usual i-th sheaf cohomology in the category of abelian sheaves.[7]
Sheaf associated to a module
Let $M$ be a module over a ring $A$. Put $X=\operatorname {Spec} (A)$ and write $D(f)=\{f\neq 0\}=\operatorname {Spec} (A[f^{-1}])$. For each pair $D(f)\subseteq D(g)$, by the universal property of localization, there is a natural map
$\rho _{g,f}:M[g^{-1}]\to M[f^{-1}]$
having the property that $\rho _{g,f}=\rho _{g,h}\circ \rho _{h,f}$. Then
$D(f)\mapsto M[f^{-1}]$
is a contravariant functor from the category whose objects are the sets D(f) and morphisms the inclusions of sets to the category of abelian groups. One can show[8] it is in fact a B-sheaf (i.e., it satisfies the gluing axiom) and thus defines the sheaf ${\widetilde {M}}$ on X called the sheaf associated to M.
The most basic example is the structure sheaf on X; i.e., ${\mathcal {O}}_{X}={\widetilde {A}}$. Moreover, ${\widetilde {M}}$ has the structure of ${\mathcal {O}}_{X}={\widetilde {A}}$-module and thus one gets the exact functor $M\mapsto {\widetilde {M}}$ from ModA, the category of modules over A to the category of modules over ${\mathcal {O}}_{X}$. It defines an equivalence from ModA to the category of quasi-coherent sheaves on X, with the inverse $\Gamma (X,-)$, the global section functor. When X is Noetherian, the functor is an equivalence from the category of finitely generated A-modules to the category of coherent sheaves on X.
The construction has the following properties: for any A-modules M, N,
• $M[f^{-1}]^{\sim }={\widetilde {M}}|_{D(f)}$.[9]
• For any prime ideal p of A, ${\widetilde {M}}_{p}\simeq M_{p}$ as Op = Ap-module.
• $(M\otimes _{A}N)^{\sim }\simeq {\widetilde {M}}\otimes _{\widetilde {A}}{\widetilde {N}}$.[10]
• If M is finitely presented, $\operatorname {Hom} _{A}(M,N)^{\sim }\simeq {\mathcal {H}}om_{\widetilde {A}}({\widetilde {M}},{\widetilde {N}})$.[10]
• $\operatorname {Hom} _{A}(M,N)\simeq \Gamma (X,{\mathcal {H}}om_{\widetilde {A}}({\widetilde {M}},{\widetilde {N}}))$, since the equivalence between ModA and the category of quasi-coherent sheaves on X.
• $(\varinjlim M_{i})^{\sim }\simeq \varinjlim {\widetilde {M_{i}}}$;[11] in particular, taking a direct sum and ~ commute.
Sheaf associated to a graded module
There is a graded analog of the construction and equivalence in the preceding section. Let R be a graded ring generated by degree-one elements as R0-algebra (R0 means the degree-zero piece) and M a graded R-module. Let X be the Proj of R (so X is a projective scheme if R is Noetherian). Then there is an O-module ${\widetilde {M}}$ such that for any homogeneous element f of positive degree of R, there is a natural isomorphism
${\widetilde {M}}|_{\{f\neq 0\}}\simeq (M[f^{-1}]_{0})^{\sim }$
as sheaves of modules on the affine scheme $\{f\neq 0\}=\operatorname {Spec} (R[f^{-1}]_{0})$;[12] in fact, this defines ${\widetilde {M}}$ by gluing.
Example: Let R(1) be the graded R-module given by R(1)n = Rn+1. Then $O(1)={\widetilde {R(1)}}$ is called Serre's twisting sheaf, which is the dual of the tautological line bundle if R is finitely generated in degree-one.
If F is an O-module on X, then, writing $F(n)=F\otimes O(n)$, there is a canonical homomorphism:
$\left(\bigoplus _{n\geq 0}\Gamma (X,F(n))\right)^{\sim }\to F,$,
which is an isomorphism if and only if F is quasi-coherent.
Computing sheaf cohomology
Main article: sheaf cohomology
Sheaf cohomology has a reputation for being difficult to calculate. Because of this, the next general fact is fundamental for any practical computation:
Theorem — Let X be a topological space, F an abelian sheaf on it and ${\mathfrak {U}}$ an open cover of X such that $\operatorname {H} ^{i}(U_{i_{0}}\cap \cdots \cap U_{i_{p}},F)=0$ for any i, p and $U_{i_{j}}$'s in ${\mathfrak {U}}$. Then for any i,
$\operatorname {H} ^{i}(X,F)=\operatorname {H} ^{i}(C^{\bullet }({\mathfrak {U}},F))$
where the right-hand side is the i-th Čech cohomology.
Serre's theorem A states that if X is a projective variety and F a coherent sheaf on it, then, for sufficiently large n, F(n) is generated by finitely many global sections. Moreover,
1. For each i, Hi(X, F) is finitely generated over R0, and
2. (Serre's theorem B) There is an integer n0, depending on F, such that
$\operatorname {H} ^{i}(X,F(n))=0,\,i\geq 1,n\geq n_{0}.$
[13][14][15]
Sheaf extension
Let (X, O) be a ringed space, and let F, H be sheaves of O-modules on X. An extension of H by F is a short exact sequence of O-modules
$0\rightarrow F\rightarrow G\rightarrow H\rightarrow 0.$
As with group extensions, if we fix F and H, then all equivalence classes of extensions of H by F form an abelian group (cf. Baer sum), which is isomorphic to the Ext group $\operatorname {Ext} _{O}^{1}(H,F)$, where the identity element in $\operatorname {Ext} _{O}^{1}(H,F)$ corresponds to the trivial extension.
In the case where H is O, we have: for any i ≥ 0,
$\operatorname {H} ^{i}(X,F)=\operatorname {Ext} _{O}^{i}(O,F),$
since both the sides are the right derived functors of the same functor $\Gamma (X,-)=\operatorname {Hom} _{O}(O,-).$
Note: Some authors, notably Hartshorne, drop the subscript O.
Assume X is a projective scheme over a Noetherian ring. Let F, G be coherent sheaves on X and i an integer. Then there exists n0 such that
$\operatorname {Ext} _{O}^{i}(F,G(n))=\Gamma (X,{\mathcal {E}}xt_{O}^{i}(F,G(n))),\,n\geq n_{0}$.[16]
See also: local-to-global Ext spectral sequence
Locally free resolutions
${\mathcal {Ext}}({\mathcal {F}},{\mathcal {G}})$ can be readily computed for any coherent sheaf ${\mathcal {F}}$ using a locally free resolution:[17] given a complex
$\cdots \to {\mathcal {L}}_{2}\to {\mathcal {L}}_{1}\to {\mathcal {L}}_{0}\to {\mathcal {F}}\to 0$
then
${\mathcal {RHom}}({\mathcal {F}},{\mathcal {G}})={\mathcal {Hom}}({\mathcal {L}}_{\bullet },{\mathcal {G}})$
hence
${\mathcal {Ext}}^{k}({\mathcal {F}},{\mathcal {G}})=h^{k}({\mathcal {Hom}}({\mathcal {L}}_{\bullet },{\mathcal {G}}))$
Hypersurface
Consider a smooth hypersurface $X$ of degree $d$. Then, we can compute a resolution
${\mathcal {O}}(-d)\to {\mathcal {O}}$
and find that
${\mathcal {Ext}}^{i}({\mathcal {O}}_{X},{\mathcal {F}})=h^{i}({\mathcal {Hom}}({\mathcal {O}}(-d)\to {\mathcal {O}},{\mathcal {F}}))$
Union of smooth complete intersections
Consider the scheme
$X={\text{Proj}}\left({\frac {\mathbb {C} [x_{0},\ldots ,x_{n}]}{(f)(g_{1},g_{2},g_{3})}}\right)\subseteq \mathbb {P} ^{n}$
where $(f,g_{1},g_{2},g_{3})$ is a smooth complete intersection and $\deg(f)=d$, $\deg(g_{i})=e_{i}$. We have a complex
${\mathcal {O}}(-d-e_{1}-e_{2}-e_{3}){\xrightarrow {\begin{bmatrix}g_{3}\\-g_{2}\\-g_{1}\end{bmatrix}}}{\begin{matrix}{\mathcal {O}}(-d-e_{1}-e_{2})\\\oplus \\{\mathcal {O}}(-d-e_{1}-e_{3})\\\oplus \\{\mathcal {O}}(-d-e_{2}-e_{3})\end{matrix}}{\xrightarrow {\begin{bmatrix}g_{2}&g_{3}&0\\-g_{1}&0&-g_{3}\\0&-g_{1}&g_{2}\end{bmatrix}}}{\begin{matrix}{\mathcal {O}}(-d-e_{1})\\\oplus \\{\mathcal {O}}(-d-e_{2})\\\oplus \\{\mathcal {O}}(-d-e_{3})\end{matrix}}{\xrightarrow {\begin{bmatrix}fg_{1}&fg_{2}&fg_{3}\end{bmatrix}}}{\mathcal {O}}$
resolving ${\mathcal {O}}_{X},$ which we can use to compute ${\mathcal {Ext}}^{i}({\mathcal {O}}_{X},{\mathcal {F}})$.
See also
• D-module (in place of O, one can also consider D, the sheaf of differential operators.)
• fractional ideal
• holomorphic vector bundle
• generic freeness
Notes
1. Vakil, Math 216: Foundations of algebraic geometry, 2.5.
2. Hartshorne, Ch. III, Proposition 2.2.
3. This cohomology functor coincides with the right derived functor of the global section functor in the category of abelian sheaves; cf. Hartshorne, Ch. III, Proposition 2.6.
4. There is a canonical homomorphism:
${\mathcal {H}}om_{O}(F,O)_{x}\to \operatorname {Hom} _{O_{x}}(F_{x},O_{x}),$
which is an isomorphism if F is of finite presentation (EGA, Ch. 0, 5.2.6.)
5. For coherent sheaves, having a tensor inverse is the same as being locally free of rank one; in fact, there is the following fact: if $F\otimes G\simeq O$ and if F is coherent, then F, G are locally free of rank one. (cf. EGA, Ch 0, 5.4.3.)
6. Hartshorne, Ch III, Lemma 2.4.
7. see also: https://math.stackexchange.com/q/447234
8. Hartshorne, Ch. II, Proposition 5.1.
9. EGA I, Ch. I, Proposition 1.3.6. harvnb error: no target: CITEREFEGA_I (help)
10. EGA I, Ch. I, Corollaire 1.3.12. harvnb error: no target: CITEREFEGA_I (help)
11. EGA I, Ch. I, Corollaire 1.3.9. harvnb error: no target: CITEREFEGA_I (help)
12. Hartshorne, Ch. II, Proposition 5.11.
13. Costa, Miró-Roig & Pons-Llopis 2021, Theorem 1.3.1
14. "Links with sheaf cohomology". Local Cohomology. 2012. pp. 438–479. doi:10.1017/CBO9781139044059.023. ISBN 9780521513630.
15. Serre 1955, §.66 Faisceaux algébriques cohérents sur les variétés projectives.
16. Hartshorne, Ch. III, Proposition 6.9.
17. Hartshorne, Robin. Algebraic Geometry. pp. 233–235.
References
• Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS. 4. doi:10.1007/bf02684778. MR 0217083.
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
• Costa, Laura; Miró-Roig, Rosa María; Pons-Llopis, Joan (2021). Ulrich Bundles. doi:10.1515/9783110647686. ISBN 9783110647686.
• "Links with sheaf cohomology". Local Cohomology. 2012. pp. 438–479. doi:10.1017/CBO9781139044059.023. ISBN 9780521513630.
• Serre, Jean-Pierre (1955), "Faisceaux algébriques cohérents (§.66 Faisceaux algébriques cohérents sur les variétés projectives.)" (PDF), Annals of Mathematics, 61 (2): 197–278, doi:10.2307/1969915, JSTOR 1969915, MR 0068874
| Wikipedia |
Tensor product
In mathematics, the tensor product $V\otimes W$ of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map $V\times W\to V\otimes W$ that maps a pair $(v,w),\ v\in V,w\in W$ to an element of $V\otimes W$ denoted $v\otimes w.$
For generalizations of this concept, see Tensor product of modules and Tensor product (disambiguation).
An element of the form $v\otimes w$ is called the tensor product of v and w. An element of $V\otimes W$ is a tensor, and the tensor product of two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span $V\otimes W$ in the sense that every element of $V\otimes W$ is a sum of elementary tensors. If bases are given for V and W, a basis of $V\otimes W$ is formed by all tensor products of a basis element of V and a basis element of W.
The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from $V\times W$ into another vector space Z factors uniquely through a linear map $V\otimes W\to Z$ (see Universal property).
Tensor products are used in many application areas, including physics and engineering. For example, in general relativity, the gravitational field is described through the metric tensor, which is a vector field of tensors, one at each point of the space-time manifold, and each belonging to the tensor product with itself of the cotangent space at the point.
Definitions and constructions
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism. There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
The tensor product can also be defined through a universal property; see § Universal property, below. As for every universal property, all objects that satisfy the property are isomorphic through a unique isomorphism that is compatible with the universal property. When this definition is used, the other definitions may be viewed as constructions of objects satisfying the universal property and as proofs that there are objects satisfying the universal property, that is that tensor products exist.
From bases
Let V and W be two vector spaces over a field F, with respective bases $B_{V}$ and $B_{W}.$
The tensor product $V\otimes W$ of V and W is a vector space which has as a basis the set of all $v\otimes w$ with $v\in B_{V}$ and $w\in B_{W}.$ This definition can be formalized in the following way (this formalization is rarely used in practice, as the preceding informal definition is generally sufficient): $V\otimes W$ is the set of the functions from the Cartesian product $B_{V}\times B_{W}$ to F that have a finite number of nonzero values. The pointwise operations make $V\otimes W$ a vector space. The function that maps $(v,w)$ to 1 and the other elements of $B_{V}\times B_{W}$ to 0 is denoted $v\otimes w.$
The set $\{v\otimes w\mid v\in B_{V},w\in B_{W}\}$ is straightforwardly a basis of $V\otimes W,$ which is called the tensor product of the bases $B_{V}$ and $B_{W}.$
The tensor product of two vectors is defined from their decomposition on the bases. More precisely, if
$x=\sum _{b\in B_{V}}x_{b}\,b\in V\quad {\text{and}}\quad y=\sum _{c\in B_{W}}y_{c}\,c\in W$
are vectors decomposed on their respective bases, then the tensor product of x and y is
${\begin{aligned}x\otimes y&=\left(\sum _{b\in B_{V}}x_{b}\,b\right)\otimes \left(\sum _{c\in B_{W}}y_{c}\,c\right)\\&=\sum _{b\in B_{V}}\sum _{c\in B_{W}}x_{b}y_{c}\,b\otimes c.\end{aligned}}$
If arranged into a rectangular array, the coordinate vector of $x\otimes y$ is the outer product of the coordinate vectors of x and y. Therefore, the tensor product is a generalization of the outer product.
It is straightforward to verify that the map $(x,y)\mapsto x\otimes y$ is a bilinear map from $V\times W$ to $V\otimes W.$
A limitation of this definition of the tensor product is that, if one changes bases, a different tensor product is defined. However, the decomposition on one basis of the elements of the other basis defines a canonical isomorphism between the two tensor products of vector spaces, which allows identifying them. Also, contrarily to the two following alternative definitions, this definition cannot be extended into a definition of the tensor product of modules over a ring.
As a quotient space
A construction of the tensor product that is basis independent can be obtained in the following way.
Let V and W be two vector spaces over a field F.
One considers first a vector space L that has the Cartesian product $V\times W$ as a basis. That is, the basis elements of L are the pairs $(v,w)$ with $v\in V$ and $w\in W.$ To get such a vector space, one can define it as the vector space of the functions $V\times W\to F$ that have a finite number of nonzero values, and identifying $(v,w)$ with the function that takes the value 1 on $(v,w)$ and 0 otherwise.
Let R be the linear subspace of L that is spanned by the relations that the tensor product must satisfy. More precisely R is spanned by the elements of one of the forms
${\begin{aligned}(v_{1}+v_{2},w)&-(v_{1},w)-(v_{2},w),\\(v,w_{1}+w_{2})&-(v,w_{1})-(v,w_{2}),\\(sv,w)&-s(v,w),\\(v,sw)&-s(v,w),\end{aligned}}$
where $v,v_{1},v_{2}\in V,$ $w,w_{1},w_{2}\in W$ and $s\in F.$
Then, the tensor product is defined as the quotient space
$V\otimes W=L/R,$
and the image of $(v,w)$ in this quotient is denoted $v\otimes w.$
It is straightforward to prove that the result of this construction satisfies the universal property considered below. (A very similar construction can be used to define the tensor product of modules.)
Universal property
In this section, the universal property satisfied by the tensor product is described. As for every universal property, two objects that satisfy the property are related by a unique isomorphism. It follows that this is a (non-constructive) way to define the tensor product of two vector spaces. In this context, the preceding constructions of tensor products may be viewed as proofs of existence of the tensor product so defined.
A consequence of this approach is that every property of the tensor product can be deduced from the universal property, and that, in practice, one may forget the method that has been used to prove its existence.
The "universal-property definition" of the tensor product of two vector spaces is the following (recall that a bilinear map is a function that is separately linear in each of its arguments):
The tensor product of two vector spaces V and W is a vector space denoted as $V\otimes W,$ together with a bilinear map ${\otimes }\colon (v,w)\mapsto v\otimes w$ from $V\times W$ to $V\otimes W,$ such that, for every bilinear map $h\colon V\times W\to Z,$ there is a unique linear map ${\tilde {h}}\colon V\otimes W\to Z,$ such that $h={\tilde {h}}\circ {\otimes }$ (that is, $h(v,w)={\tilde {h}}(v\otimes w)$ for every $v\in V$ and $w\in W$).
Linearly disjoint
Like the universal property above, the following characterization may also be used to determine whether or not a given vector space and given bilinear map form a tensor product.[1]
Theorem — Let $X,Y,$ and $Z$ be complex vector spaces and let $T:X\times Y\to Z$ be a bilinear map. Then $(Z,T)$ is a tensor product of $X$ and $Y$ if and only if[1] the image of $T$ spans all of $Z$ (that is, $\operatorname {span} \;T(X\times Y)=Z$), and also $X$ and $Y$ are $T$-linearly disjoint, which by definition means that for all positive integers $n$ and all elements $x_{1},\ldots ,x_{n}\in X$ and $y_{1},\ldots ,y_{n}\in Y$ such that $\sum _{i=1}^{n}T\left(x_{i},y_{i}\right)=0,$
1. if all $x_{1},\ldots ,x_{n}$ are linearly independent then all $y_{i}$ are $0,$ and
2. if all $y_{1},\ldots ,y_{n}$ are linearly independent then all $x_{i}$ are $0.$
Equivalently, $X$ and $Y$ are $T$-linearly disjoint if and only if for all linearly independent sequences $x_{1},\ldots ,x_{m}$ in $X$ and all linearly independent sequences $y_{1},\ldots ,y_{n}$ in $Y,$ the vectors $\left\{T\left(x_{i},y_{j}\right):1\leq i\leq m,1\leq j\leq n\right\}$ are linearly independent.
For example, it follows immediately that if $m$ and $n$ are positive integers then $Z:=\mathbb {C} ^{mn}$ and the bilinear map $T:\mathbb {C} ^{m}\times \mathbb {C} ^{n}\to \mathbb {C} ^{mn}$ defined by sending $(x,y)=\left(\left(x_{1},\ldots ,x_{m}\right),\left(y_{1},\ldots ,y_{n}\right)\right)$ to $\left(x_{i}y_{j}\right)_{\stackrel {i=1,\ldots ,m}{j=1,\ldots ,n}}$ form a tensor product of $X:=\mathbb {C} ^{m}$ and $Y:=\mathbb {C} ^{n}.$[2] Often, this map $T$ will be denoted by $\,\otimes \,$ so that $x\otimes y\;:=\;T(x,y)$ denotes this bilinear map's value at $(x,y)\in X\times Y.$
As another example, suppose that $\mathbb {C} ^{S}$ is the vector space of all complex-valued functions on a set $S$ with addition and scalar multiplication defined pointwise (meaning that $f+g$ is the map $s\mapsto f(s)+g(s)$ and $cf$ is the map $s\mapsto cf(s)$). Let $S$ and $T$ be any sets and for any $f\in \mathbb {C} ^{S}$ and $g\in \mathbb {C} ^{T},$ let $f\otimes g\in \mathbb {C} ^{S\times T}$ denote the function defined by $(s,t)\mapsto f(s)g(t).$ If $X\subseteq \mathbb {C} ^{S}$ and $Y\subseteq \mathbb {C} ^{T}$ are vector subspaces then the vector subspace $Z:=\operatorname {span} \left\{f\otimes g:f\in X,g\in Y\right\}$ of $\mathbb {C} ^{S\times T}$ together with the bilinear map
${\begin{alignedat}{4}\;&&X\times Y&&\;\to \;&Z\\[0.3ex]&&(f,g)&&\;\mapsto \;&f\otimes g\\\end{alignedat}}$
form a tensor product of $X$ and $Y.$[2]
Properties
Dimension
If V and W are vectors spaces of finite dimension, then $V\otimes W$ is finite-dimensional, and its dimension is the product of the dimensions of V and W.
This results from the fact that a basis of $V\otimes W$ is formed by taking all tensor products of a basis element of V and a basis element of W.
Associativity
The tensor product is associative in the sense that, given three vector spaces $U,V,W,$ there is a canonical isomorphism
$(U\otimes V)\otimes W\cong U\otimes (V\otimes W),$
that maps $(u\otimes v)\otimes w$ to $u\otimes (v\otimes w).$
This allows omitting parentheses in the tensor product of more than two vector spaces or vectors.
Commutativity as vector space operation
The tensor product of two vector spaces $V$ and $W$ is commutative in the sense that there is a canonical isomorphism
$V\otimes W\cong W\otimes V,$
that maps $v\otimes w$ to $w\otimes v.$
On the other hand, even when $V=W,$ the tensor product of vectors is not commutative; that is $v\otimes w\neq w\otimes v,$ in general.
The map $x\otimes y\mapsto y\otimes x$ from $V\otimes V$ to itself induces a linear automorphism that is called a braiding map. More generally and as usual (see tensor algebra), let denote $V^{\otimes n}$ the tensor product of n copies of the vector space V. For every permutation s of the first n positive integers, the map
$x_{1}\otimes \cdots \otimes x_{n}\mapsto x_{s(1)}\otimes \cdots \otimes x_{s(n)}$
induces a linear automorphism of $V^{\otimes n}\to V^{\otimes n},$ which is called a braiding map.
Tensor product of linear maps
"Tensor product of linear maps" redirects here. For the generalization for modules, see Tensor product of modules § Tensor product of linear maps and a change of base ring.
Given a linear map $f\colon U\to V,$ and a vector space W, the tensor product
$f\otimes W\colon U\otimes W\to V\otimes W$
is the unique linear map such that
$(f\otimes W)(u\otimes w)=f(u)\otimes w.$
The tensor product $W\otimes f$ is defined similarly.
Given two linear maps $f\colon U\to V$ and $g\colon W\to Z,$ their tensor product
$f\otimes g\colon U\otimes W\to V\otimes Z$
is the unique linear map that satisfies
$(f\otimes g)(u\otimes w)=f(u)\otimes g(w).$
One has
$f\otimes g=(f\otimes Z)\circ (U\otimes g)=(V\otimes g)\circ (f\otimes W).$
In terms of category theory, this means that the tensor product is a bifunctor from the category of vector spaces to itself.[3]
If f and g are both injective or surjective, then the same is true for all above defined linear maps. In particular, the tensor product with a vector space is an exact functor; this means that every exact sequence is mapped to an exact sequence (tensor products of modules do not transform injections into injections, but they are right exact functors).
By choosing bases of all vector spaces involved, the linear maps S and T can be represented by matrices. Then, depending on how the tensor $v\otimes w$ is vectorized, the matrix describing the tensor product $S\otimes T$ is the Kronecker product of the two matrices. For example, if V, X, W, and Y above are all two-dimensional and bases have been fixed for all of them, and S and T are given by the matrices
$A={\begin{bmatrix}a_{1,1}&a_{1,2}\\a_{2,1}&a_{2,2}\\\end{bmatrix}},\qquad B={\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\\end{bmatrix}},$
respectively, then the tensor product of these two matrices is
${\begin{bmatrix}a_{1,1}&a_{1,2}\\a_{2,1}&a_{2,2}\\\end{bmatrix}}\otimes {\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\\end{bmatrix}}={\begin{bmatrix}a_{1,1}{\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\\end{bmatrix}}&a_{1,2}{\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\\end{bmatrix}}\\[3pt]a_{2,1}{\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\\end{bmatrix}}&a_{2,2}{\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\\end{bmatrix}}\\\end{bmatrix}}={\begin{bmatrix}a_{1,1}b_{1,1}&a_{1,1}b_{1,2}&a_{1,2}b_{1,1}&a_{1,2}b_{1,2}\\a_{1,1}b_{2,1}&a_{1,1}b_{2,2}&a_{1,2}b_{2,1}&a_{1,2}b_{2,2}\\a_{2,1}b_{1,1}&a_{2,1}b_{1,2}&a_{2,2}b_{1,1}&a_{2,2}b_{1,2}\\a_{2,1}b_{2,1}&a_{2,1}b_{2,2}&a_{2,2}b_{2,1}&a_{2,2}b_{2,2}\\\end{bmatrix}}.$
The resultant rank is at most 4, and thus the resultant dimension is 4. Note that rank here denotes the tensor rank i.e. the number of requisite indices (while the matrix rank counts the number of degrees of freedom in the resulting array). Note $\operatorname {Tr} A\otimes B=\operatorname {Tr} A\times \operatorname {Tr} B.$
A dyadic product is the special case of the tensor product between two vectors of the same dimension.
General tensors
See also: Tensor
For non-negative integers r and s a type $(r,s)$ tensor on a vector space V is an element of
$T_{s}^{r}(V)=\underbrace {V\otimes \cdots \otimes V} _{r}\otimes \underbrace {V^{*}\otimes \cdots \otimes V^{*}} _{s}=V^{\otimes r}\otimes \left(V^{*}\right)^{\otimes s}.$
Here $V^{*}$ is the dual vector space (which consists of all linear maps f from V to the ground field K).
There is a product map, called the (tensor) product of tensors[4]
$T_{s}^{r}(V)\otimes _{K}T_{s'}^{r'}(V)\to T_{s+s'}^{r+r'}(V).$
It is defined by grouping all occurring "factors" V together: writing $v_{i}$ for an element of V and $f_{i}$ for an element of the dual space,
$(v_{1}\otimes f_{1})\otimes (v'_{1})=v_{1}\otimes v'_{1}\otimes f_{1}.$
Picking a basis of V and the corresponding dual basis of $V^{*}$ naturally induces a basis for $T_{s}^{r}(V)$ (this basis is described in the article on Kronecker products). In terms of these bases, the components of a (tensor) product of two (or more) tensors can be computed. For example, if F and G are two covariant tensors of orders m and n respectively (i.e. $F\in T_{m}^{0}$ and $G\in T_{n}^{0}$), then the components of their tensor product are given by[5]
$(F\otimes G)_{i_{1}i_{2}\cdots i_{m+n}}=F_{i_{1}i_{2}\cdots i_{m}}G_{i_{m+1}i_{m+2}i_{m+3}\cdots i_{m+n}}.$
Thus, the components of the tensor product of two tensors are the ordinary product of the components of each tensor. Another example: let U be a tensor of type (1, 1) with components $U_{\beta }^{\alpha },$ and let V be a tensor of type $(1,0)$ with components $V^{\gamma }.$ Then
$\left(U\otimes V\right)^{\alpha }{}_{\beta }{}^{\gamma }=U^{\alpha }{}_{\beta }V^{\gamma }$
and
$(V\otimes U)^{\mu \nu }{}_{\sigma }=V^{\mu }U^{\nu }{}_{\sigma }.$
Tensors equipped with their product operation form an algebra, called the tensor algebra.
Evaluation map and tensor contraction
For tensors of type (1, 1) there is a canonical evaluation map
$V\otimes V^{*}\to K$
defined by its action on pure tensors:
$v\otimes f\mapsto f(v).$
More generally, for tensors of type $(r,s),$ with r, s > 0, there is a map, called tensor contraction,
$T_{s}^{r}(V)\to T_{s-1}^{r-1}(V).$
(The copies of $V$ and $V^{*}$ on which this map is to be applied must be specified.)
On the other hand, if $V$ is finite-dimensional, there is a canonical map in the other direction (called the coevaluation map)
${\begin{cases}K\to V\otimes V^{*}\\\lambda \mapsto \sum _{i}\lambda v_{i}\otimes v_{i}^{*}\end{cases}}$
where $v_{1},\ldots ,v_{n}$ is any basis of $V,$ and $v_{i}^{*}$ is its dual basis. This map does not depend on the choice of basis.[6]
The interplay of evaluation and coevaluation can be used to characterize finite-dimensional vector spaces without referring to bases.[7]
Adjoint representation
The tensor product $T_{s}^{r}(V)$ may be naturally viewed as a module for the Lie algebra $\mathrm {End} (V)$ by means of the diagonal action: for simplicity let us assume $r=s=1,$ then, for each $u\in \mathrm {End} (V),$
$u(a\otimes b)=u(a)\otimes b-a\otimes u^{*}(b),$
where $u^{*}\in \mathrm {End} \left(V^{*}\right)$ is the transpose of u, that is, in terms of the obvious pairing on $V\otimes V^{*},$
$\langle u(a),b\rangle =\langle a,u^{*}(b)\rangle .$
There is a canonical isomorphism $T_{1}^{1}(V)\to \mathrm {End} (V)$ given by
$(a\otimes b)(x)=\langle x,b\rangle a.$
Under this isomorphism, every u in $\mathrm {End} (V)$ may be first viewed as an endomorphism of $T_{1}^{1}(V)$ and then viewed as an endomorphism of $\mathrm {End} (V).$ In fact it is the adjoint representation ad(u) of $\mathrm {End} (V).$
Linear maps as tensors
Given two finite dimensional vector spaces U, V over the same field K, denote the dual space of U as U*, and the K-vector space of all linear maps from U to V as Hom(U,V). There is an isomorphism,
$U^{*}\otimes V\cong \mathrm {Hom} (U,V),$
defined by an action of the pure tensor $f\otimes v\in U^{*}\otimes V$ on an element of $U,$
$(f\otimes v)(u)=f(u)v.$
Its "inverse" can be defined using a basis $\{u_{i}\}$ and its dual basis $\{u_{i}^{*}\}$ as in the section "Evaluation map and tensor contraction" above:
${\begin{cases}\mathrm {Hom} (U,V)\to U^{*}\otimes V\\F\mapsto \sum _{i}u_{i}^{*}\otimes F(u_{i}).\end{cases}}$
This result implies
$\dim(U\otimes V)=\dim(U)\dim(V),$
which automatically gives the important fact that $\{u_{i}\otimes v_{j}\}$ forms a basis for $U\otimes V$ where $\{u_{i}\},\{v_{j}\}$ are bases of U and V.
Furthermore, given three vector spaces U, V, W the tensor product is linked to the vector space of all linear maps, as follows:
$\mathrm {Hom} (U\otimes V,W)\cong \mathrm {Hom} (U,\mathrm {Hom} (V,W)).$
This is an example of adjoint functors: the tensor product is "left adjoint" to Hom.
Tensor products of modules over a ring
Main article: Tensor product of modules
The tensor product of two modules A and B over a commutative ring R is defined in exactly the same way as the tensor product of vector spaces over a field:
$A\otimes _{R}B:=F(A\times B)/G$
where now $F(A\times B)$ is the free R-module generated by the cartesian product and G is the R-module generated by these relations.
More generally, the tensor product can be defined even if the ring is non-commutative. In this case A has to be a right-R-module and B is a left-R-module, and instead of the last two relations above, the relation
$(ar,b)\sim (a,rb)$
is imposed. If R is non-commutative, this is no longer an R-module, but just an abelian group.
The universal property also carries over, slightly modified: the map $\varphi :A\times B\to A\otimes _{R}B$ defined by $(a,b)\mapsto a\otimes b$ is a middle linear map (referred to as "the canonical middle linear map".[8]); that is, it satisfies:[9]
${\begin{aligned}\phi (a+a',b)&=\phi (a,b)+\phi (a',b)\\\phi (a,b+b')&=\phi (a,b)+\phi (a,b')\\\phi (ar,b)&=\phi (a,rb)\end{aligned}}$
The first two properties make φ a bilinear map of the abelian group $A\times B.$ For any middle linear map $\psi $ of $A\times B,$ a unique group homomorphism f of $A\otimes _{R}B$ satisfies $\psi =f\circ \varphi ,$ and this property determines $\phi $ within group isomorphism. See the main article for details.
Tensor product of modules over a non-commutative ring
Let A be a right R-module and B be a left R-module. Then the tensor product of A and B is an abelian group defined by
$A\otimes _{R}B:=F(A\times B)/G$
where $F(A\times B)$ is a free abelian group over $A\times B$ and G is the subgroup of $F(A\times B)$ generated by relations
${\begin{aligned}&\forall a,a_{1},a_{2}\in A,\forall b,b_{1},b_{2}\in B,{\text{ for all }}r\in R:\\&(a_{1},b)+(a_{2},b)-(a_{1}+a_{2},b),\\&(a,b_{1})+(a,b_{2})-(a,b_{1}+b_{2}),\\&(ar,b)-(a,rb).\\\end{aligned}}$
The universal property can be stated as follows. Let G be an abelian group with a map $q:A\times B\to G$ that is bilinear, in the sense that
${\begin{aligned}q(a_{1}+a_{2},b)&=q(a_{1},b)+q(a_{2},b),\\q(a,b_{1}+b_{2})&=q(a,b_{1})+q(a,b_{2}),\\q(ar,b)&=q(a,rb).\end{aligned}}$
Then there is a unique map ${\overline {q}}:A\otimes B\to G$ such that ${\overline {q}}(a\otimes b)=q(a,b)$ for all $a\in A$ and $b\in B.$
Furthermore, we can give $A\otimes _{R}B$ a module structure under some extra conditions:
1. If A is a (S,R)-bimodule, then $A\otimes _{R}B$ is a left S-module where $s(a\otimes b):=(sa)\otimes b.$
2. If B is a (R,S)-bimodule, then $A\otimes _{R}B$ is a right S-module where $(a\otimes b)s:=a\otimes (bs).$
3. If A is a (S,R)-bimodule and B is a (R,T)-bimodule, then $A\otimes _{R}B$ is a (S,T)-bimodule, where the left and right actions are defined in the same way as the previous two examples.
4. If R is a commutative ring, then A and B are (R,R)-bimodules where $ra:=ar$ and $br:=rb.$ By 3), we can conclude $A\otimes _{R}B$ is a (R,R)-bimodule.
Computing the tensor product
For vector spaces, the tensor product $V\otimes W$ is quickly computed since bases of V of W immediately determine a basis of $V\otimes W,$ as was mentioned above. For modules over a general (commutative) ring, not every module is free. For example, Z/nZ is not a free abelian group (Z-module). The tensor product with Z/nZ is given by
$M\otimes _{\mathbf {Z} }\mathbf {Z} /n\mathbf {Z} =M/nM.$
More generally, given a presentation of some R-module M, that is, a number of generators $m_{i}\in M,i\in I$ together with relations
$\sum _{j\in J}a_{ji}m_{i}=0,\qquad a_{ij}\in R,$
the tensor product can be computed as the following cokernel:
$M\otimes _{R}N=\operatorname {coker} \left(N^{J}\to N^{I}\right)$
Here $N^{J}=\oplus _{j\in J}N,$ and the map $N^{J}\to N^{I}$ is determined by sending some $n\in N$ in the jth copy of $N^{J}$ to $a_{ij}n$ (in $N^{I}$). Colloquially, this may be rephrased by saying that a presentation of M gives rise to a presentation of $M\otimes _{R}N.$ This is referred to by saying that the tensor product is a right exact functor. It is not in general left exact, that is, given an injective map of R-modules $M_{1}\to M_{2},$ the tensor product
$M_{1}\otimes _{R}N\to M_{2}\otimes _{R}N$
is not usually injective. For example, tensoring the (injective) map given by multiplication with n, n : Z → Z with Z/nZ yields the zero map 0 : Z/nZ → Z/nZ, which is not injective. Higher Tor functors measure the defect of the tensor product being not left exact. All higher Tor functors are assembled in the derived tensor product.
Tensor product of algebras
Main article: Tensor product of algebras
Let R be a commutative ring. The tensor product of R-modules applies, in particular, if A and B are R-algebras. In this case, the tensor product $A\otimes _{R}B$ is an R-algebra itself by putting
$(a_{1}\otimes b_{1})\cdot (a_{2}\otimes b_{2})=(a_{1}\cdot a_{2})\otimes (b_{1}\cdot b_{2}).$
For example,
$R[x]\otimes _{R}R[y]\cong R[x,y].$
A particular example is when A and B are fields containing a common subfield R. The tensor product of fields is closely related to Galois theory: if, say, A = R[x] / f(x), where f is some irreducible polynomial with coefficients in R, the tensor product can be calculated as
$A\otimes _{R}B\cong B[x]/f(x)$
where now f is interpreted as the same polynomial, but with its coefficients regarded as elements of B. In the larger field B, the polynomial may become reducible, which brings in Galois theory. For example, if A = B is a Galois extension of R, then
$A\otimes _{R}A\cong A[x]/f(x)$
is isomorphic (as an A-algebra) to the $A^{\operatorname {deg} (f)}.$
Eigenconfigurations of tensors
Square matrices $A$ with entries in a field $K$ represent linear maps of vector spaces, say $K^{n}\to K^{n},$ and thus linear maps $\psi :\mathbb {P} ^{n-1}\to \mathbb {P} ^{n-1}$ :\mathbb {P} ^{n-1}\to \mathbb {P} ^{n-1}} of projective spaces over $K.$ If $A$ is nonsingular then $\psi $ is well-defined everywhere, and the eigenvectors of $A$ correspond to the fixed points of $\psi .$ The eigenconfiguration of $A$ consists of $n$ points in $\mathbb {P} ^{n-1},$ provided $A$ is generic and $K$ is algebraically closed. The fixed points of nonlinear maps are the eigenvectors of tensors. Let $A=(a_{i_{1}i_{2}\cdots i_{d}})$ be a $d$-dimensional tensor of format $n\times n\times \cdots \times n$ with entries $(a_{i_{1}i_{2}\cdots i_{d}})$ lying in an algebraically closed field $K$ of characteristic zero. Such a tensor $A\in (K^{n})^{\otimes d}$ defines polynomial maps $K^{n}\to K^{n}$ and $\mathbb {P} ^{n-1}\to \mathbb {P} ^{n-1}$ with coordinates
$\psi _{i}(x_{1},\ldots ,x_{n})=\sum _{j_{2}=1}^{n}\sum _{j_{3}=1}^{n}\cdots \sum _{j_{d}=1}^{n}a_{ij_{2}j_{3}\cdots j_{d}}x_{j_{2}}x_{j_{3}}\cdots x_{j_{d}}\;\;{\mbox{for }}i=1,\ldots ,n$
Thus each of the $n$ coordinates of $\psi $ is a homogeneous polynomial $\psi _{i}$ of degree $d-1$ in $\mathbf {x} =\left(x_{1},\ldots ,x_{n}\right).$ The eigenvectors of $A$ are the solutions of the constraint
${\mbox{rank}}{\begin{pmatrix}x_{1}&x_{2}&\cdots &x_{n}\\\psi _{1}(\mathbf {x} )&\psi _{2}(\mathbf {x} )&\cdots &\psi _{n}(\mathbf {x} )\end{pmatrix}}\leq 1$
and the eigenconfiguration is given by the variety of the $2\times 2$ minors of this matrix.[10]
Other examples of tensor products
Tensor product of Hilbert spaces
Main article: Tensor product of Hilbert spaces
Hilbert spaces generalize finite-dimensional vector spaces to countably-infinite dimensions. The tensor product is still defined; it is the tensor product of Hilbert spaces.
Topological tensor product
Main article: Topological tensor product
When the basis for a vector space is no longer countable, then the appropriate axiomatic formalization for the vector space is that of a topological vector space. The tensor product is still defined, it is the topological tensor product.
Tensor product of graded vector spaces
Main article: Graded vector space § Operations on graded vector spaces
Some vector spaces can be decomposed into direct sums of subspaces. In such cases, the tensor product of two spaces can be decomposed into sums of products of the subspaces (in analogy to the way that multiplication distributes over addition).
Tensor product of representations
Main article: Tensor product of representations
Vector spaces endowed with an additional multiplicative structure are called algebras. The tensor product of such algebras is described by the Littlewood–Richardson rule.
Tensor product of quadratic forms
Main article: Tensor product of quadratic forms
Tensor product of multilinear forms
Given two multilinear forms $f(x_{1},\dots ,x_{k})$ and $g(x_{1},\dots ,x_{m})$ on a vector space $V$ over the field $K$ their tensor product is the multilinear form
$(f\otimes g)(x_{1},\dots ,x_{k+m})=f(x_{1},\dots ,x_{k})g(x_{k+1},\dots ,x_{k+m}).$
[11]
This is a special case of the product of tensors if they are seen as multilinear maps (see also tensors as multilinear maps). Thus the components of the tensor product of multilinear forms can be computed by the Kronecker product.
Tensor product of sheaves of modules
Main article: Sheaf of modules
Tensor product of line bundles
Main article: Vector bundle § Operations on vector bundles
See also: tensor product bundle
Tensor product of fields
Main article: Tensor product of fields
Tensor product of graphs
It should be mentioned that, though called "tensor product", this is not a tensor product of graphs in the above sense; actually it is the category-theoretic product in the category of graphs and graph homomorphisms. However it is actually the Kronecker tensor product of the adjacency matrices of the graphs. Compare also the section Tensor product of linear maps above.
Monoidal categories
The most general setting for the tensor product is the monoidal category. It captures the algebraic essence of tensoring, without making any specific reference to what is being tensored. Thus, all tensor products can be expressed as an application of the monoidal category to some particular setting, acting on some particular objects.
Quotient algebras
A number of important subspaces of the tensor algebra can be constructed as quotients: these include the exterior algebra, the symmetric algebra, the Clifford algebra, the Weyl algebra, and the universal enveloping algebra in general.
The exterior algebra is constructed from the exterior product. Given a vector space V, the exterior product $V\wedge V$ is defined as
$V\wedge V:=V\otimes V/\{v\otimes v\mid v\in V\}.$
Note that when the underlying field of V does not have characteristic 2, then this definition is equivalent to
$V\wedge V:=V\otimes V/\{v_{1}\otimes v_{2}+v_{2}\otimes v_{1}\mid (v_{1},v_{2})\in V^{2}\}.$
The image of $v_{1}\otimes v_{2}$ in the exterior product is usually denoted $v_{1}\wedge v_{2}$ and satisfies, by construction, $v_{1}\wedge v_{2}=-v_{2}\wedge v_{1}.$ Similar constructions are possible for $V\otimes \dots \otimes V$ (n factors), giving rise to $\Lambda ^{n}V,$ the nth exterior power of V. The latter notion is the basis of differential n-forms.
The symmetric algebra is constructed in a similar manner, from the symmetric product
$V\odot V:=V\otimes V/\{v_{1}\otimes v_{2}-v_{2}\otimes v_{1}\mid (v_{1},v_{2})\in V^{2}\}.$
More generally
$\operatorname {Sym} ^{n}V:=\underbrace {V\otimes \dots \otimes V} _{n}/(\dots \otimes v_{i}\otimes v_{i+1}\otimes \dots -\dots \otimes v_{i+1}\otimes v_{i}\otimes \dots )$
That is, in the symmetric algebra two adjacent vectors (and therefore all of them) can be interchanged. The resulting objects are called symmetric tensors.
Tensor product in programming
Array programming languages
Array programming languages may have this pattern built in. For example, in APL the tensor product is expressed as ○.× (for example A ○.× B or A ○.× B ○.× C). In J the tensor product is the dyadic form of */ (for example a */ b or a */ b */ c).
Note that J's treatment also allows the representation of some tensor fields, as a and b may be functions instead of constants. This product of two functions is a derived function, and if a and b are differentiable, then a */ b is differentiable.
However, these kinds of notation are not universally present in array languages. Other array languages may require explicit treatment of indices (for example, MATLAB), and/or may not support higher-order functions such as the Jacobian derivative (for example, Fortran/APL).
See also
Look up tensor product in Wiktionary, the free dictionary.
• Dyadics – Second order tensor in vector algebra
• Extension of scalars
• Monoidal category – Category admitting tensor products
• Tensor algebra – Universal construction in multilinear algebra
• Tensor contraction – in mathematics and physics, an operation on tensorsPages displaying wikidata descriptions as a fallback
• Topological tensor product – Tensor product constructions for topological vector spaces
Notes
1. Trèves 2006, pp. 403–404.
2. Trèves 2006, pp. 407.
3. Hazewinkel, Michiel; Gubareni, Nadezhda Mikhaĭlovna; Gubareni, Nadiya; Kirichenko, Vladimir V. (2004). Algebras, rings and modules. Springer. p. 100. ISBN 978-1-4020-2690-4.
4. Bourbaki (1989), p. 244 defines the usage "tensor product of x and y", elements of the respective modules.
5. Analogous formulas also hold for contravariant tensors, as well as tensors of mixed variance. Although in many cases such as when there is an inner product defined, the distinction is irrelevant.
6. "The Coevaluation on Vector Spaces". The Unapologetic Mathematician. 2008-11-13. Archived from the original on 2017-02-02. Retrieved 2017-01-26.
7. See Compact closed category.
8. Hungerford, Thomas W. (1974). Algebra. Springer. ISBN 0-387-90518-9.
9. Chen, Jungkai Alfred (Spring 2004), "Tensor product" (PDF), Advanced Algebra II (lecture notes), National Taiwan University, archived (PDF) from the original on 2016-03-04{{citation}}: CS1 maint: location missing publisher (link)
10. Abo, H.; Seigal, A.; Sturmfels, B. (2015). "Eigenconfigurations of Tensors". arXiv:1505.05729 [math.AG].
11. Tu, L. W. (2010). An Introduction to Manifolds. Universitext. Springer. p. 25. ISBN 978-1-4419-7399-3.
References
• Bourbaki, Nicolas (1989). Elements of mathematics, Algebra I. Springer-Verlag. ISBN 3-540-64243-9.
• Gowers, Timothy. "How to lose your fear of tensor products". Archived from the original on 7 May 2021.
• Grillet, Pierre A. (2007). Abstract Algebra. Springer Science+Business Media, LLC. ISBN 978-0387715674.
• Halmos, Paul (1974). Finite dimensional vector spaces. Springer. ISBN 0-387-90093-4.
• Hungerford, Thomas W. (2003). Algebra. Springer. ISBN 0387905189.
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001
• Mac Lane, S.; Birkhoff, G. (1999). Algebra. AMS Chelsea. ISBN 0-8218-1646-2.
• Aguiar, M.; Mahajan, S. (2010). Monoidal functors, species and Hopf algebras. CRM Monograph Series Vol 29. ISBN 978-0-8218-4776-3.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• "Bibliography on the nonabelian tensor product of groups".
Tensors
Glossary of tensor theory
Scope
Mathematics
• Coordinate system
• Differential geometry
• Dyadic algebra
• Euclidean geometry
• Exterior calculus
• Multilinear algebra
• Tensor algebra
• Tensor calculus
• Physics
• Engineering
• Computer vision
• Continuum mechanics
• Electromagnetism
• General relativity
• Transport phenomena
Notation
• Abstract index notation
• Einstein notation
• Index notation
• Multi-index notation
• Penrose graphical notation
• Ricci calculus
• Tetrad (index notation)
• Van der Waerden notation
• Voigt notation
Tensor
definitions
• Tensor (intrinsic definition)
• Tensor field
• Tensor density
• Tensors in curvilinear coordinates
• Mixed tensor
• Antisymmetric tensor
• Symmetric tensor
• Tensor operator
• Tensor bundle
• Two-point tensor
Operations
• Covariant derivative
• Exterior covariant derivative
• Exterior derivative
• Exterior product
• Hodge star operator
• Lie derivative
• Raising and lowering indices
• Symmetrization
• Tensor contraction
• Tensor product
• Transpose (2nd-order tensors)
Related
abstractions
• Affine connection
• Basis
• Cartan formalism (physics)
• Connection form
• Covariance and contravariance of vectors
• Differential form
• Dimension
• Exterior form
• Fiber bundle
• Geodesic
• Levi-Civita connection
• Linear map
• Manifold
• Matrix
• Multivector
• Pseudotensor
• Spinor
• Vector
• Vector space
Notable tensors
Mathematics
• Kronecker delta
• Levi-Civita symbol
• Metric tensor
• Nonmetricity tensor
• Ricci curvature
• Riemann curvature tensor
• Torsion tensor
• Weyl tensor
Physics
• Moment of inertia
• Angular momentum tensor
• Spin tensor
• Cauchy stress tensor
• stress–energy tensor
• Einstein tensor
• EM tensor
• Gluon field strength tensor
• Metric tensor (GR)
Mathematicians
• Élie Cartan
• Augustin-Louis Cauchy
• Elwin Bruno Christoffel
• Albert Einstein
• Leonhard Euler
• Carl Friedrich Gauss
• Hermann Grassmann
• Tullio Levi-Civita
• Gregorio Ricci-Curbastro
• Bernhard Riemann
• Jan Arnoldus Schouten
• Woldemar Voigt
• Hermann Weyl
| Wikipedia |
Tensor representation
In mathematics, the tensor representations of the general linear group are those that are obtained by taking finitely many tensor products of the fundamental representation and its dual. The irreducible factors of such a representation are also called tensor representations, and can be obtained by applying Schur functors (associated to Young tableaux). These coincide with the rational representations of the general linear group.
More generally, a matrix group is any subgroup of the general linear group. A tensor representation of a matrix group is any representation that is contained in a tensor representation of the general linear group. For example, the orthogonal group O(n) admits a tensor representation on the space of all trace-free symmetric tensors of order two. For orthogonal groups, the tensor representations are contrasted with the spin representations.
The classical groups, like the symplectic group, have the property that all finite-dimensional representations are tensor representations (by Weyl's construction), while other representations (like the metaplectic representation) exist in infinite dimensions.
References
• Roe Goodman; Nolan Wallach (2009), Symmetry, representations, and invariants, Springer, chapters 9 and 10.
• Bargmann, V., & Todorov, I. T. (1977). Spaces of analytic functions on a complex cone as carriers for the symmetric tensor representations of SO(n). Journal of Mathematical Physics, 18(6), 1141–1148.
| Wikipedia |
Tensor reshaping
In multilinear algebra, a reshaping of tensors is any bijection between the set of indices of an order-$M$ tensor and the set of indices of an order-$L$ tensor, where $L<M$. The use of indices presupposes tensors in coordinate representation with respect to a basis. The coordinate representation of a tensor can be regarded as a multi-dimensional array, and a bijection from one set of indices to another therefore amounts to a rearrangement of the array elements into an array of a different shape. Such a rearrangement constitutes a particular kind of linear map between the vector space of order-$M$ tensors and the vector space of order-$L$ tensors.
Definition
Given a positive integer $M$, the notation $[M]$ refers to the set $\{1,\dots ,M\}$ of the first M positive integers.
For each integer $m$ where $1\leq m\leq M$ for a positive integer $M$, let $V_{m}$ denote an $I_{m}$-dimensional vector space over a field $F$. Then there are vector space isomorphisms (linear maps)
${\begin{aligned}V_{1}\otimes \cdots \otimes V_{M}&\simeq F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}\\&\simeq F^{I_{\pi _{1}}}\otimes \cdots \otimes F^{I_{\pi _{M}}}\\&\simeq F^{I_{\pi _{1}}I_{\pi _{2}}}\otimes F^{I_{\pi _{3}}}\otimes \cdots \otimes F^{I_{\pi _{M}}}\\&\simeq F^{I_{\pi _{1}}I_{\pi _{3}}}\otimes F^{I_{\pi _{2}}}\otimes F^{I_{\pi _{4}}}\otimes \cdots \otimes F^{I_{\pi _{M}}}\\&\,\,\,\vdots \\&\simeq F^{I_{1}I_{2}\ldots I_{M}},\end{aligned}}$
where $\pi \in {\mathfrak {S}}_{M}$ is any permutation and ${\mathfrak {S}}_{M}$ is the symmetric group on $M$ elements. Via these (and other) vector space isomorphisms, a tensor can be interpreted in several ways as an order-$L$ tensor where $L\leq M$.
Coordinate representation
The first vector space isomorphism on the list above, $V_{1}\otimes \cdots \otimes V_{M}\simeq F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}$, gives the coordinate representation of an abstract tensor. Assume that each of the $M$ vector spaces $V_{m}$ has a basis $\{v_{1}^{m},v_{2}^{m},\ldots ,v_{I_{m}}^{m}\}$. The expression of a tensor with respect to this basis has the form
${\mathcal {A}}=\sum _{i_{1}=1}^{I_{1}}\ldots \sum _{i_{M}=1}^{I_{M}}a_{i_{1},i_{2},\ldots ,i_{M}}v_{i_{1}}^{1}\otimes v_{i_{2}}^{2}\otimes \cdots \otimes v_{i_{M}}^{M},$
where the coefficients $a_{i_{1},i_{2},\ldots ,i_{M}}$ are elements of $F$. The coordinate representation of ${\mathcal {A}}$ is
$\sum _{i_{1}=1}^{I_{1}}\ldots \sum _{i_{M}=1}^{I_{M}}a_{i_{1},i_{2},\ldots ,i_{M}}\mathbf {e} _{i_{1}}^{1}\otimes \mathbf {e} _{i_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{i_{M}}^{M},$
where $\mathbf {e} _{i}^{m}$ is the $i^{\text{th}}$ standard basis vector of $F^{I_{m}}$. This can be regarded as a M-way array whose elements are the coefficients $a_{i_{1},i_{2},\ldots ,i_{M}}$.
General flattenings
For any permutation $\pi \in {\mathfrak {S}}_{M}$ there is a canonical isomorphism between the two tensor products of vector spaces $V_{1}\otimes V_{2}\otimes \cdots \otimes V_{M}$ and $V_{\pi (1)}\otimes V_{\pi (2)}\otimes \cdots \otimes V_{\pi (M)}$. Parentheses are usually omitted from such products due to the natural isomorphism between $V_{i}\otimes (V_{j}\otimes V_{k})$ and $(V_{i}\otimes V_{j})\otimes V_{k}$, but may, of course, be reintroduced to emphasize a particular grouping of factors. In the grouping,
$(V_{\pi (1)}\otimes \cdots \otimes V_{\pi (r_{1})})\otimes (V_{\pi (r_{1}+1)}\otimes \cdots \otimes V_{\pi (r_{2})})\otimes \cdots \otimes (V_{\pi (r_{L-1}+1)}\otimes \cdots \otimes V_{\pi (r_{L})}),$
there are $L$ groups with $r_{l}-r_{l-1}$ factors in the $l^{\text{th}}$ group (where $r_{0}=0$ and $r_{L}=M$).
Letting $S_{l}=(\pi (r_{l-1}+1),\pi (r_{l-1}+2),\ldots ,\pi (r_{l}))$ for each $l$ satisfying $1\leq l\leq L$, an $(S_{1},S_{2},\ldots ,S_{L})$-flattening of a tensor ${\mathcal {A}}$, denoted ${\mathcal {A}}_{(S_{1},S_{2},\ldots ,S_{L})}$, is obtained by applying the two processes above within each of the $L$ groups of factors. That is, the coordinate representation of the $l^{\text{th}}$ group of factors is obtained using the isomorphism $(V_{\pi (r_{l-1}+1)}\otimes V_{\pi (r_{l-1}+2)}\otimes \cdots \otimes V_{\pi (r_{l})})\simeq (F^{I_{\pi (r_{l-1}+1)}}\otimes F^{I_{\pi (r_{l-1}+2)}}\otimes \cdots \otimes F^{I_{\pi (r_{l})}})$, which requires specifying bases for all of the vector spaces $V_{k}$. The result is then vectorized using a bijection $\mu _{l}:[I_{\pi (r_{l-1}+1)}]\times [I_{\pi (r_{l-1}+2)}]\times \cdots \times [I_{\pi (r_{l})}]\to [I_{S_{l}}]$ to obtain an element of $F^{I_{S_{l}}}$, where $I_{S_{l}}:=\prod _{i=r_{l-1}+1}^{r_{l}}I_{\pi (i)}$, the product of the dimensions of the vector spaces in the $l^{\text{th}}$ group of factors. The result of applying these isomorphisms within each group of factors is an element of $F^{I_{S_{1}}}\otimes \cdots \otimes F^{I_{S_{L}}}$, which is a tensor of order $L$.
Vectorization
By means of a bijective map $\mu :[I_{1}]\times \cdots \times [I_{M}]\to [I_{1}\cdots I_{M}]$ :[I_{1}]\times \cdots \times [I_{M}]\to [I_{1}\cdots I_{M}]} , a vector space isomorphism between $F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}$ and $F^{I_{1}\cdots I_{M}}$ is constructed via the mapping $\mathbf {e} _{i_{1}}^{1}\otimes \cdots \mathbf {e} _{i_{m}}^{m}\otimes \cdots \otimes \mathbf {e} _{i_{M}}^{M}\mapsto \mathbf {e} _{\mu (i_{1},i_{2},\ldots ,i_{M})},$ where for every natural number $i$ such that $1\leq i\leq I_{1}\cdots I_{M}$, the vector $\mathbf {e} _{i}$ denotes the ith standard basis vector of $F^{i_{1}\cdots i_{M}}$. In such a reshaping, the tensor is simply interpreted as a vector in $F^{I_{1}\cdots I_{M}}$. This is known as vectorization, and is analogous to vectorization of matrices. A standard choice of bijection $\mu $ is such that
$\operatorname {vec} ({\mathcal {A}})={\begin{bmatrix}a_{1,1,\ldots ,1}&a_{2,1,\ldots ,1}&\cdots &a_{n_{1},1,\ldots ,1}&a_{1,2,1,\ldots ,1}&\cdots &a_{I_{1},I_{2},\ldots ,I_{M}}\end{bmatrix}}^{T},$
which is consistent with the way in which the colon operator in Matlab and GNU Octave reshapes a higher-order tensor into a vector. In general, the vectorization of ${\mathcal {A}}$ is the vector $[a_{\mu ^{-1}(i)}]_{i=1}^{I_{1}\cdots I_{M}}$.
The vectorization of ${\mathcal {A}}$ denoted with $vec({\mathcal {A}})$ or ${\mathcal {A}}_{[:]}$ is an $[S_{1},S_{2}]$-reshaping where $S_{1}=(1,2,\ldots ,M)$ and $S_{2}=\emptyset $.
Mode-m Flattening / Mode-m Matrixization
Let ${\mathcal {A}}\in F^{I_{1}}\otimes F^{I_{2}}\otimes \cdots \otimes F^{I_{M}}$ be the coordinate representation of an abstract tensor with respect to a basis. Mode-m matrixizing (a.k.a. flattening) of ${\mathcal {A}}$ is an $[S_{1},S_{2}]$-reshaping in which $S_{1}=(m)$ and $S_{2}=(1,2,\ldots ,m-1,m+1,\ldots ,M)$. Usually, a standard matrixizing is denoted by
${\bf {A}}_{[m]}={\mathcal {A}}_{[S_{1},S_{2}]}$
This reshaping is sometimes called matrixizing, matricizing, flattening or unfolding in the literature. A standard choice for the bijections $\mu _{1},\ \mu _{2}$ is the one that is consistent with the reshape function in Matlab and GNU Octave, namely
${\bf {A}}_{[m]}:={\begin{bmatrix}a_{1,1,\ldots ,1,1,1,\ldots ,1}&a_{2,1,\ldots ,1,1,1,\ldots ,1}&\cdots &a_{I_{1},I_{2},\ldots ,I_{m-1},1,I_{m+1},\ldots ,I_{M}}\\a_{1,1,\ldots ,1,2,1,\ldots ,1}&a_{2,1,\ldots ,1,2,1,\ldots ,1}&\cdots &a_{I_{1},I_{2},\ldots ,I_{m-1},2,I_{m+1},\ldots ,I_{M}}\\\vdots &\vdots &&\vdots \\a_{1,1,\ldots ,1,I_{m},1,\ldots ,1}&a_{2,1,\ldots ,1,I_{m},1,\ldots ,1}&\cdots &a_{I_{1},I_{2},\ldots ,I_{m-1},I_{m},I_{m+1},\ldots ,I_{M}}\end{bmatrix}}$
Definition Mode-m Matrixizing:[1]
$[{\bf {A}}_{[m]}]_{jk}=a_{i_{1}\dots i_{m}\dots i_{M}},$ where $j=i_{m}$ and $k=1+\sum _{n=0 \atop n\neq m}^{M}(i_{n}-1)\prod _{l=0 \atop l\neq m}^{n-1}I_{l}.$
The mode-m matrixizing of a tensor ${\mathcal {A}}\in F^{I_{1}\times ...I_{M}},$ is defined as the matrix ${\bf {A}}_{[m]}\in F^{I_{m}\times (I_{1}\dots I_{m-1}I_{m+1}\dots I_{M})}$. As the parenthetical ordering indicates, the mode-m column vectors are arranged by sweeping all the other mode indices through their ranges, with smaller mode indexes varying more rapidly than larger ones; thus
References
1. Vasilescu, M. Alex O. (2009), "Multilinear (Tensor) Algebraic Framework for Computer Graphics, Computer Vision and Machine Learning" (PDF), University of Toronto, p. 21
| Wikipedia |
Enriched category
In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a vector space of morphisms, or a topological space of morphisms. In an enriched category, the set of morphisms (the hom-set) associated with every pair of objects is replaced by an object in some fixed monoidal category of "hom-objects". In order to emulate the (associative) composition of morphisms in an ordinary category, the hom-category must have a means of composing hom-objects in an associative manner: that is, there must be a binary operation on objects giving us at least the structure of a monoidal category, though in some contexts the operation may also need to be commutative and perhaps also to have a right adjoint (i.e., making the category symmetric monoidal or even symmetric closed monoidal, respectively).
Enriched category theory thus encompasses within the same framework a wide variety of structures including
• ordinary categories where the hom-set carries additional structure beyond being a set. That is, there are operations on, or properties of morphisms that need to be respected by composition (e.g., the existence of 2-cells between morphisms and horizontal composition thereof in a 2-category, or the addition operation on morphisms in an abelian category)
• category-like entities that don't themselves have any notion of individual morphism but whose hom-objects have similar compositional aspects (e.g., preorders where the composition rule ensures transitivity, or Lawvere's metric spaces, where the hom-objects are numerical distances and the composition rule provides the triangle inequality).
In the case where the hom-object category happens to be the category of sets with the usual cartesian product, the definitions of enriched category, enriched functor, etc... reduce to the original definitions from ordinary category theory.
An enriched category with hom-objects from monoidal category M is said to be an enriched category over M or an enriched category in M, or simply an M-category. Due to Mac Lane's preference for the letter V in referring to the monoidal category, enriched categories are also sometimes referred to generally as V-categories.
Definition
Let (M, ⊗, I, α, λ, ρ) be a monoidal category. Then an enriched category C (alternatively, in situations where the choice of monoidal category needs to be explicit, a category enriched over M, or M-category), consists of
• a class ob(C) of objects of C,
• an object C(a, b) of M for every pair of objects a, b in C, used to define an arrow $f:a\rightarrow b$ in C as an arrow $f:I\rightarrow C(a,b)$ in M,
• an arrow ida : I → C(a, a) in M designating an identity for every object a in C, and
• an arrow °abc : C(b, c) ⊗ C(a, b) → C(a, c) in M designating a composition for each triple of objects a, b, c in C, used to define the composition of $f:a\rightarrow b$ and $g:b\rightarrow c$ in C as $g\circ _{\textbf {C}}f={^{\circ }}_{abc}(g\otimes f)$ together with three commuting diagrams, discussed below.
The first diagram expresses the associativity of composition:
That is, the associativity requirement is now taken over by the associator of the monoidal category M.
For the case that M is the category of sets and (⊗, I, α, λ, ρ) is the monoidal structure (×, {•}, …) given by the cartesian product, the terminal single-point set, and the canonical isomorphisms they induce, then each C(a, b) is a set whose elements may be thought of as "individual morphisms" of C, while °, now a function, defines how consecutive morphisms compose. In this case, each path leading to C(a, d) in the first diagram corresponds to one of the two ways of composing three consecutive individual morphisms a → b → c → d, i.e. elements from C(a, b), C(b, c) and C(c, d). Commutativity of the diagram is then merely the statement that both orders of composition give the same result, exactly as required for ordinary categories.
What is new here is that the above expresses the requirement for associativity without any explicit reference to individual morphisms in the enriched category C — again, these diagrams are for morphisms in monoidal category M, and not in C — thus making the concept of associativity of composition meaningful in the general case where the hom-objects C(a, b) are abstract, and C itself need not even have any notion of individual morphism.
The notion that an ordinary category must have identity morphisms is replaced by the second and third diagrams, which express identity in terms of left and right unitors:
and
Returning to the case where M is the category of sets with cartesian product, the morphisms ida: I → C(a, a) become functions from the one-point set I and must then, for any given object a, identify a particular element of each set C(a, a), something we can then think of as the "identity morphism for a in C". Commutativity of the latter two diagrams is then the statement that compositions (as defined by the functions °) involving these distinguished individual "identity morphisms in C" behave exactly as per the identity rules for ordinary categories.
Note that there are several distinct notions of "identity" being referenced here:
• the monoidal identity object I of M, being an identity for ⊗ only in the monoid-theoretic sense, and even then only up to canonical isomorphism (λ, ρ).
• the identity morphism 1C(a, b) : C(a, b) → C(a, b) that M has for each of its objects by virtue of it being (at least) an ordinary category.
• the enriched category identity ida : I → C(a, a) for each object a in C, which is again a morphism of M which, even in the case where C is deemed to have individual morphisms of its own, is not necessarily identifying a specific one.
Examples of enriched categories
• Ordinary categories are categories enriched over (Set, ×, {•}), the category of sets with Cartesian product as the monoidal operation, as noted above.
• 2-Categories are categories enriched over Cat, the category of small categories, with monoidal structure being given by cartesian product. In this case the 2-cells between morphisms a → b and the vertical-composition rule that relates them correspond to the morphisms of the ordinary category C(a, b) and its own composition rule.
• Locally small categories are categories enriched over (SmSet, ×), the category of small sets with Cartesian product as the monoidal operation. (A locally small category is one whose hom-objects are small sets.)
• Locally finite categories, by analogy, are categories enriched over (FinSet, ×), the category of finite sets with Cartesian product as the monoidal operation.
• If C is a closed monoidal category then C is enriched in itself.
• Preordered sets are categories enriched over a certain monoidal category, 2, consisting of two objects and a single nonidentity arrow between them that we can write as FALSE → TRUE, conjunction as the monoid operation, and TRUE as its monoidal identity. The hom-objects 2(a, b) then simply deny or affirm a particular binary relation on the given pair of objects (a, b); for the sake of having more familiar notation we can write this relation as a ≤ b. The existence of the compositions and identity required for a category enriched over 2 immediately translate to the following axioms respectively
b ≤ c and a ≤ b ⇒ a ≤ c (transitivity)
TRUE ⇒ a ≤ a (reflexivity)
which are none other than the axioms for ≤ being a preorder. And since all diagrams in 2 commute, this is the sole content of the enriched category axioms for categories enriched over 2.
• William Lawvere's generalized metric spaces, also known as pseudoquasimetric spaces, are categories enriched over the nonnegative extended real numbers R+∞, where the latter is given ordinary category structure via the inverse of its usual ordering (i.e., there exists a morphism r → s iff r ≥ s) and a monoidal structure via addition (+) and zero (0). The hom-objects R+∞(a, b) are essentially distances d(a, b), and the existence of composition and identity translate to
d(b, c) + d(a, b) ≥ d(a, c) (triangle inequality)
0 ≥ d(a, a)
• Categories with zero morphisms are categories enriched over (Set*, ∧), the category of pointed sets with smash product as the monoidal operation; the special point of a hom-object Hom(A, B) corresponds to the zero morphism from A to B.
• The category Ab of abelian groups and the category R-Mod of modules over a commutative ring, and the category Vect of vector spaces over a given field are enriched over themselves, where the morphisms inherit the algebraic structure "pointwise". More generally, preadditive categories are categories enriched over (Ab, ⊗) with tensor product as the monoidal operation (thinking of abelian groups as Z-modules).
Relationship with monoidal functors
If there is a monoidal functor from a monoidal category M to a monoidal category N, then any category enriched over M can be reinterpreted as a category enriched over N. Every monoidal category M has a monoidal functor M(I, –) to the category of sets, so any enriched category has an underlying ordinary category. In many examples (such as those above) this functor is faithful, so a category enriched over M can be described as an ordinary category with certain additional structure or properties.
Enriched functors
An enriched functor is the appropriate generalization of the notion of a functor to enriched categories. Enriched functors are then maps between enriched categories which respect the enriched structure.
If C and D are M-categories (that is, categories enriched over monoidal category M), an M-enriched functor T: C → D is a map which assigns to each object of C an object of D and for each pair of objects a and b in C provides a morphism in M Tab : C(a, b) → D(T(a), T(b)) between the hom-objects of C and D (which are objects in M), satisfying enriched versions of the axioms of a functor, viz preservation of identity and composition.
Because the hom-objects need not be sets in an enriched category, one cannot speak of a particular morphism. There is no longer any notion of an identity morphism, nor of a particular composition of two morphisms. Instead, morphisms from the unit to a hom-object should be thought of as selecting an identity, and morphisms from the monoidal product should be thought of as composition. The usual functorial axioms are replaced with corresponding commutative diagrams involving these morphisms.
In detail, one has that the diagram
commutes, which amounts to the equation
$T_{aa}\circ \operatorname {id} _{a}=\operatorname {id} _{T(a)},$
where I is the unit object of M. This is analogous to the rule F(ida) = idF(a) for ordinary functors. Additionally, one demands that the diagram
commute, which is analogous to the rule F(fg)=F(f)F(g) for ordinary functors.
See also
• Internal category
• Isbell conjugacy
References
• Kelly,G.M. (2005) [1982]. Basic Concepts of Enriched Category Theory. Reprints in Theory and Applications of Categories. Vol. 10.
• Mac Lane, Saunders (September 1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). Springer. ISBN 0-387-98403-8.
• Lawvere, F.W. (2002) [1973]. Metric Spaces, Generalized Logic, and Closed Categories. Reprints in Theory and Applications of Categories. Vol. 1.
• Enriched category at the nLab
Category theory
Key concepts
Key concepts
• Category
• Adjoint functors
• CCC
• Commutative diagram
• Concrete category
• End
• Exponential
• Functor
• Kan extension
• Morphism
• Natural transformation
• Universal property
Universal constructions
Limits
• Terminal objects
• Products
• Equalizers
• Kernels
• Pullbacks
• Inverse limit
Colimits
• Initial objects
• Coproducts
• Coequalizers
• Cokernels and quotients
• Pushout
• Direct limit
Algebraic categories
• Sets
• Relations
• Magmas
• Groups
• Abelian groups
• Rings (Fields)
• Modules (Vector spaces)
Constructions on categories
• Free category
• Functor category
• Kleisli category
• Opposite category
• Quotient category
• Product category
• Comma category
• Subcategory
Higher category theory
Key concepts
• Categorification
• Enriched category
• Higher-dimensional algebra
• Homotopy hypothesis
• Model category
• Simplex category
• String diagram
• Topos
n-categories
Weak n-categories
• Bicategory (pseudofunctor)
• Tricategory
• Tetracategory
• Kan complex
• ∞-groupoid
• ∞-topos
Strict n-categories
• 2-category (2-functor)
• 3-category
Categorified concepts
• 2-group
• 2-ring
• En-ring
• (Traced)(Symmetric) monoidal category
• n-group
• n-monoid
• Category
• Outline
• Glossary
| Wikipedia |
Tensors in curvilinear coordinates
Curvilinear coordinates can be formulated in tensor calculus, with important applications in physics and engineering, particularly for describing transportation of physical quantities and deformation of matter in fluid mechanics and continuum mechanics.
Vector and tensor algebra in three-dimensional curvilinear coordinates
Note: the Einstein summation convention of summing on repeated indices is used below.
Elementary vector and tensor algebra in curvilinear coordinates is used in some of the older scientific literature in mechanics and physics and can be indispensable to understanding work from the early and mid 1900s, for example the text by Green and Zerna.[1] Some useful relations in the algebra of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden,[2] Naghdi,[3] Simmonds,[4] Green and Zerna,[1] Basar and Weichert,[5] and Ciarlet.[6]
Coordinate transformations
Consider two coordinate systems with coordinate variables $(Z^{1},Z^{2},Z^{3})$ and $(Z^{\acute {1}},Z^{\acute {2}},Z^{\acute {3}})$, which we shall represent in short as just $Z^{i}$ and $Z^{\acute {i}}$ respectively and always assume our index $i$ runs from 1 through 3. We shall assume that these coordinates systems are embedded in the three-dimensional euclidean space. Coordinates $Z^{i}$ and $Z^{\acute {i}}$ may be used to explain each other, because as we move along the coordinate line in one coordinate system we can use the other to describe our position. In this way Coordinates $Z^{i}$ and $Z^{\acute {i}}$ are functions of each other
$Z^{i}=f^{i}(Z^{\acute {1}},Z^{\acute {2}},Z^{\acute {3}})$
for $i=1,2,3$
which can be written as
$Z^{i}=Z^{i}(Z^{\acute {1}},Z^{\acute {2}},Z^{\acute {3}})=Z^{i}(Z^{\acute {i}})$
for ${\acute {i}},i=1,2,3$
These three equations together are also called a coordinate transformation from $Z^{\acute {i}}$ to $Z^{i}$.Let us denote this transformation by $T$. We will therefore represent the transformation from the coordinate system with coordinate variables $Z^{\acute {i}}$ to the coordinate system with coordinates $Z^{i}$ as:
$Z=T({\acute {z}})$
Similarly we can represent $Z^{\acute {i}}$ as a function of $Z^{i}$ as follows:
$Z^{\acute {i}}=g^{\acute {i}}(Z^{1},Z^{2},Z^{3})$
for ${\acute {i}}=1,2,3$
similarly we can write the free equations more compactly as
$Z^{\acute {i}}=Z^{\acute {i}}(Z^{1},Z^{2},Z^{3})=Z^{\acute {i}}(Z^{i})$
for ${\acute {i}},i=1,2,3$
These three equations together are also called a coordinate transformation from $Z^{i}$ to $Z^{\acute {i}}$. Let us denote this transformation by $S$. We will represent the transformation from the coordinate system with coordinate variables $Z^{i}$ to the coordinate system with coordinates $Z^{\acute {i}}$ as:
${\acute {z}}=S(z)$
If the transformation $T$ is bijective then we call the image of the transformation,namely $Z^{i}$, a set of admissible coordinates for $Z^{\acute {i}}$. If $T$ is linear the coordinate system $Z^{i}$ will be called an affine coordinate system ,otherwise $Z^{i}$ is called a curvilinear coordinate system
The Jacobian
As we now see that the Coordinates $Z^{i}$ and $Z^{\acute {i}}$ are functions of each other, we can take the derivative of the coordinate variable $Z^{i}$ with respect to the coordinate variable $Z^{\acute {i}}$
consider
${\frac {\partial {Z^{i}}}{\partial {Z^{\acute {i}}}}}\;{\overset {\underset {\mathrm {def} }{}}{=}}\;J_{\acute {i}}^{i}$
for ${\acute {i}},i=1,2,3$, these derivatives can be arranged in a matrix, say $J$,in which $J_{\acute {i}}^{i}$ is the element in the $i$-th row and ${\acute {i}}$-th column
$J={\begin{pmatrix}J_{\acute {1}}^{1}&J_{\acute {2}}^{1}&J_{\acute {3}}^{1}\\J_{\acute {1}}^{2}&J_{\acute {2}}^{2}&J_{\acute {3}}^{2}\\J_{\acute {1}}^{3}&J_{\acute {2}}^{3}&J_{\acute {3}}^{3}\end{pmatrix}}={\begin{pmatrix}{\partial {Z^{1}} \over \partial {Z^{\acute {1}}}}&{\partial {Z^{1}} \over \partial {Z^{\acute {2}}}}&{\partial {Z^{1}} \over \partial {Z^{\acute {3}}}}\\{\partial {Z^{2}} \over \partial {Z^{\acute {1}}}}&{\partial {Z^{2}} \over \partial {Z^{\acute {2}}}}&{\partial {Z^{2}} \over \partial {Z^{\acute {3}}}}\\{\partial {Z^{3}} \over \partial {Z^{\acute {1}}}}&{\partial {Z^{3}} \over \partial {Z^{\acute {2}}}}&{\partial {Z^{3}} \over \partial {Z^{\acute {3}}}}\end{pmatrix}}$
The resultant matrix is called the Jacobian matrix.
Vectors in curvilinear coordinates
Let (b1, b2, b3) be an arbitrary basis for three-dimensional Euclidean space. In general, the basis vectors are neither unit vectors nor mutually orthogonal. However, they are required to be linearly independent. Then a vector v can be expressed as[4]: 27
$\mathbf {v} =v^{k}\,\mathbf {b} _{k}$
The components vk are the contravariant components of the vector v.
The reciprocal basis (b1, b2, b3) is defined by the relation [4]: 28–29
$\mathbf {b} ^{i}\cdot \mathbf {b} _{j}=\delta _{j}^{i}$
where δi j is the Kronecker delta.
The vector v can also be expressed in terms of the reciprocal basis:
$\mathbf {v} =v_{k}~\mathbf {b} ^{k}$
The components vk are the covariant components of the vector $\mathbf {v} $.
Second-order tensors in curvilinear coordinates
A second-order tensor can be expressed as
${\boldsymbol {S}}=S^{ij}~\mathbf {b} _{i}\otimes \mathbf {b} _{j}=S_{~j}^{i}~\mathbf {b} _{i}\otimes \mathbf {b} ^{j}=S_{i}^{~j}~\mathbf {b} ^{i}\otimes \mathbf {b} _{j}=S_{ij}~\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}$
The components Sij are called the contravariant components, Si j the mixed right-covariant components, Si j the mixed left-covariant components, and Sij the covariant components of the second-order tensor.
Metric tensor and relations between components
The quantities gij, gij are defined as[4]: 39
$g_{ij}=\mathbf {b} _{i}\cdot \mathbf {b} _{j}=g_{ji}~;~~g^{ij}=\mathbf {b} ^{i}\cdot \mathbf {b} ^{j}=g^{ji}$
From the above equations we have
$v^{i}=g^{ik}~v_{k}~;~~v_{i}=g_{ik}~v^{k}~;~~\mathbf {b} ^{i}=g^{ij}~\mathbf {b} _{j}~;~~\mathbf {b} _{i}=g_{ij}~\mathbf {b} ^{j}$
The components of a vector are related by[4]: 30–32
$\mathbf {v} \cdot \mathbf {b} ^{i}=v^{k}~\mathbf {b} _{k}\cdot \mathbf {b} ^{i}=v^{k}~\delta _{k}^{i}=v^{i}$
$\mathbf {v} \cdot \mathbf {b} _{i}=v_{k}~\mathbf {b} ^{k}\cdot \mathbf {b} _{i}=v_{k}~\delta _{i}^{k}=v_{i}$
Also,
$\mathbf {v} \cdot \mathbf {b} _{i}=v^{k}~\mathbf {b} _{k}\cdot \mathbf {b} _{i}=g_{ki}~v^{k}$
$\mathbf {v} \cdot \mathbf {b} ^{i}=v_{k}~\mathbf {b} ^{k}\cdot \mathbf {b} ^{i}=g^{ki}~v_{k}$
The components of the second-order tensor are related by
$S^{ij}=g^{ik}~S_{k}^{~j}=g^{jk}~S_{~k}^{i}=g^{ik}~g^{jl}~S_{kl}$
The alternating tensor
In an orthonormal right-handed basis, the third-order alternating tensor is defined as
${\boldsymbol {\mathcal {E}}}=\varepsilon _{ijk}~\mathbf {e} ^{i}\otimes \mathbf {e} ^{j}\otimes \mathbf {e} ^{k}$
In a general curvilinear basis the same tensor may be expressed as
${\boldsymbol {\mathcal {E}}}={\mathcal {E}}_{ijk}~\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}\otimes \mathbf {b} ^{k}={\mathcal {E}}^{ijk}~\mathbf {b} _{i}\otimes \mathbf {b} _{j}\otimes \mathbf {b} _{k}$
It can be shown that
${\mathcal {E}}_{ijk}=\left[\mathbf {b} _{i},\mathbf {b} _{j},\mathbf {b} _{k}\right]=(\mathbf {b} _{i}\times \mathbf {b} _{j})\cdot \mathbf {b} _{k}~;~~{\mathcal {E}}^{ijk}=\left[\mathbf {b} ^{i},\mathbf {b} ^{j},\mathbf {b} ^{k}\right]$
Now,
$\mathbf {b} _{i}\times \mathbf {b} _{j}=J~\varepsilon _{ijp}~\mathbf {b} ^{p}={\sqrt {g}}~\varepsilon _{ijp}~\mathbf {b} ^{p}$
Hence,
${\mathcal {E}}_{ijk}=J~\varepsilon _{ijk}={\sqrt {g}}~\varepsilon _{ijk}$
Similarly, we can show that
${\mathcal {E}}^{ijk}={\cfrac {1}{J}}~\varepsilon ^{ijk}={\cfrac {1}{\sqrt {g}}}~\varepsilon ^{ijk}$
Identity map
The identity map I defined by $\mathbf {I} \cdot \mathbf {v} =\mathbf {v} $ can be shown to be:[4]: 39
$\mathbf {I} =g^{ij}\mathbf {b} _{i}\otimes \mathbf {b} _{j}=g_{ij}\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}=\mathbf {b} _{i}\otimes \mathbf {b} ^{i}=\mathbf {b} ^{i}\otimes \mathbf {b} _{i}$
Scalar (dot) product
The scalar product of two vectors in curvilinear coordinates is[4]: 32
$\mathbf {u} \cdot \mathbf {v} =u^{i}v_{i}=u_{i}v^{i}=g_{ij}u^{i}v^{j}=g^{ij}u_{i}v_{j}$
Vector (cross) product
The cross product of two vectors is given by:[4]: 32–34
$\mathbf {u} \times \mathbf {v} =\varepsilon _{ijk}u_{j}v_{k}\mathbf {e} _{i}$
where εijk is the permutation symbol and ei is a Cartesian basis vector. In curvilinear coordinates, the equivalent expression is:
$\mathbf {u} \times \mathbf {v} =[(\mathbf {b} _{m}\times \mathbf {b} _{n})\cdot \mathbf {b} _{s}]u^{m}v^{n}\mathbf {b} ^{s}={\mathcal {E}}_{smn}u^{m}v^{n}\mathbf {b} ^{s}$
where ${\mathcal {E}}_{ijk}$ is the third-order alternating tensor. The cross product of two vectors is given by:
$\mathbf {u} \times \mathbf {v} =\varepsilon _{ijk}{\hat {u}}_{j}{\hat {v}}_{k}\mathbf {e} _{i}$
where εijk is the permutation symbol and $\mathbf {e} _{i}$ is a Cartesian basis vector. Therefore,
$\mathbf {e} _{p}\times \mathbf {e} _{q}=\varepsilon _{ipq}\mathbf {e} _{i}$
and
$\mathbf {b} _{m}\times \mathbf {b} _{n}={\frac {\partial \mathbf {x} }{\partial q^{m}}}\times {\frac {\partial \mathbf {x} }{\partial q^{n}}}={\frac {\partial (x_{p}\mathbf {e} _{p})}{\partial q^{m}}}\times {\frac {\partial (x_{q}\mathbf {e} _{q})}{\partial q^{n}}}={\frac {\partial x_{p}}{\partial q^{m}}}{\frac {\partial x_{q}}{\partial q^{n}}}\mathbf {e} _{p}\times \mathbf {e} _{q}=\varepsilon _{ipq}{\frac {\partial x_{p}}{\partial q^{m}}}{\frac {\partial x_{q}}{\partial q^{n}}}\mathbf {e} _{i}.$
Hence,
$(\mathbf {b} _{m}\times \mathbf {b} _{n})\cdot \mathbf {b} _{s}=\varepsilon _{ipq}{\frac {\partial x_{p}}{\partial q^{m}}}{\frac {\partial x_{q}}{\partial q^{n}}}{\frac {\partial x_{i}}{\partial q^{s}}}$
Returning to the vector product and using the relations:
${\hat {u}}_{j}={\frac {\partial x_{j}}{\partial q^{m}}}u^{m},\quad {\hat {v}}_{k}={\frac {\partial x_{k}}{\partial q^{n}}}v^{n},\quad \mathbf {e} _{i}={\frac {\partial x_{i}}{\partial q^{s}}}\mathbf {b} ^{s},$
gives us:
$\mathbf {u} \times \mathbf {v} =\varepsilon _{ijk}{\hat {u}}_{j}{\hat {v}}_{k}\mathbf {e} _{i}=\varepsilon _{ijk}{\frac {\partial x_{j}}{\partial q^{m}}}{\frac {\partial x_{k}}{\partial q^{n}}}{\frac {\partial x_{i}}{\partial q^{s}}}u^{m}v^{n}\mathbf {b} ^{s}=[(\mathbf {b} _{m}\times \mathbf {b} _{n})\cdot \mathbf {b} _{s}]u^{m}v^{n}\mathbf {b} ^{s}={\mathcal {E}}_{smn}u^{m}v^{n}\mathbf {b} ^{s}$
Identity map
The identity map ${\mathsf {I}}$ defined by ${\mathsf {I}}\cdot \mathbf {v} =\mathbf {v} $ can be shown to be[4]: 39
${\mathsf {I}}=g^{ij}\mathbf {b} _{i}\otimes \mathbf {b} _{j}=g_{ij}\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}=\mathbf {b} _{i}\otimes \mathbf {b} ^{i}=\mathbf {b} ^{i}\otimes \mathbf {b} _{i}$
Action of a second-order tensor on a vector
The action $\mathbf {v} ={\boldsymbol {S}}\mathbf {u} $ can be expressed in curvilinear coordinates as
$v^{i}\mathbf {b} _{i}=S^{ij}u_{j}\mathbf {b} _{i}=S_{j}^{i}u^{j}\mathbf {b} _{i};\qquad v_{i}\mathbf {b} ^{i}=S_{ij}u^{i}\mathbf {b} ^{i}=S_{i}^{j}u_{j}\mathbf {b} ^{i}$
Inner product of two second-order tensors
The inner product of two second-order tensors ${\boldsymbol {U}}={\boldsymbol {S}}\cdot {\boldsymbol {T}}$ can be expressed in curvilinear coordinates as
$U_{ij}\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}=S_{ik}T_{.j}^{k}\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}=S_{i}^{.k}T_{kj}\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}$
Alternatively,
${\boldsymbol {U}}=S^{ij}T_{.n}^{m}g_{jm}\mathbf {b} _{i}\otimes \mathbf {b} ^{n}=S_{.m}^{i}T_{.n}^{m}\mathbf {b} _{i}\otimes \mathbf {b} ^{n}=S^{ij}T_{jn}\mathbf {b} _{i}\otimes \mathbf {b} ^{n}$
Determinant of a second-order tensor
If ${\boldsymbol {S}}$ is a second-order tensor, then the determinant is defined by the relation
$\left[{\boldsymbol {S}}\mathbf {u} ,{\boldsymbol {S}}\mathbf {v} ,{\boldsymbol {S}}\mathbf {w} \right]=\det {\boldsymbol {S}}\left[\mathbf {u} ,\mathbf {v} ,\mathbf {w} \right]$
where $\mathbf {u} ,\mathbf {v} ,\mathbf {w} $ are arbitrary vectors and
$\left[\mathbf {u} ,\mathbf {v} ,\mathbf {w} \right]:=\mathbf {u} \cdot (\mathbf {v} \times \mathbf {w} ).$
Relations between curvilinear and Cartesian basis vectors
Let (e1, e2, e3) be the usual Cartesian basis vectors for the Euclidean space of interest and let
$\mathbf {b} _{i}={\boldsymbol {F}}\mathbf {e} _{i}$
where Fi is a second-order transformation tensor that maps ei to bi. Then,
$\mathbf {b} _{i}\otimes \mathbf {e} _{i}=({\boldsymbol {F}}\mathbf {e} _{i})\otimes \mathbf {e} _{i}={\boldsymbol {F}}(\mathbf {e} _{i}\otimes \mathbf {e} _{i})={\boldsymbol {F}}~.$
From this relation we can show that
$\mathbf {b} ^{i}={\boldsymbol {F}}^{-{\rm {T}}}\mathbf {e} ^{i}~;~~g^{ij}=[{\boldsymbol {F}}^{-{\rm {1}}}{\boldsymbol {F}}^{-{\rm {T}}}]_{ij}~;~~g_{ij}=[g^{ij}]^{-1}=[{\boldsymbol {F}}^{\rm {T}}{\boldsymbol {F}}]_{ij}$
Let $J:=\det {\boldsymbol {F}}$ be the Jacobian of the transformation. Then, from the definition of the determinant,
$\left[\mathbf {b} _{1},\mathbf {b} _{2},\mathbf {b} _{3}\right]=\det {\boldsymbol {F}}\left[\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}\right]~.$
Since
$\left[\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}\right]=1$
we have
$J=\det {\boldsymbol {F}}=\left[\mathbf {b} _{1},\mathbf {b} _{2},\mathbf {b} _{3}\right]=\mathbf {b} _{1}\cdot (\mathbf {b} _{2}\times \mathbf {b} _{3})$
A number of interesting results can be derived using the above relations.
First, consider
$g:=\det[g_{ij}]$
Then
$g=\det[{\boldsymbol {F}}^{\rm {T}}]\cdot \det[{\boldsymbol {F}}]=J\cdot J=J^{2}$
Similarly, we can show that
$\det[g^{ij}]={\cfrac {1}{J^{2}}}$
Therefore, using the fact that $[g^{ij}]=[g_{ij}]^{-1}$,
${\cfrac {\partial g}{\partial g_{ij}}}=2~J~{\cfrac {\partial J}{\partial g_{ij}}}=g~g^{ij}$
Another interesting relation is derived below. Recall that
$\mathbf {b} ^{i}\cdot \mathbf {b} _{j}=\delta _{j}^{i}\quad \Rightarrow \quad \mathbf {b} ^{1}\cdot \mathbf {b} _{1}=1,~\mathbf {b} ^{1}\cdot \mathbf {b} _{2}=\mathbf {b} ^{1}\cdot \mathbf {b} _{3}=0\quad \Rightarrow \quad \mathbf {b} ^{1}=A~(\mathbf {b} _{2}\times \mathbf {b} _{3})$
where A is a, yet undetermined, constant. Then
$\mathbf {b} ^{1}\cdot \mathbf {b} _{1}=A~\mathbf {b} _{1}\cdot (\mathbf {b} _{2}\times \mathbf {b} _{3})=AJ=1\quad \Rightarrow \quad A={\cfrac {1}{J}}$
This observation leads to the relations
$\mathbf {b} ^{1}={\cfrac {1}{J}}(\mathbf {b} _{2}\times \mathbf {b} _{3})~;~~\mathbf {b} ^{2}={\cfrac {1}{J}}(\mathbf {b} _{3}\times \mathbf {b} _{1})~;~~\mathbf {b} ^{3}={\cfrac {1}{J}}(\mathbf {b} _{1}\times \mathbf {b} _{2})$
In index notation,
$\varepsilon _{ijk}~\mathbf {b} ^{k}={\cfrac {1}{J}}(\mathbf {b} _{i}\times \mathbf {b} _{j})={\cfrac {1}{\sqrt {g}}}(\mathbf {b} _{i}\times \mathbf {b} _{j})$
where $\varepsilon _{ijk}$ is the usual permutation symbol.
We have not identified an explicit expression for the transformation tensor F because an alternative form of the mapping between curvilinear and Cartesian bases is more useful. Assuming a sufficient degree of smoothness in the mapping (and a bit of abuse of notation), we have
$\mathbf {b} _{i}={\cfrac {\partial \mathbf {x} }{\partial q^{i}}}={\cfrac {\partial \mathbf {x} }{\partial x_{j}}}~{\cfrac {\partial x_{j}}{\partial q^{i}}}=\mathbf {e} _{j}~{\cfrac {\partial x_{j}}{\partial q^{i}}}$
Similarly,
$\mathbf {e} _{i}=\mathbf {b} _{j}~{\cfrac {\partial q^{j}}{\partial x_{i}}}$
From these results we have
$\mathbf {e} ^{k}\cdot \mathbf {b} _{i}={\frac {\partial x_{k}}{\partial q^{i}}}\quad \Rightarrow \quad {\frac {\partial x_{k}}{\partial q^{i}}}~\mathbf {b} ^{i}=\mathbf {e} ^{k}\cdot (\mathbf {b} _{i}\otimes \mathbf {b} ^{i})=\mathbf {e} ^{k}$
and
$\mathbf {b} ^{k}={\frac {\partial q^{k}}{\partial x_{i}}}~\mathbf {e} ^{i}$
Vector and tensor calculus in three-dimensional curvilinear coordinates
Note: the Einstein summation convention of summing on repeated indices is used below.
Simmonds,[4] in his book on tensor analysis, quotes Albert Einstein saying[7]
The magic of this theory will hardly fail to impose itself on anybody who has truly understood it; it represents a genuine triumph of the method of absolute differential calculus, founded by Gauss, Riemann, Ricci, and Levi-Civita.
Vector and tensor calculus in general curvilinear coordinates is used in tensor analysis on four-dimensional curvilinear manifolds in general relativity,[8] in the mechanics of curved shells,[6] in examining the invariance properties of Maxwell's equations which has been of interest in metamaterials[9][10] and in many other fields.
Some useful relations in the calculus of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden,[2] Simmonds,[4] Green and Zerna,[1] Basar and Weichert,[5] and Ciarlet.[6]
Basic definitions
Let the position of a point in space be characterized by three coordinate variables $(q^{1},q^{2},q^{3})$.
The coordinate curve q1 represents a curve on which q2, q3 are constant. Let x be the position vector of the point relative to some origin. Then, assuming that such a mapping and its inverse exist and are continuous, we can write [2]: 55
$\mathbf {x} ={\boldsymbol {\varphi }}(q^{1},q^{2},q^{3})~;~~q^{i}=\psi ^{i}(\mathbf {x} )=[{\boldsymbol {\varphi }}^{-1}(\mathbf {x} )]^{i}$
The fields ψi(x) are called the curvilinear coordinate functions of the curvilinear coordinate system ψ(x) = φ−1(x).
The qi coordinate curves are defined by the one-parameter family of functions given by
$\mathbf {x} _{i}(\alpha )={\boldsymbol {\varphi }}(\alpha ,q^{j},q^{k})~,~~i\neq j\neq k$
with qj, qk fixed.
Tangent vector to coordinate curves
The tangent vector to the curve xi at the point xi(α) (or to the coordinate curve qi at the point x) is
${\cfrac {\rm {{d}\mathbf {x} _{i}}}{\rm {{d}\alpha }}}\equiv {\cfrac {\partial \mathbf {x} }{\partial q^{i}}}$
Scalar field
Let f(x) be a scalar field in space. Then
$f(\mathbf {x} )=f[{\boldsymbol {\varphi }}(q^{1},q^{2},q^{3})]=f_{\varphi }(q^{1},q^{2},q^{3})$
The gradient of the field f is defined by
$[{\boldsymbol {\nabla }}f(\mathbf {x} )]\cdot \mathbf {c} ={\cfrac {\rm {d}}{\rm {{d}\alpha }}}f(\mathbf {x} +\alpha \mathbf {c} ){\biggr |}_{\alpha =0}$
where c is an arbitrary constant vector. If we define the components ci of c are such that
$q^{i}+\alpha ~c^{i}=\psi ^{i}(\mathbf {x} +\alpha ~\mathbf {c} )$
then
$[{\boldsymbol {\nabla }}f(\mathbf {x} )]\cdot \mathbf {c} ={\cfrac {\rm {d}}{\rm {{d}\alpha }}}f_{\varphi }(q^{1}+\alpha ~c^{1},q^{2}+\alpha ~c^{2},q^{3}+\alpha ~c^{3}){\biggr |}_{\alpha =0}={\cfrac {\partial f_{\varphi }}{\partial q^{i}}}~c^{i}={\cfrac {\partial f}{\partial q^{i}}}~c^{i}$
If we set $f(\mathbf {x} )=\psi ^{i}(\mathbf {x} )$, then since $q^{i}=\psi ^{i}(\mathbf {x} )$, we have
$[{\boldsymbol {\nabla }}\psi ^{i}(\mathbf {x} )]\cdot \mathbf {c} ={\cfrac {\partial \psi ^{i}}{\partial q^{j}}}~c^{j}=c^{i}$
which provides a means of extracting the contravariant component of a vector c.
If bi is the covariant (or natural) basis at a point, and if bi is the contravariant (or reciprocal) basis at that point, then
$[{\boldsymbol {\nabla }}f(\mathbf {x} )]\cdot \mathbf {c} ={\cfrac {\partial f}{\partial q^{i}}}~c^{i}=\left({\cfrac {\partial f}{\partial q^{i}}}~\mathbf {b} ^{i}\right)\left(c^{i}~\mathbf {b} _{i}\right)\quad \Rightarrow \quad {\boldsymbol {\nabla }}f(\mathbf {x} )={\cfrac {\partial f}{\partial q^{i}}}~\mathbf {b} ^{i}$
A brief rationale for this choice of basis is given in the next section.
Vector field
A similar process can be used to arrive at the gradient of a vector field f(x). The gradient is given by
$[{\boldsymbol {\nabla }}\mathbf {f} (\mathbf {x} )]\cdot \mathbf {c} ={\cfrac {\partial \mathbf {f} }{\partial q^{i}}}~c^{i}$
If we consider the gradient of the position vector field r(x) = x, then we can show that
$\mathbf {c} ={\cfrac {\partial \mathbf {x} }{\partial q^{i}}}~c^{i}=\mathbf {b} _{i}(\mathbf {x} )~c^{i}~;~~\mathbf {b} _{i}(\mathbf {x} ):={\cfrac {\partial \mathbf {x} }{\partial q^{i}}}$
The vector field bi is tangent to the qi coordinate curve and forms a natural basis at each point on the curve. This basis, as discussed at the beginning of this article, is also called the covariant curvilinear basis. We can also define a reciprocal basis, or contravariant curvilinear basis, bi. All the algebraic relations between the basis vectors, as discussed in the section on tensor algebra, apply for the natural basis and its reciprocal at each point x.
Since c is arbitrary, we can write
${\boldsymbol {\nabla }}\mathbf {f} (\mathbf {x} )={\cfrac {\partial \mathbf {f} }{\partial q^{i}}}\otimes \mathbf {b} ^{i}$
Note that the contravariant basis vector bi is perpendicular to the surface of constant ψi and is given by
$\mathbf {b} ^{i}={\boldsymbol {\nabla }}\psi ^{i}$
Christoffel symbols of the first kind
The Christoffel symbols of the first kind are defined as
$\mathbf {b} _{i,j}={\frac {\partial \mathbf {b} _{i}}{\partial q^{j}}}:=\Gamma _{ijk}~\mathbf {b} ^{k}\quad \Rightarrow \quad \mathbf {b} _{i,j}\cdot \mathbf {b} _{l}=\Gamma _{ijl}$
To express Γijk in terms of gij we note that
${\begin{aligned}g_{ij,k}&=(\mathbf {b} _{i}\cdot \mathbf {b} _{j})_{,k}=\mathbf {b} _{i,k}\cdot \mathbf {b} _{j}+\mathbf {b} _{i}\cdot \mathbf {b} _{j,k}=\Gamma _{ikj}+\Gamma _{jki}\\g_{ik,j}&=(\mathbf {b} _{i}\cdot \mathbf {b} _{k})_{,j}=\mathbf {b} _{i,j}\cdot \mathbf {b} _{k}+\mathbf {b} _{i}\cdot \mathbf {b} _{k,j}=\Gamma _{ijk}+\Gamma _{kji}\\g_{jk,i}&=(\mathbf {b} _{j}\cdot \mathbf {b} _{k})_{,i}=\mathbf {b} _{j,i}\cdot \mathbf {b} _{k}+\mathbf {b} _{j}\cdot \mathbf {b} _{k,i}=\Gamma _{jik}+\Gamma _{kij}\end{aligned}}$
Since bi,j = bj,i we have Γijk = Γjik. Using these to rearrange the above relations gives
$\Gamma _{ijk}={\frac {1}{2}}(g_{ik,j}+g_{jk,i}-g_{ij,k})={\frac {1}{2}}[(\mathbf {b} _{i}\cdot \mathbf {b} _{k})_{,j}+(\mathbf {b} _{j}\cdot \mathbf {b} _{k})_{,i}-(\mathbf {b} _{i}\cdot \mathbf {b} _{j})_{,k}]$
Christoffel symbols of the second kind
The Christoffel symbols of the second kind are defined as
$\Gamma _{ij}^{k}=\Gamma _{ji}^{k}$
in which
${\cfrac {\partial \mathbf {b} _{i}}{\partial q^{j}}}=\Gamma _{ij}^{k}~\mathbf {b} _{k}$
This implies that
$\Gamma _{ij}^{k}={\cfrac {\partial \mathbf {b} _{i}}{\partial q^{j}}}\cdot \mathbf {b} ^{k}=-\mathbf {b} _{i}\cdot {\cfrac {\partial \mathbf {b} ^{k}}{\partial q^{j}}}$
Other relations that follow are
${\cfrac {\partial \mathbf {b} ^{i}}{\partial q^{j}}}=-\Gamma _{jk}^{i}~\mathbf {b} ^{k}~;~~{\boldsymbol {\nabla }}\mathbf {b} _{i}=\Gamma _{ij}^{k}~\mathbf {b} _{k}\otimes \mathbf {b} ^{j}~;~~{\boldsymbol {\nabla }}\mathbf {b} ^{i}=-\Gamma _{jk}^{i}~\mathbf {b} ^{k}\otimes \mathbf {b} ^{j}$
Another particularly useful relation, which shows that the Christoffel symbol depends only on the metric tensor and its derivatives, is
$\Gamma _{ij}^{k}={\frac {g^{km}}{2}}\left({\frac {\partial g_{mi}}{\partial q^{j}}}+{\frac {\partial g_{mj}}{\partial q^{i}}}-{\frac {\partial g_{ij}}{\partial q^{m}}}\right)$
Explicit expression for the gradient of a vector field
The following expressions for the gradient of a vector field in curvilinear coordinates are quite useful.
${\begin{aligned}{\boldsymbol {\nabla }}\mathbf {v} &=\left[{\cfrac {\partial v^{i}}{\partial q^{k}}}+\Gamma _{lk}^{i}~v^{l}\right]~\mathbf {b} _{i}\otimes \mathbf {b} ^{k}\\[8pt]&=\left[{\cfrac {\partial v_{i}}{\partial q^{k}}}-\Gamma _{ki}^{l}~v_{l}\right]~\mathbf {b} ^{i}\otimes \mathbf {b} ^{k}\end{aligned}}$
Representing a physical vector field
The vector field v can be represented as
$\mathbf {v} =v_{i}~\mathbf {b} ^{i}={\hat {v}}_{i}~{\hat {\mathbf {b} }}^{i}$
where $v_{i}$ are the covariant components of the field, ${\hat {v}}_{i}$ are the physical components, and (no summation)
${\hat {\mathbf {b} }}^{i}={\cfrac {\mathbf {b} ^{i}}{\sqrt {g^{ii}}}}$
is the normalized contravariant basis vector.
Second-order tensor field
The gradient of a second order tensor field can similarly be expressed as
${\boldsymbol {\nabla }}{\boldsymbol {S}}={\frac {\partial {\boldsymbol {S}}}{\partial q^{i}}}\otimes \mathbf {b} ^{i}$
Explicit expressions for the gradient
If we consider the expression for the tensor in terms of a contravariant basis, then
${\boldsymbol {\nabla }}{\boldsymbol {S}}={\frac {\partial }{\partial q^{k}}}[S_{ij}~\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}]\otimes \mathbf {b} ^{k}=\left[{\frac {\partial S_{ij}}{\partial q^{k}}}-\Gamma _{ki}^{l}~S_{lj}-\Gamma _{kj}^{l}~S_{il}\right]~\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}\otimes \mathbf {b} ^{k}$
We may also write
${\begin{aligned}{\boldsymbol {\nabla }}{\boldsymbol {S}}&=\left[{\cfrac {\partial S^{ij}}{\partial q^{k}}}+\Gamma _{kl}^{i}~S^{lj}+\Gamma _{kl}^{j}~S^{il}\right]~\mathbf {b} _{i}\otimes \mathbf {b} _{j}\otimes \mathbf {b} ^{k}\\[8pt]&=\left[{\cfrac {\partial S_{~j}^{i}}{\partial q^{k}}}+\Gamma _{kl}^{i}~S_{~j}^{l}-\Gamma _{kj}^{l}~S_{~l}^{i}\right]~\mathbf {b} _{i}\otimes \mathbf {b} ^{j}\otimes \mathbf {b} ^{k}\\[8pt]&=\left[{\cfrac {\partial S_{i}^{~j}}{\partial q^{k}}}-\Gamma _{ik}^{l}~S_{l}^{~j}+\Gamma _{kl}^{j}~S_{i}^{~l}\right]~\mathbf {b} ^{i}\otimes \mathbf {b} _{j}\otimes \mathbf {b} ^{k}\end{aligned}}$
Representing a physical second-order tensor field
The physical components of a second-order tensor field can be obtained by using a normalized contravariant basis, i.e.,
${\boldsymbol {S}}=S_{ij}~\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}={\hat {S}}_{ij}~{\hat {\mathbf {b} }}^{i}\otimes {\hat {\mathbf {b} }}^{j}$
where the hatted basis vectors have been normalized. This implies that (again no summation)
${\hat {S}}_{ij}=S_{ij}~{\sqrt {g^{ii}~g^{jj}}}$
Vector field
The divergence of a vector field ($\mathbf {v} $)is defined as
$\operatorname {div} ~\mathbf {v} ={\boldsymbol {\nabla }}\cdot \mathbf {v} ={\text{tr}}({\boldsymbol {\nabla }}\mathbf {v} )$
In terms of components with respect to a curvilinear basis
${\boldsymbol {\nabla }}\cdot \mathbf {v} ={\cfrac {\partial v^{i}}{\partial q^{i}}}+\Gamma _{\ell i}^{i}~v^{\ell }=\left[{\cfrac {\partial v_{i}}{\partial q^{j}}}-\Gamma _{ji}^{\ell }~v_{\ell }\right]~g^{ij}$
An alternative equation for the divergence of a vector field is frequently used. To derive this relation recall that
${\boldsymbol {\nabla }}\cdot \mathbf {v} ={\frac {\partial v^{i}}{\partial q^{i}}}+\Gamma _{\ell i}^{i}~v^{\ell }$
Now,
$\Gamma _{\ell i}^{i}=\Gamma _{i\ell }^{i}={\cfrac {g^{mi}}{2}}\left[{\frac {\partial g_{im}}{\partial q^{\ell }}}+{\frac {\partial g_{\ell m}}{\partial q^{i}}}-{\frac {\partial g_{il}}{\partial q^{m}}}\right]$
Noting that, due to the symmetry of ${\boldsymbol {g}}$,
$g^{mi}~{\frac {\partial g_{\ell m}}{\partial q^{i}}}=g^{mi}~{\frac {\partial g_{i\ell }}{\partial q^{m}}}$
we have
${\boldsymbol {\nabla }}\cdot \mathbf {v} ={\frac {\partial v^{i}}{\partial q^{i}}}+{\cfrac {g^{mi}}{2}}~{\frac {\partial g_{im}}{\partial q^{\ell }}}~v^{\ell }$
Recall that if [gij] is the matrix whose components are gij, then the inverse of the matrix is $[g_{ij}]^{-1}=[g^{ij}]$. The inverse of the matrix is given by
$[g^{ij}]=[g_{ij}]^{-1}={\cfrac {A^{ij}}{g}}~;~~g:=\det([g_{ij}])=\det {\boldsymbol {g}}$
where Aij are the Cofactor matrix of the components gij. From matrix algebra we have
$g=\det([g_{ij}])=\sum _{i}g_{ij}~A^{ij}\quad \Rightarrow \quad {\frac {\partial g}{\partial g_{ij}}}=A^{ij}$
Hence,
$[g^{ij}]={\cfrac {1}{g}}~{\frac {\partial g}{\partial g_{ij}}}$
Plugging this relation into the expression for the divergence gives
${\boldsymbol {\nabla }}\cdot \mathbf {v} ={\frac {\partial v^{i}}{\partial q^{i}}}+{\cfrac {1}{2g}}~{\frac {\partial g}{\partial g_{mi}}}~{\frac {\partial g_{im}}{\partial q^{\ell }}}~v^{\ell }={\frac {\partial v^{i}}{\partial q^{i}}}+{\cfrac {1}{2g}}~{\frac {\partial g}{\partial q^{\ell }}}~v^{\ell }$
A little manipulation leads to the more compact form
${\boldsymbol {\nabla }}\cdot \mathbf {v} ={\cfrac {1}{\sqrt {g}}}~{\frac {\partial }{\partial q^{i}}}(v^{i}~{\sqrt {g}})$
Second-order tensor field
The divergence of a second-order tensor field is defined using
$({\boldsymbol {\nabla }}\cdot {\boldsymbol {S}})\cdot \mathbf {a} ={\boldsymbol {\nabla }}\cdot ({\boldsymbol {S}}\mathbf {a} )$
where a is an arbitrary constant vector. [11] In curvilinear coordinates,
${\begin{aligned}{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&=\left[{\cfrac {\partial S_{ij}}{\partial q^{k}}}-\Gamma _{ki}^{l}~S_{lj}-\Gamma _{kj}^{l}~S_{il}\right]~g^{ik}~\mathbf {b} ^{j}\\[8pt]&=\left[{\cfrac {\partial S^{ij}}{\partial q^{i}}}+\Gamma _{il}^{i}~S^{lj}+\Gamma _{il}^{j}~S^{il}\right]~\mathbf {b} _{j}\\[8pt]&=\left[{\cfrac {\partial S_{~j}^{i}}{\partial q^{i}}}+\Gamma _{il}^{i}~S_{~j}^{l}-\Gamma _{ij}^{l}~S_{~l}^{i}\right]~\mathbf {b} ^{j}\\[8pt]&=\left[{\cfrac {\partial S_{i}^{~j}}{\partial q^{k}}}-\Gamma _{ik}^{l}~S_{l}^{~j}+\Gamma _{kl}^{j}~S_{i}^{~l}\right]~g^{ik}~\mathbf {b} _{j}\end{aligned}}$
Scalar field
The Laplacian of a scalar field φ(x) is defined as
$\nabla ^{2}\varphi :={\boldsymbol {\nabla }}\cdot ({\boldsymbol {\nabla }}\varphi )$ :={\boldsymbol {\nabla }}\cdot ({\boldsymbol {\nabla }}\varphi )}
Using the alternative expression for the divergence of a vector field gives us
$\nabla ^{2}\varphi ={\cfrac {1}{\sqrt {g}}}~{\frac {\partial }{\partial q^{i}}}([{\boldsymbol {\nabla }}\varphi ]^{i}~{\sqrt {g}})$
Now
${\boldsymbol {\nabla }}\varphi ={\frac {\partial \varphi }{\partial q^{l}}}~\mathbf {b} ^{l}=g^{li}~{\frac {\partial \varphi }{\partial q^{l}}}~\mathbf {b} _{i}\quad \Rightarrow \quad [{\boldsymbol {\nabla }}\varphi ]^{i}=g^{li}~{\frac {\partial \varphi }{\partial q^{l}}}$
Therefore,
$\nabla ^{2}\varphi ={\cfrac {1}{\sqrt {g}}}~{\frac {\partial }{\partial q^{i}}}\left(g^{li}~{\frac {\partial \varphi }{\partial q^{l}}}~{\sqrt {g}}\right)$
Curl of a vector field
The curl of a vector field v in covariant curvilinear coordinates can be written as
${\boldsymbol {\nabla }}\times \mathbf {v} ={\mathcal {E}}^{rst}v_{s|r}~\mathbf {b} _{t}$
where
$v_{s|r}=v_{s,r}-\Gamma _{sr}^{i}~v_{i}$
Orthogonal curvilinear coordinates
Assume, for the purposes of this section, that the curvilinear coordinate system is orthogonal, i.e.,
$\mathbf {b} _{i}\cdot \mathbf {b} _{j}={\begin{cases}g_{ii}&{\text{if }}i=j\\0&{\text{if }}i\neq j,\end{cases}}$
or equivalently,
$\mathbf {b} ^{i}\cdot \mathbf {b} ^{j}={\begin{cases}g^{ii}&{\text{if }}i=j\\0&{\text{if }}i\neq j,\end{cases}}$
where $g^{ii}=g_{ii}^{-1}$. As before, $\mathbf {b} _{i},\mathbf {b} _{j}$ are covariant basis vectors and bi, bj are contravariant basis vectors. Also, let (e1, e2, e3) be a background, fixed, Cartesian basis. A list of orthogonal curvilinear coordinates is given below.
Metric tensor in orthogonal curvilinear coordinates
Main article: Metric tensor
Let r(x) be the position vector of the point x with respect to the origin of the coordinate system. The notation can be simplified by noting that x = r(x). At each point we can construct a small line element dx. The square of the length of the line element is the scalar product dx • dx and is called the metric of the space. Recall that the space of interest is assumed to be Euclidean when we talk of curvilinear coordinates. Let us express the position vector in terms of the background, fixed, Cartesian basis, i.e.,
$\mathbf {x} =\sum _{i=1}^{3}x_{i}~\mathbf {e} _{i}$
Using the chain rule, we can then express dx in terms of three-dimensional orthogonal curvilinear coordinates (q1, q2, q3) as
$\mathrm {d} \mathbf {x} =\sum _{i=1}^{3}\sum _{j=1}^{3}\left({\cfrac {\partial x_{i}}{\partial q^{j}}}~\mathbf {e} _{i}\right)\mathrm {d} q^{j}$
Therefore, the metric is given by
$\mathrm {d} \mathbf {x} \cdot \mathrm {d} \mathbf {x} =\sum _{i=1}^{3}\sum _{j=1}^{3}\sum _{k=1}^{3}{\cfrac {\partial x_{i}}{\partial q^{j}}}~{\cfrac {\partial x_{i}}{\partial q^{k}}}~\mathrm {d} q^{j}~\mathrm {d} q^{k}$
The symmetric quantity
$g_{ij}(q^{i},q^{j})=\sum _{k=1}^{3}{\cfrac {\partial x_{k}}{\partial q^{i}}}~{\cfrac {\partial x_{k}}{\partial q^{j}}}=\mathbf {b} _{i}\cdot \mathbf {b} _{j}$
is called the fundamental (or metric) tensor of the Euclidean space in curvilinear coordinates.
Note also that
$g_{ij}={\cfrac {\partial \mathbf {x} }{\partial q^{i}}}\cdot {\cfrac {\partial \mathbf {x} }{\partial q^{j}}}=\left(\sum _{k}h_{ki}~\mathbf {e} _{k}\right)\cdot \left(\sum _{m}h_{mj}~\mathbf {e} _{m}\right)=\sum _{k}h_{ki}~h_{kj}$
where hij are the Lamé coefficients.
If we define the scale factors, hi, using
$\mathbf {b} _{i}\cdot \mathbf {b} _{i}=g_{ii}=\sum _{k}h_{ki}^{2}=:h_{i}^{2}\quad \Rightarrow \quad \left|{\cfrac {\partial \mathbf {x} }{\partial q^{i}}}\right|=\left|\mathbf {b} _{i}\right|={\sqrt {g_{ii}}}=h_{i}$
we get a relation between the fundamental tensor and the Lamé coefficients.
Example: Polar coordinates
If we consider polar coordinates for R2, note that
$(x,y)=(r\cos \theta ,r\sin \theta )$
(r, θ) are the curvilinear coordinates, and the Jacobian determinant of the transformation (r,θ) → (r cos θ, r sin θ) is r.
The orthogonal basis vectors are br = (cos θ, sin θ), bθ = (−r sin θ, r cos θ). The normalized basis vectors are er = (cos θ, sin θ), eθ = (−sin θ, cos θ) and the scale factors are hr = 1 and hθ= r. The fundamental tensor is g11 =1, g22 =r2, g12 = g21 =0.
Line and surface integrals
If we wish to use curvilinear coordinates for vector calculus calculations, adjustments need to be made in the calculation of line, surface and volume integrals. For simplicity, we again restrict the discussion to three dimensions and orthogonal curvilinear coordinates. However, the same arguments apply for $n$-dimensional problems though there are some additional terms in the expressions when the coordinate system is not orthogonal.
Line integrals
Normally in the calculation of line integrals we are interested in calculating
$\int _{C}f\,ds=\int _{a}^{b}f(\mathbf {x} (t))\left|{\partial \mathbf {x} \over \partial t}\right|\;dt$
where x(t) parametrizes C in Cartesian coordinates. In curvilinear coordinates, the term
$\left|{\partial \mathbf {x} \over \partial t}\right|=\left|\sum _{i=1}^{3}{\partial \mathbf {x} \over \partial q^{i}}{\partial q^{i} \over \partial t}\right|$
by the chain rule. And from the definition of the Lamé coefficients,
${\partial \mathbf {x} \over \partial q^{i}}=\sum _{k}h_{ki}~\mathbf {e} _{k}$
and thus
${\begin{aligned}\left|{\partial \mathbf {x} \over \partial t}\right|&=\left|\sum _{k}\left(\sum _{i}h_{ki}~{\cfrac {\partial q^{i}}{\partial t}}\right)\mathbf {e} _{k}\right|\\[8pt]&={\sqrt {\sum _{i}\sum _{j}\sum _{k}h_{ki}~h_{kj}{\cfrac {\partial q^{i}}{\partial t}}{\cfrac {\partial q^{j}}{\partial t}}}}={\sqrt {\sum _{i}\sum _{j}g_{ij}~{\cfrac {\partial q^{i}}{\partial t}}{\cfrac {\partial q^{j}}{\partial t}}}}\end{aligned}}$
Now, since $g_{ij}=0$ when $i\neq j$, we have
$\left|{\partial \mathbf {x} \over \partial t}\right|={\sqrt {\sum _{i}g_{ii}~\left({\cfrac {\partial q^{i}}{\partial t}}\right)^{2}}}={\sqrt {\sum _{i}h_{i}^{2}~\left({\cfrac {\partial q^{i}}{\partial t}}\right)^{2}}}$
and we can proceed normally.
Surface integrals
Likewise, if we are interested in a surface integral, the relevant calculation, with the parameterization of the surface in Cartesian coordinates is:
$\int _{S}f\,dS=\iint _{T}f(\mathbf {x} (s,t))\left|{\partial \mathbf {x} \over \partial s}\times {\partial \mathbf {x} \over \partial t}\right|\,ds\,dt$
Again, in curvilinear coordinates, we have
$\left|{\partial \mathbf {x} \over \partial s}\times {\partial \mathbf {x} \over \partial t}\right|=\left|\left(\sum _{i}{\partial \mathbf {x} \over \partial q^{i}}{\partial q^{i} \over \partial s}\right)\times \left(\sum _{j}{\partial \mathbf {x} \over \partial q^{j}}{\partial q^{j} \over \partial t}\right)\right|$
and we make use of the definition of curvilinear coordinates again to yield
${\partial \mathbf {x} \over \partial q^{i}}{\partial q^{i} \over \partial s}=\sum _{k}\left(\sum _{i=1}^{3}h_{ki}~{\partial q^{i} \over \partial s}\right)\mathbf {e} _{k}~;~~{\partial \mathbf {x} \over \partial q^{j}}{\partial q^{j} \over \partial t}=\sum _{m}\left(\sum _{j=1}^{3}h_{mj}~{\partial q^{j} \over \partial t}\right)\mathbf {e} _{m}$
Therefore,
${\begin{aligned}\left|{\partial \mathbf {x} \over \partial s}\times {\partial \mathbf {x} \over \partial t}\right|&=\left|\sum _{k}\sum _{m}\left(\sum _{i=1}^{3}h_{ki}~{\partial q^{i} \over \partial s}\right)\left(\sum _{j=1}^{3}h_{mj}~{\partial q^{j} \over \partial t}\right)\mathbf {e} _{k}\times \mathbf {e} _{m}\right|\\[8pt]&=\left|\sum _{p}\sum _{k}\sum _{m}{\mathcal {E}}_{kmp}\left(\sum _{i=1}^{3}h_{ki}~{\partial q^{i} \over \partial s}\right)\left(\sum _{j=1}^{3}h_{mj}~{\partial q^{j} \over \partial t}\right)\mathbf {e} _{p}\right|\end{aligned}}$
where ${\mathcal {E}}$ is the permutation symbol.
In determinant form, the cross product in terms of curvilinear coordinates will be:
${\begin{vmatrix}\mathbf {e} _{1}&\mathbf {e} _{2}&\mathbf {e} _{3}\\&&\\\sum _{i}h_{1i}{\partial q^{i} \over \partial s}&\sum _{i}h_{2i}{\partial q^{i} \over \partial s}&\sum _{i}h_{3i}{\partial q^{i} \over \partial s}\\&&\\\sum _{j}h_{1j}{\partial q^{j} \over \partial t}&\sum _{j}h_{2j}{\partial q^{j} \over \partial t}&\sum _{j}h_{3j}{\partial q^{j} \over \partial t}\end{vmatrix}}$
Grad, curl, div, Laplacian
In orthogonal curvilinear coordinates of 3 dimensions, where
$\mathbf {b} ^{i}=\sum _{k}g^{ik}~\mathbf {b} _{k}~;~~g^{ii}={\cfrac {1}{g_{ii}}}={\cfrac {1}{h_{i}^{2}}}$
one can express the gradient of a scalar or vector field as
$\nabla \varphi =\sum _{i}{\partial \varphi \over \partial q^{i}}~\mathbf {b} ^{i}=\sum _{i}\sum _{j}{\partial \varphi \over \partial q^{i}}~g^{ij}~\mathbf {b} _{j}=\sum _{i}{\cfrac {1}{h_{i}^{2}}}~{\partial f \over \partial q^{i}}~\mathbf {b} _{i}~;~~\nabla \mathbf {v} =\sum _{i}{\cfrac {1}{h_{i}^{2}}}~{\partial \mathbf {v} \over \partial q^{i}}\otimes \mathbf {b} _{i}$
For an orthogonal basis
$g=g_{11}~g_{22}~g_{33}=h_{1}^{2}~h_{2}^{2}~h_{3}^{2}\quad \Rightarrow \quad {\sqrt {g}}=h_{1}h_{2}h_{3}$
The divergence of a vector field can then be written as
${\boldsymbol {\nabla }}\cdot \mathbf {v} ={\cfrac {1}{h_{1}h_{2}h_{3}}}~{\frac {\partial }{\partial q^{i}}}(h_{1}h_{2}h_{3}~v^{i})$
Also,
$v^{i}=g^{ik}~v_{k}\quad \Rightarrow v^{1}=g^{11}~v_{1}={\cfrac {v_{1}}{h_{1}^{2}}}~;~~v^{2}=g^{22}~v_{2}={\cfrac {v_{2}}{h_{2}^{2}}}~;~~v^{3}=g^{33}~v_{3}={\cfrac {v_{3}}{h_{3}^{2}}}$
Therefore,
${\boldsymbol {\nabla }}\cdot \mathbf {v} ={\cfrac {1}{h_{1}h_{2}h_{3}}}~\sum _{i}{\frac {\partial }{\partial q^{i}}}\left({\cfrac {h_{1}h_{2}h_{3}}{h_{i}^{2}}}~v_{i}\right)$
We can get an expression for the Laplacian in a similar manner by noting that
$g^{li}~{\frac {\partial \varphi }{\partial q^{l}}}=\left\{g^{11}~{\frac {\partial \varphi }{\partial q^{1}}},g^{22}~{\frac {\partial \varphi }{\partial q^{2}}},g^{33}~{\frac {\partial \varphi }{\partial q^{3}}}\right\}=\left\{{\cfrac {1}{h_{1}^{2}}}~{\frac {\partial \varphi }{\partial q^{1}}},{\cfrac {1}{h_{2}^{2}}}~{\frac {\partial \varphi }{\partial q^{2}}},{\cfrac {1}{h_{3}^{2}}}~{\frac {\partial \varphi }{\partial q^{3}}}\right\}$
Then we have
$\nabla ^{2}\varphi ={\cfrac {1}{h_{1}h_{2}h_{3}}}~\sum _{i}{\frac {\partial }{\partial q^{i}}}\left({\cfrac {h_{1}h_{2}h_{3}}{h_{i}^{2}}}~{\frac {\partial \varphi }{\partial q^{i}}}\right)$
The expressions for the gradient, divergence, and Laplacian can be directly extended to n-dimensions.
The curl of a vector field is given by
$\nabla \times \mathbf {v} ={\frac {1}{h_{1}h_{2}h_{3}}}\sum _{i=1}^{n}\mathbf {e} _{i}\sum _{jk}\varepsilon _{ijk}h_{i}{\frac {\partial (h_{k}v_{k})}{\partial q^{j}}}$
where εijk is the Levi-Civita symbol.
Example: Cylindrical polar coordinates
For cylindrical coordinates we have
$(x_{1},x_{2},x_{3})=\mathbf {x} ={\boldsymbol {\varphi }}(q^{1},q^{2},q^{3})={\boldsymbol {\varphi }}(r,\theta ,z)=\{r\cos \theta ,r\sin \theta ,z\}$
and
$\{\psi ^{1}(\mathbf {x} ),\psi ^{2}(\mathbf {x} ),\psi ^{3}(\mathbf {x} )\}=(q^{1},q^{2},q^{3})\equiv (r,\theta ,z)=\{{\sqrt {x_{1}^{2}+x_{2}^{2}}},\tan ^{-1}(x_{2}/x_{1}),x_{3}\}$
where
$0<r<\infty ~,~~0<\theta <2\pi ~,~~-\infty <z<\infty $
Then the covariant and contravariant basis vectors are
${\begin{aligned}\mathbf {b} _{1}&=\mathbf {e} _{r}=\mathbf {b} ^{1}\\\mathbf {b} _{2}&=r~\mathbf {e} _{\theta }=r^{2}~\mathbf {b} ^{2}\\\mathbf {b} _{3}&=\mathbf {e} _{z}=\mathbf {b} ^{3}\end{aligned}}$
where $\mathbf {e} _{r},\mathbf {e} _{\theta },\mathbf {e} _{z}$ are the unit vectors in the $r,\theta ,z$ directions.
Note that the components of the metric tensor are such that
$g^{ij}=g_{ij}=0(i\neq j)~;~~{\sqrt {g^{11}}}=1,~{\sqrt {g^{22}}}={\cfrac {1}{r}},~{\sqrt {g^{33}}}=1$
which shows that the basis is orthogonal.
The non-zero components of the Christoffel symbol of the second kind are
$\Gamma _{12}^{2}=\Gamma _{21}^{2}={\cfrac {1}{r}}~;~~\Gamma _{22}^{1}=-r$
Representing a physical vector field
The normalized contravariant basis vectors in cylindrical polar coordinates are
${\hat {\mathbf {b} }}^{1}=\mathbf {e} _{r}~;~~{\hat {\mathbf {b} }}^{2}=\mathbf {e} _{\theta }~;~~{\hat {\mathbf {b} }}^{3}=\mathbf {e} _{z}$
and the physical components of a vector v are
$({\hat {v}}_{1},{\hat {v}}_{2},{\hat {v}}_{3})=(v_{1},v_{2}/r,v_{3})=:(v_{r},v_{\theta },v_{z})$
Gradient of a scalar field
The gradient of a scalar field, f(x), in cylindrical coordinates can now be computed from the general expression in curvilinear coordinates and has the form
${\boldsymbol {\nabla }}f={\cfrac {\partial f}{\partial r}}~\mathbf {e} _{r}+{\cfrac {1}{r}}~{\cfrac {\partial f}{\partial \theta }}~\mathbf {e} _{\theta }+{\cfrac {\partial f}{\partial z}}~\mathbf {e} _{z}$
Gradient of a vector field
Similarly, the gradient of a vector field, v(x), in cylindrical coordinates can be shown to be
${\begin{aligned}{\boldsymbol {\nabla }}\mathbf {v} &={\cfrac {\partial v_{r}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left({\cfrac {\partial v_{r}}{\partial \theta }}-v_{\theta }\right)~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\cfrac {\partial v_{r}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\\[8pt]&+{\cfrac {\partial v_{\theta }}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left({\cfrac {\partial v_{\theta }}{\partial \theta }}+v_{r}\right)~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\cfrac {\partial v_{\theta }}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\\[8pt]&+{\cfrac {\partial v_{z}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}{\cfrac {\partial v_{z}}{\partial \theta }}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\cfrac {\partial v_{z}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\end{aligned}}$
Divergence of a vector field
Using the equation for the divergence of a vector field in curvilinear coordinates, the divergence in cylindrical coordinates can be shown to be
${\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {v} &={\cfrac {\partial v_{r}}{\partial r}}+{\cfrac {1}{r}}\left({\cfrac {\partial v_{\theta }}{\partial \theta }}+v_{r}\right)+{\cfrac {\partial v_{z}}{\partial z}}\end{aligned}}$
Laplacian of a scalar field
The Laplacian is more easily computed by noting that ${\boldsymbol {\nabla }}^{2}f={\boldsymbol {\nabla }}\cdot {\boldsymbol {\nabla }}f$. In cylindrical polar coordinates
$\mathbf {v} ={\boldsymbol {\nabla }}f=\left[v_{r}~~v_{\theta }~~v_{z}\right]=\left[{\cfrac {\partial f}{\partial r}}~~{\cfrac {1}{r}}{\cfrac {\partial f}{\partial \theta }}~~{\cfrac {\partial f}{\partial z}}\right]$
Hence,
${\boldsymbol {\nabla }}\cdot \mathbf {v} ={\boldsymbol {\nabla }}^{2}f={\cfrac {\partial ^{2}f}{\partial r^{2}}}+{\cfrac {1}{r}}\left({\cfrac {1}{r}}{\cfrac {\partial ^{2}f}{\partial \theta ^{2}}}+{\cfrac {\partial f}{\partial r}}\right)+{\cfrac {\partial ^{2}f}{\partial z^{2}}}={\cfrac {1}{r}}\left[{\cfrac {\partial }{\partial r}}\left(r{\cfrac {\partial f}{\partial r}}\right)\right]+{\cfrac {1}{r^{2}}}{\cfrac {\partial ^{2}f}{\partial \theta ^{2}}}+{\cfrac {\partial ^{2}f}{\partial z^{2}}}$
Representing a physical second-order tensor field
The physical components of a second-order tensor field are those obtained when the tensor is expressed in terms of a normalized contravariant basis. In cylindrical polar coordinates these components are:
${\begin{aligned}{\hat {S}}_{11}&=S_{11}=:S_{rr},&{\hat {S}}_{12}&={\frac {S_{12}}{r}}=:S_{r\theta },&{\hat {S}}_{13}&=S_{13}=:S_{rz}\\[6pt]{\hat {S}}_{21}&={\frac {S_{21}}{r}}=:S_{\theta r},&{\hat {S}}_{22}&={\frac {S_{22}}{r^{2}}}=:S_{\theta \theta },&{\hat {S}}_{23}&={\frac {S_{23}}{r}}=:S_{\theta z}\\[6pt]{\hat {S}}_{31}&=S_{31}=:S_{zr},&{\hat {S}}_{32}&={\frac {S_{32}}{r}}=:S_{z\theta },&{\hat {S}}_{33}&=S_{33}=:S_{zz}\end{aligned}}$
Gradient of a second-order tensor field
Using the above definitions we can show that the gradient of a second-order tensor field in cylindrical polar coordinates can be expressed as
${\begin{aligned}{\boldsymbol {\nabla }}{\boldsymbol {S}}&={\frac {\partial S_{rr}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{rr}}{\partial \theta }}-(S_{\theta r}+S_{r\theta })\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{rr}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{r\theta }}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{r\theta }}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{r\theta }}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{rz}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{rz}}{\partial \theta }}-S_{\theta z}\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{rz}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{\theta r}}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta r}}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{\theta r}}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{\theta \theta }}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta \theta }}{\partial \theta }}+(S_{r\theta }+S_{\theta r})\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{\theta \theta }}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{\theta z}}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta z}}{\partial \theta }}+S_{rz}\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{\theta z}}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{zr}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{zr}}{\partial \theta }}-S_{z\theta }\right]~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{zr}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{z\theta }}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{z\theta }}{\partial \theta }}+S_{zr}\right]~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{z\theta }}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{zz}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}~{\frac {\partial S_{zz}}{\partial \theta }}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{zz}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}\end{aligned}}$
Divergence of a second-order tensor field
The divergence of a second-order tensor field in cylindrical polar coordinates can be obtained from the expression for the gradient by collecting terms where the scalar product of the two outer vectors in the dyadic products is nonzero. Therefore,
${\begin{aligned}{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&={\frac {\partial S_{rr}}{\partial r}}~\mathbf {e} _{r}+{\frac {\partial S_{r\theta }}{\partial r}}~\mathbf {e} _{\theta }+{\frac {\partial S_{rz}}{\partial r}}~\mathbf {e} _{z}\\[8pt]&+{\cfrac {1}{r}}\left[{\frac {\partial S_{r\theta }}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta \theta }}{\partial \theta }}+(S_{r\theta }+S_{\theta r})\right]~\mathbf {e} _{\theta }+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta z}}{\partial \theta }}+S_{rz}\right]~\mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{zr}}{\partial z}}~\mathbf {e} _{r}+{\frac {\partial S_{z\theta }}{\partial z}}~\mathbf {e} _{\theta }+{\frac {\partial S_{zz}}{\partial z}}~\mathbf {e} _{z}\end{aligned}}$
See also
• Covariance and contravariance
• Basic introduction to the mathematics of curved spacetime
• Orthogonal coordinates
• Frenet–Serret formulas
• Covariant derivative
• Tensor derivative (continuum mechanics)
• Curvilinear perspective
• Del in cylindrical and spherical coordinates
References
Notes
1. Green, A. E.; Zerna, W. (1968). Theoretical Elasticity. Oxford University Press. ISBN 0-19-853486-8.
2. Ogden, R. W. (2000). Nonlinear elastic deformations. Dover.
3. Naghdi, P. M. (1972). "Theory of shells and plates". In S. Flügge (ed.). Handbook of Physics. Vol. VIa/2. pp. 425–640.
4. Simmonds, J. G. (1994). A brief on tensor analysis. Springer. ISBN 0-387-90639-8.
5. Basar, Y.; Weichert, D. (2000). Numerical continuum mechanics of solids: fundamental concepts and perspectives. Springer.
6. Ciarlet, P. G. (2000). Theory of Shells. Vol. 1. Elsevier Science.
7. Einstein, A. (1915). "Contribution to the Theory of General Relativity". In Laczos, C. (ed.). The Einstein Decade. p. 213.
8. Misner, C. W.; Thorne, K. S.; Wheeler, J. A. (1973). Gravitation. W. H. Freeman and Co. ISBN 0-7167-0344-0.
9. Greenleaf, A.; Lassas, M.; Uhlmann, G. (2003). "Anisotropic conductivities that cannot be detected by EIT". Physiological Measurement. 24 (2): 413–419. doi:10.1088/0967-3334/24/2/353. PMID 12812426. S2CID 250813768.
10. Leonhardt, U.; Philbin, T. G. (2006). "General relativity in electrical engineering". New Journal of Physics. 8 (10): 247. arXiv:cond-mat/0607418. Bibcode:2006NJPh....8..247L. doi:10.1088/1367-2630/8/10/247. S2CID 12100599.
11. "The divergence of a tensor field". Introduction to Elasticity/Tensors. Wikiversity. Retrieved 2010-11-26.
Further reading
• Spiegel, M. R. (1959). Vector Analysis. New York: Schaum's Outline Series. ISBN 0-07-084378-3.
• Arfken, George (1995). Mathematical Methods for Physicists. Academic Press. ISBN 0-12-059877-9.
External links
• Derivation of Unit Vectors in Curvilinear Coordinates
• MathWorld's page on Curvilinear Coordinates
• Prof. R. Brannon's E-Book on Curvilinear Coordinates
Orthogonal coordinate systems
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| Wikipedia |
Tensor sketch
In statistics, machine learning and algorithms, a tensor sketch is a type of dimensionality reduction that is particularly efficient when applied to vectors that have tensor structure.[1][2] Such a sketch can be used to speed up explicit kernel methods, bilinear pooling in neural networks and is a cornerstone in many numerical linear algebra algorithms.[3]
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Mathematical definition
Mathematically, a dimensionality reduction or sketching matrix is a matrix $M\in \mathbb {R} ^{k\times d}$, where $k<d$, such that for any vector $x\in \mathbb {R} ^{d}$
$|\|Mx\|_{2}-\|x\|_{2}|<\varepsilon \|x\|_{2}$
with high probability. In other words, $M$ preserves the norm of vectors up to a small error.
A tensor sketch has the extra property that if $x=y\otimes z$ for some vectors $y\in \mathbb {R} ^{d_{1}},z\in \mathbb {R} ^{d_{2}}$ such that $d_{1}d_{2}=d$, the transformation $M(y\otimes z)$ can be computed more efficiently. Here $\otimes $ denotes the Kronecker product, rather than the outer product, though the two are related by a flattening.
The speedup is achieved by first rewriting $M(y\otimes z)=M'y\circ M''z$, where $\circ $ denotes the elementwise (Hadamard) product. Each of $M'y$ and $M''z$ can be computed in time $O(kd_{1})$ and $O(kd_{2})$, respectively; including the Hadamard product gives overall time $O(d_{1}d_{2}+kd_{1}+kd_{2})$. In most use cases this method is significantly faster than the full $M(y\otimes z)$ requiring $O(kd)=O(kd_{1}d_{2})$ time.
For higher-order tensors, such as $x=y\otimes z\otimes t$, the savings are even more impressive.
History
The term tensor sketch was coined in 2013[4] describing a technique by Rasmus Pagh[5] from the same year. Originally it was understood using the fast Fourier transform to do fast convolution of count sketches. Later research works generalized it to a much larger class of dimensionality reductions via Tensor random embeddings.
Tensor random embeddings were introduced in 2010 in a paper[6] on differential privacy and were first analyzed by Rudelson et al. in 2012 in the context of sparse recovery.[7]
Avron et al.[8] were the first to study the subspace embedding properties of tensor sketches, particularly focused on applications to polynomial kernels. In this context, the sketch is required not only to preserve the norm of each individual vector with a certain probability but to preserve the norm of all vectors in each individual linear subspace. This is a much stronger property, and it requires larger sketch sizes, but it allows the kernel methods to be used very broadly as explored in the book by David Woodruff.[3]
Tensor random projections
The face-splitting product is defined as the tensor products of the rows (was proposed by V. Slyusar[9] in 1996[10][11][12][13][14] for radar and digital antenna array applications). More directly, let $\mathbf {C} \in \mathbb {R} ^{3\times 3}$ and $\mathbf {D} \in \mathbb {R} ^{3\times 3}$ be two matrices. Then the face-splitting product $\mathbf {C} \bullet \mathbf {D} $ is[10][11][12][13] $\mathbf {C} \bullet \mathbf {D} =\left[{\begin{array}{c }\mathbf {C} _{1}\otimes \mathbf {D} _{1}\\\hline \mathbf {C} _{2}\otimes \mathbf {D} _{2}\\\hline \mathbf {C} _{3}\otimes \mathbf {D} _{3}\\\end{array}}\right]=\left[{\begin{array}{c c c c c c c c c }\mathbf {C} _{1,1}\mathbf {D} _{1,1}&\mathbf {C} _{1,1}\mathbf {D} _{1,2}&\mathbf {C} _{1,1}\mathbf {D} _{1,3}&\mathbf {C} _{1,2}\mathbf {D} _{1,1}&\mathbf {C} _{1,2}\mathbf {D} _{1,2}&\mathbf {C} _{1,2}\mathbf {D} _{1,3}&\mathbf {C} _{1,3}\mathbf {D} _{1,1}&\mathbf {C} _{1,3}\mathbf {D} _{1,2}&\mathbf {C} _{1,3}\mathbf {D} _{1,3}\\\hline \mathbf {C} _{2,1}\mathbf {D} _{2,1}&\mathbf {C} _{2,1}\mathbf {D} _{2,2}&\mathbf {C} _{2,1}\mathbf {D} _{2,3}&\mathbf {C} _{2,2}\mathbf {D} _{2,1}&\mathbf {C} _{2,2}\mathbf {D} _{2,2}&\mathbf {C} _{2,2}\mathbf {D} _{2,3}&\mathbf {C} _{2,3}\mathbf {D} _{2,1}&\mathbf {C} _{2,3}\mathbf {D} _{2,2}&\mathbf {C} _{2,3}\mathbf {D} _{2,3}\\\hline \mathbf {C} _{3,1}\mathbf {D} _{3,1}&\mathbf {C} _{3,1}\mathbf {D} _{3,2}&\mathbf {C} _{3,1}\mathbf {D} _{3,3}&\mathbf {C} _{3,2}\mathbf {D} _{3,1}&\mathbf {C} _{3,2}\mathbf {D} _{3,2}&\mathbf {C} _{3,2}\mathbf {D} _{3,3}&\mathbf {C} _{3,3}\mathbf {D} _{3,1}&\mathbf {C} _{3,3}\mathbf {D} _{3,2}&\mathbf {C} _{3,3}\mathbf {D} _{3,3}\end{array}}\right].$ The reason this product is useful is the following identity:
$(\mathbf {C} \bullet \mathbf {D} )(x\otimes y)=\mathbf {C} x\circ \mathbf {D} y=\left[{\begin{array}{c }(\mathbf {C} x)_{1}(\mathbf {D} y)_{1}\\(\mathbf {C} x)_{2}(\mathbf {D} y)_{2}\\\vdots \end{array}}\right],$
where $\circ $ is the element-wise (Hadamard) product. Since this operation can be computed in linear time, $\mathbf {C} \bullet \mathbf {D} $ can be multiplied on vectors with tensor structure much faster than normal matrices.
Construction with fast Fourier transform
The tensor sketch of Pham and Pagh[4] computes $C^{(1)}x\ast C^{(2)}y$, where $C^{(1)}$ and $C^{(2)}$ are independent count sketch matrices and $\ast $ is vector convolution. They show that, amazingly, this equals $C(x\otimes y)$ – a count sketch of the tensor product!
It turns out that this relation can be seen in terms of the face-splitting product as
$C^{(1)}x\ast C^{(2)}y={\mathcal {F}}^{-1}({\mathcal {F}}C^{(1)}x\circ {\mathcal {F}}C^{(2)}y)$, where ${\mathcal {F}}$ is the Fourier transform matrix.
Since ${\mathcal {F}}$ is an orthonormal matrix, ${\mathcal {F}}^{-1}$ doesn't impact the norm of $Cx$ and may be ignored. What's left is that $C\sim {\mathcal {C}}^{(1)}\bullet {\mathcal {C}}^{(2)}$.
On the other hand,
${\mathcal {F}}(C^{(1)}x\ast C^{(2)}y)={\mathcal {F}}C^{(1)}x\circ {\mathcal {F}}C^{(2)}y=({\mathcal {F}}C^{(1)}\bullet {\mathcal {F}}C^{(2)})(x\otimes y)$.
Application to general matrices
The problem with the original tensor sketch algorithm was that it used count sketch matrices, which aren't always very good dimensionality reductions.
In 2020[15] it was shown that any matrices with random enough independent rows suffice to create a tensor sketch. This allows using matrices with stronger guarantees, such as real Gaussian Johnson Lindenstrauss matrices.
In particular, we get the following theorem
Consider a matrix $T$ with i.i.d. rows $T_{1},\dots ,T_{m}\in \mathbb {R} ^{d}$, such that $E[(T_{1}x)^{2}]=\|x\|_{2}^{2}$ and $E[(T_{1}x)^{p}]^{1/p}\leq {\sqrt {ap}}\|x\|_{2}$. Let $T^{(1)},\dots ,T^{(c)}$ be independent consisting of $T$ and $M=T^{(1)}\bullet \dots \bullet T^{(c)}$.
Then $|\|Mx\|_{2}-\|x\|_{2}|<\varepsilon \|x\|_{2}$ with probability $1-\delta $ for any vector $x$ if
$m=(4a)^{2c}\varepsilon ^{-2}\log 1/\delta +(2ae)\varepsilon ^{-1}(\log 1/\delta )^{c}$.
In particular, if the entries of $T$ are $\pm 1$ we get $m=O(\varepsilon ^{-2}\log 1/\delta +\varepsilon ^{-1}({\tfrac {1}{c}}\log 1/\delta )^{c})$ which matches the normal Johnson Lindenstrauss theorem of $m=O(\varepsilon ^{-2}\log 1/\delta )$ when $\varepsilon $ is small.
The paper[15] also shows that the dependency on $\varepsilon ^{-1}({\tfrac {1}{c}}\log 1/\delta )^{c}$ is necessary for constructions using tensor randomized projections with Gaussian entries.
Variations
Recursive construction
Because of the exponential dependency on $c$ in tensor sketches based on the face-splitting product, a different approach was developed in 2020[15] which applies
$M(x\otimes y\otimes \cdots )=M^{(1)}(x\otimes (M^{(2)}y\otimes \cdots ))$
We can achieve such an $M$ by letting
$M=M^{(c)}(M^{(c-1)}\otimes I_{d})(M^{(c-2)}\otimes I_{d^{2}})\cdots (M^{(1)}\otimes I_{d^{c-1}})$.
With this method, we only apply the general tensor sketch method to order 2 tensors, which avoids the exponential dependency in the number of rows.
It can be proved[15] that combining $c$ dimensionality reductions like this only increases $\varepsilon $ by a factor ${\sqrt {c}}$.
Fast constructions
The fast Johnson–Lindenstrauss transform is a dimensionality reduction matrix
Given a matrix $M\in \mathbb {R} ^{k\times d}$, computing the matrix vector product $Mx$ takes $kd$ time. The Fast Johnson Lindenstrauss Transform (FJLT),[16] was introduced by Ailon and Chazelle in 2006.
A version of this method takes $M=\operatorname {SHD} $ where
1. $D$ is a diagonal matrix where each diagonal entry $D_{i,i}$ is $\pm 1$ independently.
The matrix-vector multiplication $Dx$ can be computed in $O(d)$ time.
1. $H$ is a Hadamard matrix, which allows matrix-vector multiplication in time $O(d\log d)$
2. $S$ is a $k\times d$ sampling matrix which is all zeros, except a single 1 in each row.
If the diagonal matrix is replaced by one which has a tensor product of $\pm 1$ values on the diagonal, instead of being fully independent, it is possible to compute $\operatorname {SHD} (x\otimes y)$ fast.
For an example of this, let $\rho ,\sigma \in \{-1,1\}^{2}$ be two independent $\pm 1$ vectors and let $D$ be a diagonal matrix with $\rho \otimes \sigma $ on the diagonal. We can then split up $\operatorname {SHD} (x\otimes y)$ as follows:
${\begin{aligned}&\operatorname {SHD} (x\otimes y)\\&\quad ={\begin{bmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\end{bmatrix}}{\begin{bmatrix}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\end{bmatrix}}{\begin{bmatrix}\sigma _{1}\rho _{1}&0&0&0\\0&\sigma _{1}\rho _{2}&0&0\\0&0&\sigma _{2}\rho _{1}&0\\0&0&0&\sigma _{2}\rho _{2}\\\end{bmatrix}}{\begin{bmatrix}x_{1}y_{1}\\x_{2}y_{1}\\x_{1}y_{2}\\x_{2}y_{2}\end{bmatrix}}\\[5pt]&\quad =\left({\begin{bmatrix}1&0\\0&1\\1&0\end{bmatrix}}\bullet {\begin{bmatrix}1&0\\1&0\\0&1\end{bmatrix}}\right)\left({\begin{bmatrix}1&1\\1&-1\end{bmatrix}}\otimes {\begin{bmatrix}1&1\\1&-1\end{bmatrix}}\right)\left({\begin{bmatrix}\sigma _{1}&0\\0&\sigma _{2}\\\end{bmatrix}}\otimes {\begin{bmatrix}\rho _{1}&0\\0&\rho _{2}\\\end{bmatrix}}\right)\left({\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\otimes {\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}\right)\\[5pt]&\quad =\left({\begin{bmatrix}1&0\\0&1\\1&0\end{bmatrix}}\bullet {\begin{bmatrix}1&0\\1&0\\0&1\end{bmatrix}}\right)\left({\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}\sigma _{1}&0\\0&\sigma _{2}\\\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\,\otimes \,{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}\rho _{1}&0\\0&\rho _{2}\\\end{bmatrix}}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}\right)\\[5pt]&\quad ={\begin{bmatrix}1&0\\0&1\\1&0\end{bmatrix}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}\sigma _{1}&0\\0&\sigma _{2}\\\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\,\circ \,{\begin{bmatrix}1&0\\1&0\\0&1\end{bmatrix}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}\rho _{1}&0\\0&\rho _{2}\\\end{bmatrix}}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}.\end{aligned}}$
In other words, $\operatorname {SHD} =S^{(1)}HD^{(1)}\bullet S^{(2)}HD^{(2)}$, splits up into two Fast Johnson–Lindenstrauss transformations, and the total reduction takes time $O(d_{1}\log d_{1}+d_{2}\log d_{2})$ rather than $d_{1}d_{2}\log(d_{1}d_{2})$ as with the direct approach.
The same approach can be extended to compute higher degree products, such as $\operatorname {SHD} (x\otimes y\otimes z)$
Ahle et al.[15] shows that if $\operatorname {SHD} $ has $\varepsilon ^{-2}(\log 1/\delta )^{c+1}$ rows, then $|\|\operatorname {SHD} x\|_{2}-\|x\||\leq \varepsilon \|x\|_{2}$ for any vector $x\in \mathbb {R} ^{d^{c}}$ with probability $1-\delta $, while allowing fast multiplication with degree $c$ tensors.
Jin et al.,[17] the same year, showed a similar result for the more general class of matrices call RIP, which includes the subsampled Hadamard matrices. They showed that these matrices allow splitting into tensors provided the number of rows is $\varepsilon ^{-2}(\log 1/\delta )^{2c-1}\log d$. In the case $c=2$ this matches the previous result.
These fast constructions can again be combined with the recursion approach mentioned above, giving the fastest overall tensor sketch.
Data aware sketching
It is also possible to do so-called "data aware" tensor sketching. Instead of multiplying a random matrix on the data, the data points are sampled independently with a certain probability depending on the norm of the point.[18]
Applications
Explicit polynomial kernels
Kernel methods are popular in machine learning as they give the algorithm designed the freedom to design a "feature space" in which to measure the similarity of their data points. A simple kernel-based binary classifier is based on the following computation:
${\hat {y}}(\mathbf {x'} )=\operatorname {sgn} \sum _{i=1}^{n}y_{i}k(\mathbf {x} _{i},\mathbf {x'} ),$
where $\mathbf {x} _{i}\in \mathbb {R} ^{d}$ are the data points, $y_{i}$ is the label of the $i$th point (either −1 or +1), and ${\hat {y}}(\mathbf {x'} )$ is the prediction of the class of $\mathbf {x'} $. The function $k:\mathbb {R} ^{d}\times \mathbb {R} ^{d}\to \mathbb {R} $ is the kernel. Typical examples are the radial basis function kernel, $k(x,x')=\exp(-\|x-x'\|_{2}^{2})$, and polynomial kernels such as $k(x,x')=(1+\langle x,x'\rangle )^{2}$.
When used this way, the kernel method is called "implicit". Sometimes it is faster to do an "explicit" kernel method, in which a pair of functions $f,g:\mathbb {R} ^{d}\to \mathbb {R} ^{D}$ are found, such that $k(x,x')=\langle f(x),g(x')\rangle $. This allows the above computation to be expressed as
${\hat {y}}(\mathbf {x'} )=\operatorname {sgn} \sum _{i=1}^{n}y_{i}\langle f(\mathbf {x} _{i}),g(\mathbf {x'} )\rangle =\operatorname {sgn} \left\langle \left(\sum _{i=1}^{n}y_{i}f(\mathbf {x} _{i})\right),g(\mathbf {x'} )\right\rangle ,$
where the value $\sum _{i=1}^{n}y_{i}f(\mathbf {x} _{i})$ can be computed in advance.
The problem with this method is that the feature space can be very large. That is $D>>d$. For example, for the polynomial kernel $k(x,x')=\langle x,x'\rangle ^{3}$ we get $f(x)=x\otimes x\otimes x$ and $g(x')=x'\otimes x'\otimes x'$, where $\otimes $ is the tensor product and $f(x),g(x')\in \mathbb {R} ^{D}$ where $D=d^{3}$. If $d$ is already large, $D$ can be much larger than the number of data points ($n$) and so the explicit method is inefficient.
The idea of tensor sketch is that we can compute approximate functions $f',g':\mathbb {R} ^{d}\to \mathbb {R} ^{t}$ where $t$ can even be smaller than $d$, and which still have the property that $\langle f'(x),g'(x')\rangle \approx k(x,x')$.
This method was shown in 2020[15] to work even for high degree polynomials and radial basis function kernels.
Compressed matrix multiplication
Assume we have two large datasets, represented as matrices $X,Y\in \mathbb {R} ^{n\times d}$, and we want to find the rows $i,j$ with the largest inner products $\langle X_{i},Y_{j}\rangle $. We could compute $Z=XY^{T}\in \mathbb {R} ^{n\times n}$ and simply look at all $n^{2}$ possibilities. However, this would take at least $n^{2}$ time, and probably closer to $n^{2}d$ using standard matrix multiplication techniques.
The idea of Compressed Matrix Multiplication is the general identity
$XY^{T}=\sum _{i=1}^{d}X_{i}\otimes Y_{i}$
where $\otimes $ is the tensor product. Since we can compute a (linear) approximation to $X_{i}\otimes Y_{i}$ efficiently, we can sum those up to get an approximation for the complete product.
Compact multilinear pooling
Bilinear pooling is the technique of taking two input vectors, $x,y$ from different sources, and using the tensor product $x\otimes y$ as the input layer to a neural network.
In[19] the authors considered using tensor sketch to reduce the number of variables needed.
In 2017 another paper[20] takes the FFT of the input features, before they are combined using the element-wise product. This again corresponds to the original tensor sketch.
References
1. "Low-rank Tucker decomposition of large tensors using: Tensor Sketch" (PDF). amath.colorado.edu. Boulder, Colorado: University of Colorado Boulder.
2. Ahle, Thomas; Knudsen, Jakob (2019-09-03). "Almost Optimal Tensor Sketch". ResearchGate. Retrieved 2020-07-11.
3. Woodruff, David P. "Sketching as a Tool for Numerical Linear Algebra Archived 2022-10-22 at the Wayback Machine." Theoretical Computer Science 10.1-2 (2014): 1–157.
4. Ninh, Pham; Pagh, Rasmus (2013). Fast and scalable polynomial kernels via explicit feature maps. SIGKDD international conference on Knowledge discovery and data mining. Association for Computing Machinery. doi:10.1145/2487575.2487591.
5. Pagh, Rasmus (2013). "Compressed matrix multiplication". ACM Transactions on Computation Theory. Association for Computing Machinery. 5 (3): 1–17. arXiv:1108.1320. doi:10.1145/2493252.2493254. S2CID 47560654.
6. Kasiviswanathan, Shiva Prasad, et al. "The price of privately releasing contingency tables and the spectra of random matrices with correlated rows Archived 2022-10-22 at the Wayback Machine." Proceedings of the forty-second ACM symposium on Theory of computing. 2010.
7. Rudelson, Mark, and Shuheng Zhou. "Reconstruction from anisotropic random measurements Archived 2022-10-17 at the Wayback Machine." Conference on Learning Theory. 2012.
8. Avron, Haim; Nguyen, Huy; Woodruff, David (2013). "Subspace Embeddings for the Polynomial Kernel". NIPS'14: Proceedings of the 27th International Conference on Neural Information Processing Systems. Association for Computing Machinery. arXiv:1108.1320. doi:10.1145/2493252.2493254. S2CID 47560654.
9. Anna Esteve, Eva Boj & Josep Fortiana (2009): Interaction Terms in Distance-Based Regression, Communications in Statistics – Theory and Methods, 38:19, P. 3501 Archived 2021-04-26 at the Wayback Machine
10. Slyusar, V. I. (1998). "End products in matrices in radar applications" (PDF). Radioelectronics and Communications Systems. 41 (3): 50–53.
11. Slyusar, V. I. (1997-05-20). "Analytical model of the digital antenna array on a basis of face-splitting matrix products" (PDF). Proc. ICATT-97, Kyiv: 108–109.
12. Slyusar, V. I. (1997-09-15). "New operations of matrices product for applications of radars" (PDF). Proc. Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-97), Lviv.: 73–74.
13. Slyusar, V. I. (March 13, 1998). "A Family of Face Products of Matrices and its Properties" (PDF). Cybernetics and Systems Analysis C/C of Kibernetika I Sistemnyi Analiz. – 1999. 35 (3): 379–384. doi:10.1007/BF02733426. S2CID 119661450.
14. Slyusar, V. I. (2003). "Generalized face-products of matrices in models of digital antenna arrays with nonidentical channels" (PDF). Radioelectronics and Communications Systems. 46 (10): 9–17.
15. Ahle, Thomas; Kapralov, Michael; Knudsen, Jakob; Pagh, Rasmus; Velingker, Ameya; Woodruff, David; Zandieh, Amir (2020). Oblivious Sketching of High-Degree Polynomial Kernels. ACM-SIAM Symposium on Discrete Algorithms. Association for Computing Machinery. doi:10.1137/1.9781611975994.9.
16. Ailon, Nir; Chazelle, Bernard (2006). "Approximate nearest neighbors and the fast Johnson–Lindenstrauss transform". Proceedings of the 38th Annual ACM Symposium on Theory of Computing. New York: ACM Press. pp. 557–563. doi:10.1145/1132516.1132597. ISBN 1-59593-134-1. MR 2277181. S2CID 490517.
17. Jin, Ruhui, Tamara G. Kolda, and Rachel Ward. "Faster Johnson–Lindenstrauss Transforms via Kronecker Products." arXiv preprint arXiv:1909.04801 (2019).
18. Wang, Yining; Tung, Hsiao-Yu; Smola, Alexander; Anandkumar, Anima. Fast and Guaranteed Tensor Decomposition via Sketching. Advances in Neural Information Processing Systems 28 (NIPS 2015). arXiv:1506.04448.
19. Gao, Yang, et al. "Compact bilinear pooling Archived 2022-01-20 at the Wayback Machine." Proceedings of the IEEE conference on computer vision and pattern recognition. 2016.
20. Algashaam, Faisal M., et al. "Multispectral periocular classification with multimodal compact multi-linear pooling Archived 2022-10-22 at the Wayback Machine." IEEE Access 5 (2017): 14572–14578.
Further reading
• Ahle, Thomas; Knudsen, Jakob (2019-09-03). "Almost Optimal Tensor Sketch". ResearchGate. Retrieved 2020-07-11.
• Slyusar, V. I. (1998). "End products in matrices in radar applications" (PDF). Radioelectronics and Communications Systems. 41 (3): 50–53.
• Slyusar, V. I. (1997-05-20). "Analytical model of the digital antenna array on a basis of face-splitting matrix products" (PDF). Proc. ICATT-97, Kyiv: 108–109.
• Slyusar, V. I. (1997-09-15). "New operations of matrices product for applications of radars" (PDF). Proc. Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-97), Lviv.: 73–74.
• Slyusar, V. I. (March 13, 1998). "A Family of Face Products of Matrices and its Properties" (PDF). Cybernetics and Systems Analysis C/C of Kibernetika I Sistemnyi Analiz.- 1999. 35 (3): 379–384. doi:10.1007/BF02733426. S2CID 119661450.
| Wikipedia |
Teo Mora
Ferdinando 'Teo' Mora[lower-alpha 1] is an Italian mathematician, and since 1990 until 2019 a professor of algebra at the University of Genoa.
Life and work
Mora's degree is in mathematics from the University of Genoa in 1974.[1] Mora's publications span forty years; his notable contributions in computer algebra are the tangent cone algorithm[2][3] and its extension of Buchberger theory of Gröbner bases and related algorithm earlier[4] to non-commutative polynomial rings[5] and more recently[6] to effective rings; less significant[7] the notion of Gröbner fan; marginal, with respect to the other authors, his contribution to the FGLM algorithm.
Mora is on the managing-editorial-board of the journal AAECC published by Springer,[8] and was also formerly an editor of the Bulletin of the Iranian Mathematical Society.[lower-alpha 2]
He is the author of the tetralogy Solving Polynomial Equation Systems:
• Solving Polynomial Equation Systems I: The Kronecker-Duval Philosophy, on equations in one variable[9]
• Solving Polynomial Equation Systems II: Macaulay's paradigm and Gröbner technology, on equations in several variables[10][9]
• Solving Polynomial Equation Systems III: Algebraic Solving,
• Solving Polynomial Equation Systems IV: Buchberger Theory and Beyond, on the Buchberger algorithm
Personal life
Mora lives in Genoa.[11] Mora published a book trilogy in 1977-1978 (reprinted 2001-2003) called Storia del cinema dell'orrore on the history of horror films.[11] Italian television said in 2014 that the books are an "authoritative guide with in-depth detailed descriptions and analysis."[12]
See also
• FGLM algorithm, Buchberger's algorithm
• Gröbner fan, Gröbner basis
• Algebraic geometry#Computational algebraic geometry, System of polynomial equations
References
1. University of Genoa faculty-page.
2. An algorithm to compute the equations of tangent cones; An introduction to the tangent cone algorithm.
3. Better algorithms due to Greuel-Pfister and Gräbe are currently available.
4. Gröbner bases for non-commutative polynomial rings.
5. Extending the proposal set by George M. Bergman.
6. De Nugis Groebnerialium 4: Zacharias, Spears, Möller, Buchberger–Weispfenning theory for effective associative rings; see also Seven variations on standard bases.
7. The result is a weaker version of the result presented in the same issue of the journal by Bayer and Morrison.
8. Springer-Verlag website.
9. David P. Roberts (UMN) (September 14, 2006). "[Review of the book] Solving Polynomial Equation Systems I: The Kronecker-Duval Philosophy [and also Solving Polynomial Equation Systems II: Macaulay's Paradigm and Gröbner Technology]". Mathematical Association of America Press.
10. S. C. Coutinho (UFRJ) (March 2009). "Review of solving polynomial equation systems II: Macaulay's paradigm and Gröbner technology by Teo Mora (Cambridge University Press 2005)" (PDF). SIGACT Newsletter. 40 (1): 14–17. doi:10.1145/1515698.1515702. S2CID 12448065 – via Publisher's site. {{cite journal}}: External link in |via= (help)
11. Giovanni Bogani (December 11, 2002). "O tempora, O... Teo Mora". Genoa, Italy: Repubblica.it. ...Teo Mora vive a Genova. ...scritto libri come La madre di tutte le dualità: l'algoritmo di Moeller, Il teorema di Kalkbrenner, o L'algoritmo di Buchberger ... Negli [1977] anni '70, Mora aveva scritto una monumentale Storia del cinema horror. ... la [2001] ripropone, in una nuova edizione, riveduta, corretta e completamente aggiornata. ...Nel primo volume... fino al 1957... Nosferatu, attori come Boris Karloff e Bela Lugosi... film come Il gabinetto del dottor Caligari. ...Nel secondo volume si arriva fino al 1966... Roger Corman... Il terzo volume arriva fino al 1978... Brian De Palma, David Cronenberg, George Romero, Dario Argento, Mario Bava. ... Translation: "...Teo Mora lives in Genoa. ...written works include The Mother of All Dualities: The Möller Algorithm, The Kalkbrenner Theorem, and The Buchberger Algorithm ... In the 1970s, Mora wrote the monumental History of Horror Cinema. ...reprinted [in 2001], as a new edition: revised, corrected, and completely updated. Two volume are already out, the third [volume] will be released in late January [2002], the fourth [volume] in spring 2003. ... In the first volume... [covering] through 1957... Nosferatu, actors like Boris Karloff and Bela Lugosi... films like The Cabinet of Dr. Caligari. ...The second volume covers until 1966... Roger Corman, director ...The third volume covers through 1978... Brian De Palma, David Cronenberg, George Romero, Dario Argento, Mario Bava. ..."
12. "Mostri Universal" [The Universal Pictures monsters]. No. 20. RAI 4, Radiotelevisione Italiana. September 12, 2014. ...[text:] L'intervista — Teo Mora: Professore di Algebra presso il dipartimento di Informatica e Scienze dell'Informazione dell'Università di Genova, è anche un noto esperto di cinema horror. Ha curato Storia del cinema dell'orrore, un'autorevole guida in tre volumi con approfondimenti, schede e analisi dettagliate sui film, i registi e gli attori... [multimedia: video content] ... Translation: "...[text:] professor of Algebra in the Computer and Information Science department of the University of Genoa, also a well-known expert on horror films. His book Storia del cinema dell'orrore is an authoritative guide with in-depth detailed descriptions and analysis of films, directors, and actors... [multimedia: video content] ..."
Notes
1. Teo Mora is his nickname, but used in most of his post-1980s publications; he has also used the pen name Theo Moriarty.[1]
2. See previous faculty-page.
Further reading
• Teo Mora (1977). Storia del cinema dell'orrore. Vol. 1. Fanucci. ISBN 978-88-347-0800-2.. "Second". and "third". volumes: ISBN 88-347-0850-4, ISBN 88-347-0897-0. Reprinted 2001.
• George M Bergman (1978). "The diamond lemma for ring theory". Advances in Mathematics. 29 (2): 178–218. doi:10.1016/0001-8708(78)90010-5.
• F. Mora (1982). "An algorithm to compute the equations of tangent cones". Computer Algebra. pp. 158–165. doi:10.1007/3-540-11607-9_18. ISBN 978-3-540-11607-3. {{cite book}}: |journal= ignored (help)
• F. Mora (1986). "Groebner bases for non-commutative polynomial rings". Gröbner bases for non-commutative polynomial rings (PDF). pp. 353–362. doi:10.1007/3-540-16776-5_740. ISBN 978-3-540-16776-1. {{cite book}}: |journal= ignored (help)
• David Bayer; Ian Morrison (1988). "Standard bases and geometric invariant theory I. Initial ideals and state polytopes". Journal of Symbolic Computation. 6 (2–3): 209–218. doi:10.1016/S0747-7171(88)80043-9.
• also in: Lorenzo Robbiano, ed. (1989). Computational Aspects of Commutative Algebra. Vol. 6. London: Academic Press.
• Teo Mora (1988). "Seven variations on standard bases" – via Bibliography. {{cite news}}: External link in |via= (help)
• Gerhard Pfister; T.Mora; Carlo Traverso (1992). Christoph M Hoffmann (ed.). "An introduction to the tangent cone algorithm". Issues in Robotics and Nonlinear Geometry (Advances in Computing Research). 6: 199–270.
• T. Mora (1994). "An introduction to commutative and non-commutative Gröbner bases". Theoretical Computer Science. 134: 131–173. doi:10.1016/0304-3975(94)90283-6 – via Publisher's site. {{cite journal}}: External link in |via= (help)
• Hans-Gert Gräbe (1995). "Algorithms in Local Algebra". Journal of Symbolic Computation. 19 (6): 545–557. doi:10.1006/jsco.1995.1031.
• Gert-Martin Greuel; G. Pfister (1996). "Advances and improvements in the theory of standard bases and syzygies". CiteSeerX 10.1.1.49.1231.
• M.Caboara, T.Mora (2002). "The Chen-Reed-Helleseth-Truong Decoding Algorithm and the Gianni-Kalkbrenner Gröbner Shape Theorem". Journal of Applicable Algebra. 13 (3): 209–232. doi:10.1007/s002000200097. S2CID 2505343 – via Publisher's site. Author's site. {{cite journal}}: External link in |via= (help)
• M.E. Alonso; M.G. Marinari; M.T. Mora (2003). "The Big Mother of All the Dualities, I: Möller Algorithm". Communications in Algebra. 31 (2): 783–818. CiteSeerX 10.1.1.57.7799. doi:10.1081/AGB-120017343. S2CID 120556267 – via Publisher's site. Author's site. {{cite journal}}: External link in |via= (help)
• Teo Mora (March 1, 2003). Solving Polynomial Equation Systems I: The Kronecker-Duval Philosophy (PDF). Encyclopedia of Mathematics and its Applications Series. Vol. 88. Cambridge University Press. doi:10.1017/cbo9780511542831. ISBN 9780521811545. S2CID 118216321. Archived from the original (PDF) on April 6, 2017 – via Publisher's website. Excerpt. {{cite book}}: External link in |via= (help)
• T. Mora (2005). Solving Polynomial Equation Systems II: Macaulay's Paradigm and Gröbner Technology. Encyclopedia of Mathematics and its Applications. Vol. 99. Cambridge University Press.
• T. Mora (2015). Solving Polynomial Equation Systems III: Algebraic Solving. Encyclopedia of Mathematics and its Applications. Vol. 157. Cambridge University Press.
• T Mora (2016). Solving Polynomial Equation Systems IV: Buchberger Theory and Beyond. Encyclopedia of Mathematics and its Applications. Vol. 158. Cambridge University Press. ISBN 9781107109636.
• T. Mora (2015). De Nugis Groebnerialium 4: Zacharias, Spears, Möller. pp. 283–290. doi:10.1145/2755996.2756640. ISBN 9781450334358. S2CID 14654596. {{cite book}}: |journal= ignored (help)
• Michela Ceria; Teo Mora (2016). "Buchberger–Weispfenning theory for effective associative rings". Journal of Symbolic Computation. 83: 112–146. arXiv:1611.08846. doi:10.1016/j.jsc.2016.11.008. S2CID 10363249.
• T Mora (2016). Solving Polynomial Equation Systems IV: Buchberger Theory and Beyond. Encyclopedia of Mathematics and its Applications. Vol. 158. Cambridge University Press. ISBN 9781107109636.
External links
• Official page
• Teo Mora and Michela Ceria, Do It Yourself: Buchberger and Janet bases over effective rings, Part 1: Buchberger Algorithm via Spear’s Theorem, Zacharias’ Representation, Weisspfenning Multiplication, Part 2: Moeller Lifting Theorem vs Buchberger Criteria, Part 3: What happens to involutive bases?. Invited talk at ICMS 2020 International Congress on Mathematical Software , Braunschweig, 13-16 July 2020
Authority control
International
• ISNI
• VIAF
National
• Germany
• Italy
• Israel
• Belgium
• United States
• Sweden
• Netherlands
Academics
• CiNii
• zbMATH
People
• Deutsche Biographie
| Wikipedia |
Teragon
A teragon is a polygon with an infinite number of sides, the most famous example being the Koch snowflake ("triadic Koch teragon"). The term was coined by Benoît Mandelbrot from the words Classical Greek τέρας (teras, monster) + γωνία (gōnía, corner).[2] Typically, a teragon will be bounded by one or more self-similar fractal curves, which are created by replacing each line segment in an initial figure with multiple connected segments, then replacing each of those segments with the same pattern of segments, then repeating the process an infinite number of times for every line segment in the figure.
Other examples
The horned triangle, created by erecting a series of smaller triangles on one corner of an equilateral triangle, is another example of a teragon. It is also an example of a rep-tile, or shape that can be completely dissected into smaller copies of itself.
References
1. Albeverio, Sergio; Andrey, Sergio; Giordano, Paolo; and Vancheri, Alberto (1997). The Dynamics of Complex Urban Systems, p.222. Springer. ISBN 9783790819373.
2. Larson, Ron; Hostetler, Robert P.; and Edwards, Bruce H. (1998). Calculus, p.546. 6th edition. Houghton Mifflin. ISBN 9780395869741.
Further reading
• Mandelbrot, B. B. (1982). The Fractal Geometry of Nature. W.H. Freeman and Company. ISBN 0-7167-1186-9.
Fractals
Characteristics
• Fractal dimensions
• Assouad
• Box-counting
• Higuchi
• Correlation
• Hausdorff
• Packing
• Topological
• Recursion
• Self-similarity
Iterated function
system
• Barnsley fern
• Cantor set
• Koch snowflake
• Menger sponge
• Sierpinski carpet
• Sierpinski triangle
• Apollonian gasket
• Fibonacci word
• Space-filling curve
• Blancmange curve
• De Rham curve
• Minkowski
• Dragon curve
• Hilbert curve
• Koch curve
• Lévy C curve
• Moore curve
• Peano curve
• Sierpiński curve
• Z-order curve
• String
• T-square
• n-flake
• Vicsek fractal
• Hexaflake
• Gosper curve
• Pythagoras tree
• Weierstrass function
Strange attractor
• Multifractal system
L-system
• Fractal canopy
• Space-filling curve
• H tree
Escape-time
fractals
• Burning Ship fractal
• Julia set
• Filled
• Newton fractal
• Douady rabbit
• Lyapunov fractal
• Mandelbrot set
• Misiurewicz point
• Multibrot set
• Newton fractal
• Tricorn
• Mandelbox
• Mandelbulb
Rendering techniques
• Buddhabrot
• Orbit trap
• Pickover stalk
Random fractals
• Brownian motion
• Brownian tree
• Brownian motor
• Fractal landscape
• Lévy flight
• Percolation theory
• Self-avoiding walk
People
• Michael Barnsley
• Georg Cantor
• Bill Gosper
• Felix Hausdorff
• Desmond Paul Henry
• Gaston Julia
• Helge von Koch
• Paul Lévy
• Aleksandr Lyapunov
• Benoit Mandelbrot
• Hamid Naderi Yeganeh
• Lewis Fry Richardson
• Wacław Sierpiński
Other
• "How Long Is the Coast of Britain?"
• Coastline paradox
• Fractal art
• List of fractals by Hausdorff dimension
• The Fractal Geometry of Nature (1982 book)
• The Beauty of Fractals (1986 book)
• Chaos: Making a New Science (1987 book)
• Kaleidoscope
• Chaos theory
| Wikipedia |
Terence Tao
Terence Chi-Shen Tao FAA FRS (Chinese: 陶哲軒; born 17 July 1975) is an Australian mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes topics in harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing and analytic number theory.[4]
Terence Tao
FAA FRS
Tao in 2021
Born (1975-07-17) 17 July 1975
Adelaide, South Australia, Australia
Citizenship
• Australia
• United States[1]
Alma mater
• Flinders University (BS, MSc)
Princeton University (PhD)
Known forPartial Differential Equations Analytic Number Theory Random matrices Compressed Sensing Combinatorics Dynamical Systems
SpouseLaura Tao
Children2
AwardsFields Medal (2006)
List
• Salem Prize (2000)
• Bôcher Memorial Prize (2002)
• Clay Research Award (2003)
• Australian Mathematical Society Medal (2005)
• Ostrowski Prize (2005)
• Levi L. Conant Prize (2005)
• MacArthur Award (2006)
• SASTRA Ramanujan Prize (2006)
• Sloan Fellowship (2006)
• Fellow of the Royal Society (2007)
• Alan T. Waterman Award (2008)
• Onsager Medal (2008)
• Convocation Award (2008)
• King Faisal International Prize (2010)[2]
• Nemmers Prize in Mathematics (2010)
• Pólya Prize (2010)[3]
• Crafoord Prize (2012)
• Simons Investigator (2012)
• Breakthrough Prize in Mathematics (2014)
• Royal Medal (2014)
• PROSE Award (2015)
• Riemann Prize (2019)
• Princess of Asturias Award (2020)
• Bolyai Prize (2020)
• IEEE Jack S. Kilby Signal Processing Medal (2021)
• USIA Award (2021)
• Education & Research award finalist (2022)
• Global Australian of the Year Award (2022)
• Research.com Mathematics in United States Leader Award (2022)
• Grande Médaille (2023)
• Research.com Mathematics in United States Leader Award (2023)
Scientific career
FieldsHarmonic analysis
InstitutionsUniversity of California, Los Angeles
ThesisThree Regularity Results in Harmonic Analysis[1] (1996)
Doctoral advisorElias M. Stein
Doctoral studentsMonica Vișan
Website
• www.math.ucla.edu/~tao/
• terrytao.wordpress.com
mathstodon.xyz/@tao
Terence Tao
Traditional Chinese陶哲軒
Simplified Chinese陶哲轩
Transcriptions
Standard Mandarin
Hanyu PinyinTáo Zhéxuān
Wu
SuzhouneseDau Tseh-shie
Yue: Cantonese
Yale RomanizationTòuh Jit-hīn
JyutpingTou4 Zit3-hin1
IPA[tʰou˩ tsiːt̚˧.hiːn˥]
Tao was born to ethnic Chinese immigrant parents and raised in Adelaide. Tao won the Fields Medal in 2006 and won the Royal Medal and Breakthrough Prize in Mathematics in 2014. He is also a 2006 MacArthur Fellow. Tao has been the author or co-author of over three hundred research papers.[5] He is widely regarded as one of the greatest living mathematicians and has been referred to as the "Mozart of mathematics".[6][7][8][9][10]
Life and career
Family
Tao's parents are first-generation immigrants from Hong Kong to Australia.[11] Tao's father, Billy Tao,[lower-alpha 1] was a Chinese paediatrician who was born in Shanghai and earned his medical degree (MBBS) from the University of Hong Kong in 1969.[12] Tao's mother, Grace Leong,[lower-alpha 2] was born in Hong Kong; she received a first-class honours degree in mathematics and physics at the University of Hong Kong.[10] She was a secondary school teacher of mathematics and physics in Hong Kong.[13] Billy and Grace met as students at the University of Hong Kong.[14] They then emigrated from Hong Kong to Australia in 1972.[11][10]
Tao also has two brothers, Trevor and Nigel, who are living in Australia. Both formerly represented the states at the International Mathematical Olympiad.[15] Furthermore, Trevor has been representing Australia internationally in chess and holds the title of Chess International Master.[16] Tao speaks Cantonese but cannot write Chinese. Tao is married to Laura Tao, an electrical engineer at NASA's Jet Propulsion Laboratory.[10][17] They live in Los Angeles, California, and have two children: Riley[lower-alpha 3] and daughter Madeleine.[18][19][20]
Childhood
A child prodigy,[21] Tao exhibited extraordinary mathematical abilities from an early age, attending university-level mathematics courses at the age of 9. He is one of only three children in the history of the Johns Hopkins' Study of Exceptional Talent program to have achieved a score of 700 or greater on the SAT math section while just eight years old; Tao scored a 760.[22][6] Julian Stanley, Director of the Study of Mathematically Precocious Youth, stated that Tao had the greatest mathematical reasoning ability he had found in years of intensive searching.[23]
Tao was the youngest participant to date in the International Mathematical Olympiad, first competing at the age of ten; in 1986, 1987, and 1988, he won a bronze, silver, and gold medal, respectively. Tao remains the youngest winner of each of the three medals in the Olympiad's history, having won the gold medal at the age of 13 in 1988.[24]
Career
At age 14, Tao attended the Research Science Institute, a summer program for secondary students. In 1991, he received his bachelor's and master's degrees at the age of 16 from Flinders University under the direction of Garth Gaudry.[25] In 1992, he won a postgraduate Fulbright Scholarship to undertake research in mathematics at Princeton University in the United States. From 1992 to 1996, Tao was a graduate student at Princeton University under the direction of Elias Stein, receiving his PhD at the age of 21.[25] In 1996, he joined the faculty of the University of California, Los Angeles. In 1999, when he was 24, he was promoted to full professor at UCLA and remains the youngest person ever appointed to that rank by the institution.[25]
He is known for his collaborative mindset; by 2006, Tao had worked with over 30 others in his discoveries,[6] reaching 68 co-authors by October 2015.
Tao has had a particularly extensive collaboration with British mathematician Ben J. Green; together they proved the Green–Tao theorem, which is well-known among both amateur and professional mathematicians. This theorem states that there are arbitrarily long arithmetic progressions of prime numbers. The New York Times described it this way:[26][27]
In 2004, Dr. Tao, along with Ben Green, a mathematician now at the University of Cambridge in England, solved a problem related to the Twin Prime Conjecture by looking at prime number progressions—series of numbers equally spaced. (For example, 3, 7 and 11 constitute a progression of prime numbers with a spacing of 4; the next number in the sequence, 15, is not prime.) Dr. Tao and Dr. Green proved that it is always possible to find, somewhere in the infinity of integers, a progression of prime numbers of equal spacing and any length.
Many other results of Tao have received mainstream attention in the scientific press, including:
• his establishment of finite time blowup for a modification of the Navier–Stokes existence and smoothness Millennium Problem[28]
• his 2015 resolution of the Erdős discrepancy problem, which used entropy estimates within analytic number theory[29]
• his 2019 progress on the Collatz conjecture, in which he proved the probabilistic claim that almost all Collatz orbits attain almost bounded values.[30]
Tao has also resolved or made progress on a number of conjectures. In 2012, Green and Tao announced proofs of the conjectured "orchard-planting problem," which asks for the maximum number of lines through exactly 3 points in a set of n points in the plane, not all on a line. In 2018, with Brad Rodgers, Tao showed that the de Bruijn–Newman constant, the nonpositivity of which is equivalent to the Riemann hypothesis, is nonnegative.[31] In 2020, Tao proved Sendov's conjecture, concerning the locations of the roots and critical points of a complex polynomial, in the special case of polynomials with sufficiently high degree.[32]
Recognition
British mathematician and Fields medalist Timothy Gowers remarked on Tao's breadth of knowledge:[33]
Tao's mathematical knowledge has an extraordinary combination of breadth and depth: he can write confidently and authoritatively on topics as diverse as partial differential equations, analytic number theory, the geometry of 3-manifolds, nonstandard analysis, group theory, model theory, quantum mechanics, probability, ergodic theory, combinatorics, harmonic analysis, image processing, functional analysis, and many others. Some of these are areas to which he has made fundamental contributions. Others are areas that he appears to understand at the deep intuitive level of an expert despite officially not working in those areas. How he does all this, as well as writing papers and books at a prodigious rate, is a complete mystery. It has been said that David Hilbert was the last person to know all of mathematics, but it is not easy to find gaps in Tao's knowledge, and if you do then you may well find that the gaps have been filled a year later.
An article by New Scientist[34] writes of his ability:
Such is Tao's reputation that mathematicians now compete to interest him in their problems, and he is becoming a kind of Mr Fix-it for frustrated researchers. "If you're stuck on a problem, then one way out is to interest Terence Tao," says Charles Fefferman [professor of mathematics at Princeton University].[35]
Tao has won numerous mathematician honours and awards over the years.[36] He is a Fellow of the Royal Society, the Australian Academy of Science (Corresponding Member), the National Academy of Sciences (Foreign member), the American Academy of Arts and Sciences, the American Philosophical Society,[37] and the American Mathematical Society.[38] In 2006 he received the Fields Medal; he was the first Australian, the first UCLA faculty member, and one of the youngest mathematicians to receive the award.[35][39] He was also awarded the MacArthur Fellowship. He has been featured in The New York Times, CNN, USA Today, Popular Science, and many other media outlets.[40] In 2014, Tao received a CTY Distinguished Alumni Honor from Johns Hopkins Center for Gifted and Talented Youth in front of 979 attendees in 8th and 9th grade that are in the same program from which Tao graduated. In 2021, President Joe Biden announced Tao had been selected as one of 30 members of his President's Council of Advisors on Science and Technology, a body bringing together America's most distinguished leaders in science and technology.[41] In 2021, Tao was awarded the Riemann Prize Week as recipient of the inaugural Riemann Prize 2019 by the Riemann International School of Mathematics at the University of Insubria.[42] Tao was a finalist to become Australian of the Year in 2007.[43]
As of 2022, Tao has published over three hundred articles, along with sixteen books.[44] He has an Erdős number of 2.[45] He is a highly cited researcher.[46][47]
Research contributions
Dispersive partial differential equations
From 2001 to 2010, Tao was part of a well-known collaboration with James Colliander, Markus Keel, Gigliola Staffilani, and Hideo Takaoka. They found a number of novel results, many to do with the well-posedness of weak solutions, for Schrödinger equations, KdV equations, and KdV-type equations.[C+03]
Michael Christ, Colliander, and Tao developed methods of Carlos Kenig, Gustavo Ponce, and Luis Vega to establish ill-posedness of certain Schrödinger and KdV equations for Sobolev data of sufficiently low exponents.[CCT03][48] In many cases these results were sharp enough to perfectly complement well-posedness results for sufficiently large exponents as due to Bourgain, Colliander−Keel−Staffilani−Takaoka−Tao, and others. Further such notable results for Schrödinger equations were found by Tao in collaboration with Ioan Bejenaru.[BT06]
A particularly notable result of the Colliander−Keel−Staffilani−Takaoka−Tao collaboration established the long-time existence and scattering theory of a power-law Schrödinger equation in three dimensions.[C+08] Their methods, which made use of the scale-invariance of the simple power law, were extended by Tao in collaboration with Monica Vișan and Xiaoyi Zhang to deal with nonlinearities in which the scale-invariance is broken.[TVZ07] Rowan Killip, Tao, and Vișan later made notable progress on the two-dimensional problem in radial symmetry.[KTV09]
A technical tour de force by Tao in 2001 considered the wave maps equation with two-dimensional domain and spherical range.[T01a] He built upon earlier innovations of Daniel Tataru, who considered wave maps valued in Minkowski space.[49] Tao proved the global well-posedness of solutions with sufficiently small initial data. The fundamental difficulty is that Tao considers smallness relative to the critical Sobolev norm, which typically requires sophisticated techniques. Tao later adapted some of his work on wave maps to the setting of the Benjamin–Ono equation; Alexandru Ionescu and Kenig later obtained improved results with Tao's methods.[T04a][50]
In 2016, Tao constructed a variant of the Navier–Stokes equations which possess solutions exhibiting irregular behavior in finite time.[T16] Due to structural similarities between Tao's system and the Navier–Stokes equations themselves, it follows that any positive resolution of the Navier–Stokes existence and smoothness problem must take into account the specific nonlinear structure of the equations. In particular, certain previously-proposed resolutions of the problem could not be legitimate.[51] Tao speculated that the Navier–Stokes equations might be able to simulate a Turing complete system, and that as a consequence it might be possible to (negatively) resolve the existence and smoothness problem using a modification of his results.[6][28] However, such results remain (as of 2022) conjectural.
Harmonic analysis
Bent Fuglede introduced the Fuglede conjecture in the 1970s, positing a tile-based characterisation of those Euclidean domains for which a Fourier ensemble provides a basis of L2.[52] Tao resolved the conjecture in the negative for dimensions larger than 5, based upon the construction of an elementary counterexample to an analogous problem in the setting of finite groups.[T04b]
With Camil Muscalu and Christoph Thiele, Tao considered certain multilinear singular integral operators with the multiplier allowed to degenerate on a hyperplane, identifying conditions which ensure operator continuity relative to Lp spaces.[MTT02] This unified and extended earlier notable results of Ronald Coifman, Carlos Kenig, Michael Lacey, Yves Meyer, Elias Stein, and Thiele, among others.[53][54][55][56][57][58] Similar problems were analyzed by Tao in 2001 in the context of Bourgain spaces, rather than the usual Lp spaces.[T01b] Such estimates are used in establishing well-posedness results for dispersive partial differential equations, following famous earlier work of Jean Bourgain, Kenig, Gustavo Ponce, and Luis Vega, among others.[59][60]
A number of Tao's results deal with "restriction" phenomena in Fourier analysis, which have been widely studied since seminal articles of Charles Fefferman, Robert Strichartz, and Peter Tomas in the 1970s.[61][62][63] Here one studies the operation which restricts input functions on Euclidean space to a submanifold and outputs the product of the Fourier transforms of the corresponding measures. It is of major interest to identify exponents such that this operation is continuous relative to Lp spaces. Such multilinear problems originated in the 1990s, including in notable work of Jean Bourgain, Sergiu Klainerman, and Matei Machedon.[64][65][66] In collaboration with Ana Vargas and Luis Vega, Tao made some foundational contributions to the study of the bilinear restriction problem, establishing new exponents and drawing connections to the linear restriction problem. They also found analogous results for the bilinear Kakeya problem which is based upon the X-ray transform instead of the Fourier transform.[TVV98] In 2003, Tao adapted ideas developed by Thomas Wolff for bilinear restriction to conical sets into the setting of restriction to quadratic hypersurfaces.[T03][67] The multilinear setting for these problems was further developed by Tao in collaboration with Jonathan Bennett and Anthony Carbery; their work was extensively used by Bourgain and Larry Guth in deriving estimates for general oscillatory integral operators.[BCT06][68]
Compressed sensing and statistics
In collaboration with Emmanuel Candes and Justin Romberg, Tao has made notable contributions to the field of compressed sensing. In mathematical terms, most of their results identify settings in which a convex optimisation problem correctly computes the solution of an optimisation problem which seems to lack a computationally tractable structure. These problems are of the nature of finding the solution of an underdetermined linear system with the minimal possible number of nonzero entries, referred to as "sparsity". Around the same time, David Donoho considered similar problems from the alternative perspective of high-dimensional geometry.[69]
Motivated by striking numerical experiments, Candes, Romberg, and Tao first studied the case where the matrix is given by the discrete Fourier transform.[CRT06a] Candes and Tao abstracted the problem and introduced the notion of a "restricted linear isometry," which is a matrix that is quantitatively close to an isometry when restricted to certain subspaces.[CT05] They showed that it is sufficient for either exact or optimally approximate recovery of sufficiently sparse solutions. Their proofs, which involved the theory of convex duality, were markedly simplified in collaboration with Romberg, to use only linear algebra and elementary ideas of harmonic analysis.[CRT06b] These ideas and results were later improved by Candes.[70] Candes and Tao also considered relaxations of the sparsity condition, such as power-law decay of coefficients.[CT06] They complemented these results by drawing on a large corpus of past results in random matrix theory to show that, according to the Gaussian ensemble, a large number of matrices satisfy the restricted isometry property.[CT06]
In 2009, Candes and Benjamin Recht considered an analogous problem for recovering a matrix from knowledge of only a few of its entries and the information that the matrix is of low rank.[71] They formulated the problem in terms of convex optimisation, studying minimisation of the nuclear norm. Candes and Tao, in 2010, developed further results and techniques for the same problem.[CT10] Improved results were later found by Recht.[72] Similar problems and results have also been considered by a number of other authors.[73][74][75][76][77]
In 2007, Candes and Tao introduced a novel statistical estimator for linear regression, which they called the "Dantzig selector." They proved a number of results on its success as an estimator and model selector, roughly in parallel to their earlier work on compressed sensing.[CT07] A number of other authors have since studied the Dantzig selector, comparing it to similar objects such as the statistical lasso introduced in the 1990s.[78] Trevor Hastie, Robert Tibshirani, and Jerome H. Friedman conclude that it is "somewhat unsatisfactory" in a number of cases.[79] Nonetheless it remains of significant interest in the statistical literature.
Random matrices
In the 1950s, Eugene Wigner initiated the study of random matrices and their eigenvalues.[80][81] Wigner studied the case of hermitian and symmetric matrices, proving a "semicircle law" for their eigenvalues. In 2010, Tao and Van Vu made a major contribution to the study of non-symmetric random matrices. They showed that if n is large and the entries of a n × n matrix A are selected randomly according to any fixed probability distribution of average 0 and standard deviation 1, then the eigenvalues of A will tend to be uniformly scattered across the disk of radius n1/2 around the origin; this can be made precise using the language of measure theory.[TV10] This gave a proof of the long-conjectured circular law, which had previously been proved in weaker formulations by many other authors. In Tao and Vu's formulation, the circular law becomes an immediate consequence of a "universality principle" stating that the distribution of the eigenvalues can depend only on the average and standard deviation of the given component-by-component probability distribution, thereby providing a reduction of the general circular law to a calculation for specially-chosen probability distributions.
In 2011, Tao and Vu established a "four moment theorem", which applies to random hermitian matrices whose components are independently distributed, each with average 0 and standard deviation 1, and which are exponentially unlikely to be large (as for a Gaussian distribution). If one considers two such random matrices which agree on the average value of any quadratic polynomial in the diagonal entries and on the average value of any quartic polynomial in the off-diagonal entries, then Tao and Vu show that the expected value of a large number of functions of the eigenvalues will also coincide, up to an error which is uniformly controllable by the size of the matrix and which becomes arbitrarily small as the size of the matrix increases.[TV11] Similar results were obtained around the same time by László Erdös, Horng-Tzer Yau, and Jun Yin.[82][83]
Analytic number theory and arithmetic combinatorics
In 2004, Tao, together with Jean Bourgain and Nets Katz, studied the additive and multiplicative structure of subsets of finite fields of prime order.[BKT04] It is well known that there are no nontrivial subrings of such a field. Bourgain, Katz, and Tao provided a quantitative formulation of this fact, showing that for any subset of such a field, the number of sums and products of elements of the subset must be quantitatively large, as compared to the size of the field and the size of the subset itself. Improvements of their result were later given by Bourgain, Alexey Glibichuk, and Sergei Konyagin.[84][85]
Tao and Ben Green proved the existence of arbitrarily long arithmetic progressions in the prime numbers; this result is generally referred to as the Green–Tao theorem, and is among Tao's most well-known results.[GT08] The source of Green and Tao's arithmetic progressions is Endre Szemerédi's seminal 1975 theorem on existence of arithmetic progressions in certain sets of integers. Green and Tao showed that one can use a "transference principle" to extend the validity of Szemerédi's theorem to further sets of integers. The Green–Tao theorem then arises as a special case, although it is not trivial to show that the prime numbers satisfy the conditions of Green and Tao's extension of the Szemerédi theorem.
In 2010, Green and Tao gave a multilinear extension of Dirichlet's celebrated theorem on arithmetic progressions. Given a k × n matrix A and a k × 1 matrix v whose components are all integers, Green and Tao give conditions on when there exist infinitely many n × 1 matrices x such that all components of Ax + v are prime numbers.[GT10] The proof of Green and Tao was incomplete, as it was conditioned upon unproven conjectures. Those conjectures were proved in later work of Green, Tao, and Tamar Ziegler.[GTZ12]
Notable awards
• 1999 – Packard Fellowship
• 2000 – Salem Prize for:[86]
"his work in Lp harmonic analysis and on related questions in geometric measure theory and partial differential equations."
• 2002 – Bôcher Memorial Prize for:
Global regularity of wave maps I. Small critical Sobolev norm in high dimensions. Internat. Math. Res. Notices (2001), no. 6, 299-328.
Global regularity of wave maps II. Small energy in two dimensions. Comm. Math. Phys. 2244 (2001), no. 2, 443-544.
in addition to "his remarkable series of papers, written in collaboration with J. Colliander, M. Keel, G. Staffilani, and H. Takaoka, on global regularity in optimal Sobolev spaces for KdV and other equations, as well as his many deep contributions to Strichartz and bilinear estimates."
• 2003 – Clay Research Award for:[87]
his restriction theorems in Fourier analysis, his work on wave maps, his global existence theorems for KdV-type equations, and for his solution with Allen Knutson of Horn's conjecture
• 2005 – Australian Mathematical Society Medal
• 2005 – Ostrowski Prize (with Ben Green) for:
"their exceptional achievements in the area of analytic and combinatorial number theory"
• 2005 – Levi L.Conant Prize (with Allen Knutson) for:
their expository article "Honeycombs and Sums of Hermitian Matrices" (Notices of the AMS. 48 (2001), 175–186.)
• 2006 – Fields Medal for:
"his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory"
• 2006 – MacArthur Award
• 2006 – SASTRA Ramanujan Prize[88]
• 2006 – Sloan Fellowship
• 2007 – Fellow of the Royal Society[89]
• 2008 – Alan T. Waterman Award for:[90]
"his surprising and original contributions to many fields of mathematics, including number theory, differential equations, algebra, and harmonic analysis"
• 2008 – Onsager Medal[91] for:
"his combination of mathematical depth, width and volume in a manner unprecedented in contemporary mathematics". His Lars Onsager lecture was entitled "Structure and randomness in the prime numbers" at NTNU, Norway.[92]
• 2009 – Inducted into the American Academy of Arts and Sciences[93]
• 2010 – King Faisal International Prize
• 2010 – Nemmers Prize in Mathematics[94]
• 2010 – Polya Prize (with Emmanuel Candès)
• 2012 – Crafoord Prize[95][96]
• 2012 – Simons Investigator[97]
• 2014 – Breakthrough Prize in Mathematics
"For numerous breakthrough contributions to harmonic analysis, combinatorics, partial differential equations and analytic number theory."
• 2014 – Royal Medal
• 2015 – PROSE award in the category of "Mathematics" for:[98]
"Hilbert's Fifth Problem and Related Topics" ISBN 978-1-4704-1564-8
• 2019 – Riemann Prize[99]
• 2020 – Princess of Asturias Award for Technical and Scientific Research,[100] with Emmanuel Candès, for their work on compressed sensing
• 2020 – Bolyai Prize[101]
• 2021 – IEEE Jack S. Kilby Signal Processing Medal[102]
• 2021 – USIA Award
• 2022 – Education & Research award finalist
• 2022 - Global Australian of the Year (Advance Global Australians; Advance.org)[103][104]
• 2022 - Research.com Mathematics in United States Leader Award
• 2023 _ Grande Médaille
• 2023 _ Research.com Mathematics in United States Leader Award
Major publications
Textbooks
• — (2006). Solving mathematical problems. A personal perspective (Second edition of 1992 original ed.). Oxford: Oxford University Press. ISBN 978-0-19-920560-8. MR 2265113. Zbl 1098.00006.
• — (2006). Nonlinear dispersive equations. Local and global analysis. CBMS Regional Conference Series in Mathematics. Vol. 106. Providence, RI: American Mathematical Society. doi:10.1090/cbms/106. ISBN 0-8218-4143-2. MR 2233925. Zbl 1106.35001.
• —; Vu, Van H. (2006). Additive combinatorics. Cambridge Studies in Advanced Mathematics. Vol. 105. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511755149. ISBN 978-0-521-85386-6. MR 2289012. Zbl 1127.11002.[105][106]
• — (2008). Structure and randomness. Pages from year one of a mathematical blog. Providence, RI: American Mathematical Society. doi:10.1090/mbk/059. ISBN 978-0-8218-4695-7. MR 2459552. Zbl 1245.00024.
• — (2009). Poincaré's legacies, pages from year two of a mathematical blog. Part I. Providence, RI: American Mathematical Society. doi:10.1090/mbk/066. ISBN 978-0-8218-4883-8. MR 2523047. Zbl 1171.00003.
• — (2009). Poincaré's legacies, pages from year two of a mathematical blog. Part II. Providence, RI: American Mathematical Society. doi:10.1090/mbk/067. ISBN 978-0-8218-4885-2. MR 2541289. Zbl 1175.00010.
• — (2010). An epsilon of room, I: real analysis. Pages from year three of a mathematical blog (PDF). Graduate Studies in Mathematics. Vol. 117. Providence, RI: American Mathematical Society. doi:10.1090/gsm/117. ISBN 978-0-8218-5278-1. MR 2760403. Zbl 1216.46002.[107]
• — (2010). An epsilon of room, II. Pages from year three of a mathematical blog (PDF). Providence, RI: American Mathematical Society. doi:10.1090/mbk/077. ISBN 978-0-8218-5280-4. MR 2780010. Zbl 1218.00001.
• — (2011). An introduction to measure theory (PDF). Graduate Studies in Mathematics. Vol. 126. Providence, RI: American Mathematical Society. doi:10.1090/gsm/126. ISBN 978-0-8218-6919-2. MR 2827917. Zbl 1231.28001.[108]
• — (2012). Topics in random matrix theory (PDF). Graduate Studies in Mathematics. Vol. 132. Providence, RI: American Mathematical Society. doi:10.1090/gsm/132. ISBN 978-0-8218-7430-1. MR 2906465. Zbl 1256.15020.
• — (2012). Higher order Fourier analysis (PDF). Graduate Studies in Mathematics. Vol. 142. Providence, RI: American Mathematical Society. doi:10.1090/gsm/142. ISBN 978-0-8218-8986-2. MR 2931680. Zbl 1277.11010.
• — (2013). Compactness and contradiction (PDF). Providence, RI: American Mathematical Society. doi:10.1090/mbk/081. ISBN 978-0-8218-9492-7. MR 3026767. Zbl 1276.00007.
• — (2014). Analysis. I. Texts and Readings in Mathematics. Vol. 37 (Third edition of 2006 original ed.). New Delhi: Hindustan Book Agency. ISBN 978-93-80250-64-9. MR 3309891. Zbl 1300.26002.
• — (2014). Analysis. II. Texts and Readings in Mathematics. Vol. 38 (Third edition of 2006 original ed.). New Delhi: Hindustan Book Agency. ISBN 978-93-80250-65-6. MR 3310023. Zbl 1300.26003.
• — (2014). Hilbert's fifth problem and related topics. Graduate Studies in Mathematics. Vol. 153. Providence, RI: American Mathematical Society. doi:10.1090/gsm/153. ISBN 978-1-4704-1564-8. MR 3237440. Zbl 1298.22001.
• — (2015). Expansion in finite simple groups of Lie type. Graduate Studies in Mathematics. Vol. 164. Providence, RI: American Mathematical Society. doi:10.1090/gsm/164. ISBN 978-1-4704-2196-0. MR 3309986. S2CID 118288443. Zbl 1336.20015.[109]
Research articles. Tao is the author of over 300 articles. The following, among the most cited, are surveyed above.
KT98.
Keel, Markus; Tao, Terence (1998). "Endpoint Strichartz estimates". American Journal of Mathematics. 120 (5): 955–980. CiteSeerX 10.1.1.599.1892. doi:10.1353/ajm.1998.0039. JSTOR 25098630. MR 1646048. S2CID 13012479. Zbl 0922.35028.
TVV98.
Tao, Terence; Vargas, Ana; Vega, Luis (1998). "A bilinear approach to the restriction and Kakeya conjectures". Journal of the American Mathematical Society. 11 (4): 967–1000. doi:10.1090/S0894-0347-98-00278-1. MR 1625056. Zbl 0924.42008.
KT99.
Knutson, Allen; Tao, Terence (1999). "The honeycomb model of $GL_{n}(\mathbb {C} )$ tensor products. I. Proof of the saturation conjecture". Journal of the American Mathematical Society. 12 (4): 1055–1090. doi:10.1090/S0894-0347-99-00299-4. MR 1671451. Zbl 0944.05097.
C+01.
Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. (2001). "Global well-posedness for Schrödinger equations with derivative". SIAM Journal on Mathematical Analysis. 33 (3): 649–669. arXiv:math/0101263. doi:10.1137/S0036141001384387. MR 1871414. Zbl 1002.35113.
T01a.
Tao, Terence (2001). "Global regularity of wave maps. II. Small energy in two dimensions". Communications in Mathematical Physics. 224 (2): 443–544. arXiv:math/0011173. Bibcode:2001CMaPh.224..443T. doi:10.1007/PL00005588. MR 1869874. S2CID 119634411. Zbl 1020.35046. (Erratum: )
T01b.
Tao, Terence (2001). "Multilinear weighted convolution of L2-functions, and applications to nonlinear dispersive equations". American Journal of Mathematics. 123 (5): 839–908. arXiv:math/0005001. doi:10.1353/ajm.2001.0035. JSTOR 25099087. MR 1854113. S2CID 984131. Zbl 0998.42005.
C+02a.
Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. (2002). "A refined global well-posedness result for Schrödinger equations with derivative". SIAM Journal on Mathematical Analysis. 34 (1): 64–86. arXiv:math/0110026. doi:10.1137/S0036141001394541. MR 1950826. S2CID 9007785. Zbl 1034.35120.
C+02b.
Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. (2002). "Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation". Mathematical Research Letters. 9 (5–6): 659–682. doi:10.4310/MRL.2002.v9.n5.a9. MR 1906069. Zbl 1152.35491.
MTT02.
Muscalu, Camil; Tao, Terence; Thiele, Christoph (2002). "Multi-linear operators given by singular multipliers". Journal of the American Mathematical Society. 15 (2): 469–496. doi:10.1090/S0894-0347-01-00379-4. MR 1887641. Zbl 0994.42015.
CCT03.
Christ, Michael; Colliander, James; Tao, Terrence (2003). "Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations". American Journal of Mathematics. 125 (6): 1235–1293. arXiv:math/0203044. doi:10.1353/ajm.2003.0040. MR 2018661. S2CID 11001499. Zbl 1048.35101.
C+03.
Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. (2003). "Sharp global well-posedness for KdV and modified KdV on $\mathbb {R} $ and $\mathbb {T} $". Journal of the American Mathematical Society. 16 (3): 705–749. doi:10.1090/S0894-0347-03-00421-1. MR 1969209. Zbl 1025.35025.
T03.
Tao, T. (2003). "A sharp bilinear restrictions estimate for paraboloids". Geometric and Functional Analysis. 13 (6): 1359–1384. arXiv:math/0210084. doi:10.1007/s00039-003-0449-0. MR 2033842. S2CID 15873489. Zbl 1068.42011.
BKT04.
Bourgain, J.; Katz, N.; Tao, T. (2004). "A sum-product estimate in finite fields, and applications". Geometric and Functional Analysis. 14 (1): 27–57. arXiv:math/0301343. doi:10.1007/s00039-004-0451-1. MR 2053599. S2CID 14097626. Zbl 1145.11306.
C+04.
Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. (2004). "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ℝ3". Communications on Pure and Applied Mathematics. 57 (8): 987–1014. arXiv:math/0301260. doi:10.1002/cpa.20029. MR 2053757. S2CID 16423475. Zbl 1060.35131.
KTW04.
Knutson, Allen; Tao, Terence; Woodward, Christopher (2004). "The honeycomb model of $GL_{n}(\mathbb {C} )$ tensor products. II. Puzzles determine facets of the Littlewood–Richardson cone". Journal of the American Mathematical Society. 17 (1): 19–48. doi:10.1090/S0894-0347-03-00441-7. MR 2015329. Zbl 1043.05111.
T04a.
Tao, Terence (2004). "Global well-posedness of the Benjamin–Ono equation in H1(ℝ)". Journal of Hyperbolic Differential Equations. 1 (1): 27–49. arXiv:math/0307289. doi:10.1142/S0219891604000032. MR 2052470. Zbl 1055.35104.
T04b.
Tao, Terence (2004). "Fuglede's conjecture is false in 5 and higher dimensions". Mathematical Research Letters. 11 (2–3): 251–258. doi:10.4310/MRL.2004.v11.n2.a8. MR 2067470. Zbl 1092.42014.
CT05.
Candes, Emmanuel J.; Tao, Terence (2005). "Decoding by linear programming". IEEE Transactions on Information Theory. 51 (12): 4203–4215. arXiv:math/0502327. doi:10.1109/TIT.2005.858979. MR 2243152. S2CID 12605120. Zbl 1264.94121.
BT06.
Bejenaru, Ioan; Tao, Terence (2006). "Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation". Journal of Functional Analysis. 233 (1): 228–259. doi:10.1016/j.jfa.2005.08.004. MR 2204680. Zbl 1090.35162.
BCT06.
Bennett, Jonathan; Carbery, Anthony; Tao, Terence (2006). "On the multilinear restriction and Kakeya conjectures". Acta Mathematica. 196 (2): 261–302. doi:10.1007/s11511-006-0006-4. MR 2275834. Zbl 1203.42019.
CRT06a.
Candès, Emmanuel J.; Romberg, Justin K.; Tao, Terence (2006). "Stable signal recovery from incomplete and inaccurate measurements". Communications on Pure and Applied Mathematics. 59 (8): 1207–1223. arXiv:math/0503066. doi:10.1002/cpa.20124. MR 2230846. S2CID 119159284. Zbl 1098.94009.
CRT06b.
Candès, Emmanuel J.; Romberg, Justin; Tao, Terence (2006). "Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information". IEEE Transactions on Information Theory. 52 (2): 489–509. arXiv:math/0409186. doi:10.1109/TIT.2005.862083. MR 2236170. S2CID 7033413. Zbl 1231.94017.
CT06.
Candes, Emmanuel J.; Tao, Terence (2006). "Near-optimal signal recovery from random projections: universal encoding strategies?". IEEE Transactions on Information Theory. 52 (12): 5406–5425. arXiv:math/0410542. doi:10.1109/TIT.2006.885507. MR 2300700. S2CID 1431305. Zbl 1309.94033.
CT07.
Candes, Emmanuel; Tao, Terence (2007). "The Dantzig selector: statistical estimation when p is much larger than n". Annals of Statistics. 35 (6): 2313–2351. doi:10.1214/009053606000001523. MR 2382644. Zbl 1139.62019.
TVZ07.
Tao, Terence; Visan, Monica; Zhang, Xiaoyi (2007). "The nonlinear Schrödinger equation with combined power-type nonlinearities". Communications in Partial Differential Equations. 32 (7–9): 1281–1343. arXiv:math/0511070. doi:10.1080/03605300701588805. MR 2354495. S2CID 15109526. Zbl 1187.35245.
C+08.
Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. (2008). "Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ℝ3". Annals of Mathematics. Second Series. 167 (3): 767–865. doi:10.4007/annals.2008.167.767. MR 2415387. Zbl 1178.35345.
GT08.
Green, Ben; Tao, Terence (2008). "The primes contain arbitrarily long arithmetic progressions". Annals of Mathematics. Second Series. 167 (2): 481–547. doi:10.4007/annals.2008.167.481. MR 2415379. Zbl 1191.11025.
KTV09.
Killip, Rowan; Tao, Terence; Visan, Monica (2009). "The cubic nonlinear Schrödinger equation in two dimensions with radial data". Journal of the European Mathematical Society. 11 (6): 1203–1258. doi:10.4171/JEMS/180. MR 2557134. Zbl 1187.35237.
CT10.
Candès, Emmanuel J.; Tao, Terence (2010). "The power of convex relaxation: near-optimal matrix completion". IEEE Transactions on Information Theory. 56 (5): 2053–2080. arXiv:0903.1476. doi:10.1109/TIT.2010.2044061. MR 2723472. S2CID 1255437. Zbl 1366.15021.
GT10.
Green, Ben; Tao, Terence (2008). "The primes contain arbitrarily long arithmetic progressions". Annals of Mathematics. Second Series. 167 (2): 481–547. doi:10.4007/annals.2008.167.481. MR 2415379. Zbl 1191.11025.
TV10.
Tao, Terence; Vu, Van (2010). With an appendix by Manjunath Krishnapur. "Random matrices: universality of ESDs and the circular law". Annals of Probability. 38 (5): 2023–2065. doi:10.1214/10-AOP534. MR 2722794. Zbl 1203.15025.
TV11.
Tao, Terence; Vu, Van (2011). "Random matrices: universality of local eigenvalue statistics". Acta Mathematica. 206 (1): 127–204. doi:10.1007/s11511-011-0061-3. MR 2784665. Zbl 1217.15043.
GTZ12.
Green, Ben; Tao, Terence; Ziegler, Tamar (2012). "An inverse theorem for the Gowers Us+1[N]-norm". Annals of Mathematics. Second Series. 176 (2): 1231–1372. doi:10.4007/annals.2012.176.2.11. MR 2950773. Zbl 1282.11007.
T16.
Tao, Terence (2016). "Finite time blowup for an averaged three-dimensional Navier–Stokes equation". Journal of the American Mathematical Society. 29 (3): 601–674. doi:10.1090/jams/838. MR 3486169. Zbl 1342.35227.
Notes
1. Chinese: 陶象國; pinyin: Táo Xiàngguó
2. Chinese: 梁蕙蘭; Jyutping: Loeng⁴ Wai⁶-laan⁴
3. Being non-binary, Riley's pronouns are they/them.
See also
• Erdős discrepancy problem
• Inscribed square problem
• Goldbach's weak conjecture
• Cramer conjecture
References
1. "Vitae and Bibliography for Terence Tao". 12 October 2009. Retrieved 21 January 2010.
2. King Faisal Foundation – retrieved 11 January 2010.
3. "SIAM: George Pólya Prize". Archived from the original on 23 October 2021. Retrieved 5 September 2015.
4. "Mathematician Proves Huge Result on 'Dangerous' Problem". 11 December 2019. Archived from the original on 23 October 2021.
5. "Search | arXiv e-print repository".
6. Cook, Gareth (24 July 2015). "The Singular Mind of Terry Tao (Published 2015)". The New York Times. ISSN 0362-4331. Retrieved 15 February 2021.
7. "Primed for Success". 2 October 2007.
8. "PRESIDENT'S COUNCIL OF ADVISORS ON SCIENCE AND TECHNOLOGY: Terence Tao, PhD". 2021.
9. "Terence Tao, 'Mozart of Math,' is first UCLA math prof to win Fields Medal". 8 August 2006.
10. Terence Tao: the Mozart of maths, 7 March 2015, Stephanie Wood, The Sydney Morning Herald.
11. Wen Wei Po, Page A4, 24 August 2006.
12. Dr Billy Tao, Healthshare.
13. Oriental Daily, Page A29, 24 August 2006.
14. Terence Chi-Shen Tao, MacTutor History of Mathematics archive, School of Mathematics and Statistics, University of St Andrews, Scotland.
15. Nigel makes Waves: Google's bid to overthrow email, Asher Moses, Sydney Morning Herald, 2 October 2009
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17. "History, Travel, Arts, Science, People, Places – Smithsonian". Archived from the original on 10 September 2012. Retrieved 5 September 2015.
18. Wood, Stephanie (4 March 2015). "Terence Tao: the Mozart of maths". The Sydney Morning Herald. Retrieved 13 February 2023.
19. "Winners of Our Fourth Annual Podcast Contest". The New York Times. 1 July 2021. Retrieved 26 March 2023.
20. "Coming Out From Narnia". Soundcloud. 1 July 2021. Retrieved 8 April 2023.
21. Clements, M. A. (Ken) (1984), "Terence Tao", Educational Studies in Mathematics, 15 (3): 213–238, doi:10.1007/BF00312075, JSTOR 3482178, S2CID 189827772.
22. Radical acceleration in Australia: Terence Tao
23. "Radical Acceleration in Australia: Terence Tao". www.davidsongifted.org. Archived from the original on 23 October 2021.
24. "International Mathematical Olympiad".
25. It's prime time as numbers man Tao tops his Field Stephen Cauchi, 23 August 2006. Retrieved 31 August 2006.
26. Kenneth Chang (13 March 2007). "Journeys to the Distant Fields of Prime". The New York Times. Archived from the original on 23 October 2021.
27. "Corrections: For the Record". The New York Times. 13 March 2007. Archived from the original on 23 October 2021.
28. "Quanta Magazine". 24 February 2014.
29. "Terence Tao's Answer to the Erdős Discrepancy Problem". Quanta Magazine. October 2015. Archived from the original on 26 February 2019.
30. Tao, Terence (2019). "Almost all orbits of the Collatz map attain almost bounded values". arXiv:1909.03562 [math.PR].
31. Rodgers, Brad; Tao, Terence (6 April 2020). "The De Bruijn-Newman constant is non-negative". Forum of Mathematics, Pi. 8: e6. arXiv:1801.05914. doi:10.1017/fmp.2020.6..
32. Tao, Terence (2020). "Sendov's conjecture for sufficiently high degree polynomials". arXiv:2012.04125 [math.CV].
33. Mathematical Reviews MR2523047, Review by Timothy Gowers of Terence Tao's Poincaré's legacies, part I, http://mathscinet
34. NewScientist.com, Prestigious Fields Medals for mathematics awarded, 22 August 2006.
35. "2006 Fields Medals awarded" (PDF). Notices of the American Mathematical Society. 53 (9): 1037–1044. October 2006. Archived from the original (PDF) on 2 November 2006.
36. "Vitae". UCLA. Retrieved 5 September 2015.
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41. "President Biden Announces Members of President's Council of Advisors on Science and Technology". White House. 22 September 2021. Archived from the original on 23 October 2021.
42. "Terence Tao, il matematico con il QI più alto al mondo: "Non so cantare e ho fallito un paio di esami""". Huffington Post Italy. 21 September 2021. Archived from the original on 23 October 2021.
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47. "Highly Cited Researchers". publons.com. Archived from the original on 23 October 2021. Retrieved 6 February 2021.
48. Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis. On the ill-posedness of some canonical dispersive equations. Duke Math. J. 106 (2001), no. 3, 617–633.
49. Tataru, Daniel. On global existence and scattering for the wave maps equation. Amer. J. Math. 123 (2001), no. 1, 37–77.
50. Ionescu, Alexandru D.; Kenig, Carlos E. Global well-posedness of the Benjamin-Ono equation in low-regularity spaces. J. Amer. Math. Soc. 20 (2007), no. 3, 753–798.
51. Lemarié-Rieusset, Pierre Gilles (2016). The Navier–Stokes problem in the 21st century. Boca Raton, FL: CRC Press. doi:10.1201/b19556. ISBN 978-1-4665-6621-7. MR 3469428. Zbl 1342.76029.
52. Fuglede, Bent. Commuting self-adjoint partial differential operators and a group theoretic problem. J. Functional Analysis 16 (1974), 101–121.
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55. Coifman, Ronald R.; Meyer, Yves Au delà des opérateurs pseudo-différentiels. Astérisque, 57. Société Mathématique de France, Paris, 1978. i+185 pp.
56. Lacey, Michael; Thiele, Christoph. Lp estimates on the bilinear Hilbert transform for 2<p<∞. Ann. of Math. (2) 146 (1997), no. 3, 693–724.
57. Lacey, Michael; Thiele, Christoph On Calderón's conjecture. Ann. of Math. (2) 149 (1999), no. 2, 475–496.
58. Kenig, Carlos E.; Stein, Elias M. Multilinear estimates and fractional integration. Math. Res. Lett. 6 (1999), no. 1, 1–15.
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60. Ginibre, Jean. Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain). Séminaire Bourbaki, Vol. 1994/95. Astérisque No. 237 (1996), Exp. No. 796, 4, 163–187.
61. Fefferman, Charles. Inequalities for strongly singular convolution operators. Acta Math. 124 (1970), 9–36.
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63. Strichartz, Robert S. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44 (1977), no. 3, 705–714.
64. Bourgain, J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 3 (1993), no. 2, 107–156
65. Bourgain, J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation. Geom. Funct. Anal. 3 (1993), no. 3, 209–262.
66. Klainerman, S.; Machedon, M. Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46 (1993), no. 9, 1221–1268.
67. Wolff, Thomas. A sharp bilinear cone restriction estimate. Ann. of Math. (2) 153 (2001), no. 3, 661–698.
68. Bourgain, Jean; Guth, Larry. Bounds on oscillatory integral operators based on multilinear estimates. Geom. Funct. Anal. 21 (2011), no. 6, 1239–1295.
69. Donoho, David L. Compressed sensing. IEEE Trans. Inform. Theory 52 (2006), no. 4, 1289–1306.
70. Candès, Emmanuel J. The restricted isometry property and its implications for compressed sensing. C. R. Math. Acad. Sci. Paris 346 (2008), no. 9-10, 589–592.
71. Candès, Emmanuel J.; Recht, Benjamin Exact matrix completion via convex optimization. Found. Comput. Math. 9 (2009), no. 6, 717–772.
72. Recht, Benjamin A simpler approach to matrix completion. J. Mach. Learn. Res. 12 (2011), 3413–3430.
73. Keshavan, Raghunandan H.; Montanari, Andrea; Oh, Sewoong Matrix completion from a few entries. IEEE Trans. Inform. Theory 56 (2010), no. 6, 2980–2998.
74. Recht, Benjamin; Fazel, Maryam; Parrilo, Pablo A. Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52 (2010), no. 3, 471–501.
75. Candès, Emmanuel J.; Plan, Yaniv Tight oracle inequalities for low-rank matrix recovery from a minimal number of noisy random measurements. IEEE Trans. Inform. Theory 57 (2011), no. 4, 2342–2359.
76. Koltchinskii, Vladimir; Lounici, Karim; Tsybakov, Alexandre B. Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion. Ann. Statist. 39 (2011), no. 5, 2302–2329.
77. Gross, David Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. Inform. Theory 57 (2011), no. 3, 1548–1566.
78. Bickel, Peter J.; Ritov, Ya'acov; Tsybakov, Alexandre B. Simultaneous analysis of lasso and Dantzig selector. Ann. Statist. 37 (2009), no. 4, 1705–1732.
79. Hastie, Trevor; Tibshirani, Robert; Friedman, Jerome The elements of statistical learning. Data mining, inference, and prediction. Second edition. Springer Series in Statistics. Springer, New York, 2009. xxii+745 pp. ISBN 978-0-387-84857-0
80. Wigner, Eugene P. Characteristic vectors of bordered matrices with infinite dimensions. Annals of Mathematics (2) 62 (1955), 548–564.
81. Wigner, Eugene P. On the distribution of the roots of certain symmetric matrices. Ann. of Math. (2) 67 (1958), 325–327.
82. Erdős, László; Yau, Horng-Tzer; Yin, Jun (2012). "Rigidity of eigenvalues of generalized Wigner matrices". Advances in Mathematics. 229 (3): 1435–1515. arXiv:1007.4652. doi:10.1016/j.aim.2011.12.010.
83. Erdős, László; Yau, Horng-Tzer; Yin, Jun. Bulk universality for generalized Wigner matrices. Probab. Theory Related Fields 154 (2012), no. 1-2, 341–407.
84. Bourgain, J. More on the sum-product phenomenon in prime fields and its applications. Int. J. Number Theory 1 (2005), no. 1, 1–32.
85. Bourgain, J.; Glibichuk, A.A.; Konyagin, S.V. Estimates for the number of sums and products and for exponential sums in fields of prime order. J. London Math. Soc. (2) 73 (2006), no. 2, 380–398.
86. Mathematics People. Notices of the AMS
87. Clay Research Awards.
88. Alladi, Krishnaswami (9 December 2019). "Ramanujan's legacy: the work of the SASTRA prize winners". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. The Royal Society. 378 (2163): 20180438. doi:10.1098/rsta.2018.0438. ISSN 1364-503X. PMID 31813370. S2CID 198231874.
89. Fellows and Foreign Members of the Royal Society, retrieved 9 June 2010.
90. National Science Foundation, Alan T. Waterman Award. Retrieved 18 April 2008.
91. "The Lars Onsager Lecture and Professorship – IMF". Archived from the original on 3 February 2009. Retrieved 13 January 2009.
92. NTNU's Onsager Lecture, by Terence Tao on YouTube
93. "Alphabetical Index of Active AAAS Members" (PDF). amacad.org. Archived from the original (PDF) on 5 October 2013. Retrieved 21 November 2013.
His 2009 induction ceremony is here.
94. "Major Math and Science Awards Announced: Northwestern University News". Archived from the original on 16 April 2010. Retrieved 5 September 2015.
95. "The Crafoord Prize in Mathematics 2012 and The Crafoord Prize in Astronomy 2012". Royal Swedish Academy of Sciences. 19 January 2012. Archived from the original on 23 October 2021. Retrieved 13 November 2014.
96. "4 Scholars Win Crafoord Prizes in Astronomy and Math – The Ticker – Blogs – The Chronicle of Higher Education". 19 January 2012. Archived from the original on 23 October 2021. Retrieved 5 September 2015.
97. "Simons Investigators Awardees". Simons Foundation. Archived from the original on 23 October 2021. Retrieved 9 September 2017.
98. PROSE 2015 winners
99. "Riemann Prize laureate 2019: Terence Tao". Archived from the original on 20 December 2019. Retrieved 23 November 2019.
100. "Yves Meyer, Ingrid Daubechies, Terence Tao and Emmanuel Candès, Princess of Asturias Award for Technical and Scientific Research 2020". Princess of Asturias Foundation. Archived from the original on 26 June 2020. Retrieved 23 June 2020.
101. "Vitae and Bibliography for Terence Tao". UCLA. Retrieved 13 November 2020.
102. "IEEE Awards". IEEE Awards. 27 June 2022. Retrieved 10 September 2022.
103. World’s greatest mathematician named 2022 Global Australian of the Year, Advance.org, media release 2022-09-08, accessed 2022-09-14
104. Why this maths genius refuses to work for a hedge fund, Tess Bennett, Australian Financial Review, 2022-09-07, accessed 2022-09-14
105. Green, Ben (2009). "Review: Additive combinatorics by Terence C. Tao and Van H. Vu" (PDF). Bull. Amer. Math. Soc. (N.S.). 46 (3): 489–497. doi:10.1090/s0273-0979-09-01231-2. Archived from the original (PDF) on 11 March 2012.
106. Vestal, Donald L. (6 June 2007). "Review of Additive Combinatorics by Terence Tao and Van H. Vu". MAA Reviews, Mathematical Association of America.
107. Stenger, Allen (4 March 2011). "Review of A Epsilon of Room, I: Real Analysis: Pages from year three of a mathematical blog by Terence Tao". MAA Reviews, Mathematical Association of America.
108. Poplicher, Mihaela (14 April 2012). "Review of An Introduction to Measure Theory by Terence Tao". MAA Reviews, Mathematical Association of America.
109. Lubotzky, Alexander (25 January 2018). "Review of Expansion in finite simple groups of Lie type by Terence Tao". Bull. Amer. Math. Soc. (N.S.): 1. doi:10.1090/bull/1610; review published electronically{{cite journal}}: CS1 maint: postscript (link)
External links
Wikiquote has quotations related to Terence Tao.
Wikimedia Commons has media related to Terence Tao.
• Terence Tao's home page
• Tao's research blog
• Tao's MathOverflow page
• O'Connor, John J.; Robertson, Edmund F., "Terence Tao", MacTutor History of Mathematics Archive, University of St Andrews
• Terence Tao at the Mathematics Genealogy Project
• Terence Tao's entry in the Numericana Hall of Fame
• Terence Tao's results at International Mathematical Olympiad
• "Terence Tao: Structure and Randomness in the Prime Numbers, UCLA". YouTube. UCLA. 22 January 2009.
• "Terence Tao: Nilsequences and the Primes, UCLA". YouTube. UCLA. 30 January 2009.
• "Minerva Lectures 2013 – Terence Tao Talk 1: Sets with few ordinary lines". YouTube. princetonmathematics. 24 May 2013.
• "Minerva Lectures 2013 – Terence Tao Talk 2: Polynomial expanders and an algebrai regularity lemma". YouTube. princetonmathematics. 24 May 2013.
• "Minerva Lectures 2013 – Terence Tao Talk 3: Universality for Wigner random matrices". YouTube. princetonmathematics. 24 May 2013.
• "Ultraproducts as a Bridge Between Discrete and Continuous Mathematics". YouTube. Simons Institute. 12 December 2013.
• "Terry Tao, Ph.D. Small and Large Gaps Between the Primes". YouTube. UCLA. 7 October 2014.
• "Terence Tao: 2015 Breakthrough Prize in Mathematics Symposium". YouTube. Breakthrough. 4 December 2014.
• "Can the Navier-Stokes Equations Blow Up in Finite Time? | Prof. Terence Tao". YouTube. The Israel Academy of Science and Humanities. 25 March 2015.
• "Terence Tao: The Erdős Discrepancy Problem". YouTube. Institute for Pure & Applied Mathematics (IPAM). 9 October 2015.
• "Terence Tao: An integration approach to the Toeplitz square peg problem". YouTube. Centre International de Recontres Mathématiques. 13 October 2017.
• "Terence Tao: Vaporizing and freezing the Riemann zeta function". YouTube. Università degli Studi di Milano – Bicocca. 27 July 2018.
• "RIA Hamilton Lecture 2020 – Professor Terence Tao". YouTube. The Royal Irish Academy. 16 October 2020.
• "An Interview with Terence Tao". YouTube. Young Scientists Journal. 16 December 2020.
• "Terence Tao (UCLA): Pseudorandomness of the Louisville function". YouTube. Hausdorr Center for Mathematics. 3 May 2021.
• "Day 2 – The notorious Collatz conjecture – Terence Tao". YouTube. Università degli Studi dell'Insubria. 30 October 2021. (See Collatz conjecture.)
• "Day 3 – Sendov's conjecture for sufficiently high degree polynomials – Terence Tao". YouTube. Università degli Studi dell'Insubria. 30 October 2021. (See Sendov's conjecture.)
• "Day 3 – Interview to Terence Tao – Umberto Bottazzini". YouTube. Università degli Studi dell'Insubria. 30 October 2021.
• "Day 3 – Singmaster's conjecture in the interior of Pascal's triangle – Terence Tao". YouTube. Università degli Studi dell'Insubria. 30 October 2021. (See Singmaster's conjecture.)
Fellows of the Royal Society elected in 2007
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Breakthrough Prize laureates
Mathematics
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• Clifford P. Brangwynne, Anthony A. Hyman, Demis Hassabis, John Jumper, Emmanuel Mignot, Masashi Yanagisawa (2023)
Laureates of the Prince or Princess of Asturias Award for Technical and Scientific Research
Prince of Asturias Award for Technical and Scientific Research
1980s
• 1981: Alberto Sols
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Princess of Asturias Award for Technical and Scientific Research
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• 2015: Emmanuelle Charpentier and Jennifer Doudna
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• 2018: Svante Pääbo
• 2019: Joanne Chory and Sandra Myrna Díaz
2020s
• 2020: Yves Meyer, Ingrid Daubechies, Terence Tao and Emmanuel Candès
• 2021: Katalin Karikó, Drew Weissman, Philip Felgner, Uğur Şahin, Özlem Türeci, Derrick Rossi and Sarah Gilbert
• 2022: Geoffrey Hinton, Yann LeCun, Yoshua Bengio and Demis Hassabis
• 2023: Jeffrey I. Gordon, Everett Peter Greenberg and Bonnie Bassler
Recipients of SASTRA Ramanujan Prize
• Manjul Bhargava (2005)
• Kannan Soundararajan (2005)
• Terence Tao (2006)
• Ben Green (2007)
• Akshay Venkatesh (2008)
• Kathrin Bringmann (2009)
• Wei Zhang (2010)
• Roman Holowinsky (2011)
• Zhiwei Yun (2012)
• Peter Scholze (2013)
• James Maynard (2014)
• Jacob Tsimerman (2015)
• Kaisa Matomäki (2016)
• Maksym Radziwill (2016)
• Maryna Viazovska (2017)
• Yifeng Liu (2018)
• Jack Thorne (2018)
• Adam Harper (2019)
• Shai Evra (2020)
• Will Sawin (2021)
• Yunqing Tang (2022)
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| Wikipedia |
Teresa Cohen
Teresa Cohen (February 14, 1892 – August 10, 1992) was an American mathematician.[1]
Teresa Cohen
Born(1892-02-14)February 14, 1892
Baltimore, Maryland, U.S.
DiedAugust 10, 1992(1992-08-10) (aged 100)
OccupationMathematician
NationalityAmerican
EducationGoucher College (BA)
Johns Hopkins University (MA)
Early life and education
She was born in Baltimore, Maryland to Rebecca Sinsheimer and Benjamin Cohen.[2][3] She graduated in 1909[2] from the Friends School of Baltimore whose teachers she credited with her interest in mathematics and teaching.[3] She earned her Bachelor of Arts degree in mathematics and physics at Goucher College in 1912. Cohen was resident fellow at Goucher from 1912 to 1913.[2] In 1915, she earned a Master of Arts degree from Johns Hopkins University[2] where she later earned her PhD in 1918.[3] She was one of the first women in the United States to earn a doctorate in Mathematics.[4] She completed her dissertation entitled "Investigations on the Plane Quartic" under doctoral advisor Frank Morley. Cohen also acknowledged the support of professors Cohen and Arthur Byron Coble of Johns Hopkins, and Clara Latimer Bacon and Florence Lewis of Goucher College.[2]
Career
Dr. Cohen was invited to join the faculty of The Pennsylvania State University in 1920 and became the first woman to serve on the Mathematics faculty.[4] She advanced to the rank of full professor, one of only a handful of women to have that status at Penn State at that time. Due to University regulations she officially retired in 1962, but she maintained an office in the Department of Mathematics and tutored students for free until 1985 at the age of 94, when an accident forced her to return to her native Baltimore and enter a nursing home.
She had been a member of the American Mathematical Society, the Mathematical Association of America, Pi Mu Epsilon, and Sigma Delta Epsilon, the national honor society for women in science.
The works she published included four papers on investigations of the plane quartic, and a co-authored paper with William Knight about the convergence and divergence of the p-series
$\sum _{n=1}^{\infty }n^{-p},$
in which they gave proofs that could be understood by persons not familiar with the integral test for convergence of a series.
Personal life
Aside from teaching, mathematics, and her local synagogue, Dr. Cohen's main interest was music. She was an amateur violinist. Cohen died of pneumonia in Baltimore in 1992 at the age of 100. She was survived by her sister, nieces, and a nephew. At the time of her death, Cohen was the oldest surviving Goucher College alumna and member of the Mathematical Association of America. The Teresa Cohen Tutorial Endowment Fund at Pennsylvania State University was established in her honor. She was interred at Temple Oheb Shalom cemetery.[4]
References
1. Green, Judy; LaDuke, Jeanne (2009). Pioneering Women in American Mathematics — The Pre-1940 PhD's. History of Mathematics. Vol. 34. American Mathematical Society, The London Mathematical Society. ISBN 978-0-8218-4376-5. Biography on p.128-130 of the Supplementary Material at AMS
2. Cohen, Teresa (1919). Investigations on the plane quartic. Baltimore: American journal of mathematics. pp. 191–211. hdl:2027/mdp.39015079994953.
3. Teresa Cohen Biography.
4. "Obituary. Teresa Cohen". The Baltimore Sun. 12 August 1992. Retrieved 2018-03-29.
Authority control
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| Wikipedia |
Teresa Melo
Maria Teresa Rocha de Magalhães Melo (born 1966) is a Portuguese mathematician and operations researcher who works as a professor in the business school of the Saarland University of Applied Sciences (htw saar), a Fachhochschule in Saarbrücken, Germany.[1] Her research interests include facility location, supply chain management, and in-hospital patient transport.[2]
Education and career
Melo was born in 1966 in Lisbon,[2] and studied applied mathematics at the University of Lisbon, graduating in 1989 and earning a master's degree there in statistics and operations research in 1991. Next, she went to the Econometric Institute of Erasmus University Rotterdam for doctoral study, completing a PhD in operations research in 1996[1] with a dissertation concerning the economic lot scheduling problem.[2]
She took postdoctoral research positions at the Forschungszentrum Jülich and Fraunhofer Institute for Industrial Mathematics in 1997 and 1999 respectively, and became an invited lecturer at the Technical University of Kaiserslautern in 2000.[1] She moved to htw saar as an associate professor in 2007,[1][2] and became a full professor in 2011. At htw saar, she is also founding co-director of the Institute for Supply Chain and Operations Management.[1]
References
1. Curriculum vitae (PDF), January 2020, retrieved 2021-05-20
2. Neue Professorin für Mathematik und Statistik an der HTW (in German), Ministerium für Bildung und Kultur Saarland, 27 March 2007, retrieved 2021-05-20 – via bildungsklick
External links
• Teresa Melo publications indexed by Google Scholar
• Faculty profile
Authority control: Academics
• DBLP
• Google Scholar
• MathSciNet
• ORCID
• Scopus
• zbMATH
| Wikipedia |
Teri Perl
Teri Perl (born November 19, 1926)[1] is an American mathematics educator, author of mathematics resource books, and a co-founder of The Learning Company.[2]
Teri Perl
Born (1926-11-19) November 19, 1926
New York City, New York
Known forco-found The Learning Company
Academic background
Education
• Brooklyn College
• San Jose State University
• Stanford University
ThesisDiscriminating Factors and Sex Differences in Electing Mathematics (1979)
Academic work
DisciplineMathematics education
Education and career
Perl was born in New York City, New York. In 1947, she received a bachelor's degree in economics from Brooklyn College in New York. She did post-baccalaureate work at San Jose State University in California and earned a secondary mathematics teaching credential there in 1969. While doing substitute teaching in the San Francisco Bay Area, Perl discovered that there weren't adequate resources for substitute teachers. She began a long-distance collaboration with Miriam K. Freedman to produce A Sourcebook for Substitutes and Other Teachers. Freedman developed the material for teachers of English, foreign languages, and social science and Perl for teachers of mathematics and science.[3] Perl was a mathematics consultant and resource teacher at Ventura Elementary School in Palo Alto, California from 1971 to 1978.
After her youngest child was in high school, Perl attended Stanford University in Palo Alto, California and obtained a Ph.D. in mathematics education in 1979. The title of her dissertation was "Discriminating Factors and Sex Differences in Electing Mathematics". From 1971 to 1979, while working on her doctorate at Stanford, Perl taught part-time at San Francisco State University.[3]
In 1980, Perl, Ann McCormick, Leslie Grimm, and Warren Robinett founded The Learning Company (TLC), an educational software company that developed grade-based learning software, as well as edutainment games and productivity tools. Perl was the content designer for TLC's Math Rabbit.[4] She was responsible for writing users'/teachers' guides to accompany several software packages produced by TLC. In 1995, TLC was taken over by SoftKey, who changed their name to The Learning Company. After several sales, The Learning Company brand was used by Houghton Mifflin Harcourt through 2018.
In 1974, Perl, together with a group of San Francisco Bay area women, founded the Math/Science Network[5][6] to encourage and inspire middle and high school girls to pursue careers in STEM. The network was rebranded as the Expanding Your Horizons (EYH) Network in 1982 when EYH obtained nonprofit status. The MSN/EYH developed a one-day workshop for middle and high school girls with hands-on workshops and interactions with women in STEM careers. Perl served as president of the EYH Network from 1999 to 2007.[3] EYH has run over 80 Career Days, with up to 25,000 girls participating each year.[7] In 2010, the EYH Network was honored by the National Science Board with their Public Service Award "for its decades-long commitment to the early development of interest in mathematics and science among young women, making significant strides toward its goal to develop a pool of qualified women to undertake careers in mathematics, science and engineering."[8]
In March 1979 , Perl's book Math Equals: Biographies of Women Mathematicians + Related Activities received a very positive review by the American Library Association's publication Choice.
"An excellent, documented historical account of contributions made by somen interested in mathematics presented in a most unusual and readable format. The attention of a reader in mathematics is captivated immediately..."[9]
In 2005, GirlSource[10] honored Perl with its WAVE (Women of Achievement, Vision and Excellence) award.[3][11] In 2022 she became a fellow of the Association for Women in Mathematics, "For amazing and tireless efforts over five decades to promote women in mathematics and related fields; particularly, for co-founding what we now know as Expanding Your Horizons, her biographies of women mathematicians, and her influential role in The Learning Company, which have together inspired generations of women and girls."[12]
Selected publications
• Perl, Teri (February 1978). Math Equals: Biographies of Women Mathematicians + Related Activities. Addison Wesley Innovative Series. Menlo Park, CA: Addison-Wesley. p. 250. ISBN 978-0201057096.
• Morrow, Charlene; Perl, Teri, eds. (May 30, 1998). Notable Women in Mathematics: A Biographical Dictionary. Westport, CT: Greenwood. p. 320. ISBN 978-0313291319.
• Book review: Forgasz, Helen (1999). "The Lives and Contributions of Women Mathematicians and Educators". Mathematics Education Research Journal. 11 (2): 154–155. Bibcode:1999MEdRJ..11..154F. doi:10.1007/BF03217068. S2CID 178797322. Retrieved 14 March 2021.
• Freedman, Miriam K.; Perl, Teri (1974). A Sourcebook for Substitutes and Other Teachers. Parsippany, NJ: Dale Seymour Publications/Pearson Learning. ISBN 978-0201057867.
• Perl, Teri Hoch (1979). "The Ladies' Diary or Woman's Almanack". Historia Mathematica. 6 (1): 36–53. doi:10.1016/0315-0860(79)90103-4. MR 0518839.
References
1. "Birth date from ISNI authority control file". ISNI. Retrieved 11 February 2021.
2. Who's who of American Women, 1991-1992. Marquis Who's Who. 1991. p. 771. ISBN 9780837904177.
3. Dewar, Jacqueline M. (2017). "Celebrating the Contributions of Three Women to Mathematics Teaching and Learning". In Beery, J.; Greenwald, S.; Jensen-Vallin, J.; Mast, M. (eds.). Women in Mathematics: Celebrating the Centennial of the Mathematical Association of America. Association for Women in Mathematics Series. Vol. 10. New York: Springer International Publishing. pp. 141–156. doi:10.1007/978-3-319-66694-5_8. ISBN 978-3-319-66693-8. ISSN 2364-5733.
4. "Teri Perl". SDSC Staff Home Pages. Retrieved 2021-02-10.
5. Lenore Blum. "Celebrating 30 years of effective change" (PDF). Carnegie Mellon University. Retrieved 2 March 2021.
6. "Math Science Network Reviews and Ratings - Oakland, CA - Donate, Volunteer, Review". GreatNonprofits. Retrieved 2021-02-10.
7. "EYH History : Techbridge – Inspiring Girls in Science, Technology and Engineering". Home. Retrieved 2021-02-10.
8. "NSB and NSF Recognize Extraordinary Science, Service with Annual Awards". NSF. 2010-05-06. Retrieved 2021-02-10.
9. "Perl, Teri. Math equals: biographies of women mathematicians and related activities". Choice. American Library Association: 114. March 1979.
10. "Girlsource, Inc. Reviews and Ratings - San Francisco, CA - Donate, Volunteer, Review". GreatNonprofits. Retrieved 2021-02-10.
11. "2005 WAVE Award Winners". GirlSource. Archived from the original on 2005-10-27. Retrieved 13 March 2021.
12. "2022 AWM Fellows". Association for Women in Mathematics. Retrieved 6 December 2022.
Authority control
International
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| Wikipedia |
Term discrimination
Term discrimination is a way to rank keywords in how useful they are for information retrieval.
Overview
This is a method similar to tf-idf but it deals with finding keywords suitable for information retrieval and ones that are not. Please refer to Vector Space Model first.
This method uses the concept of Vector Space Density that the less dense an occurrence matrix is, the better an information retrieval query will be.
An optimal index term is one that can distinguish two different documents from each other and relate two similar documents. On the other hand, a sub-optimal index term can not distinguish two different document from two similar documents.
The discrimination value is the difference in the occurrence matrix's vector-space density versus the same matrix's vector-space without the index term's density.
Let:
$A$ be the occurrence matrix
$A_{k}$ be the occurrence matrix without the index term $k$
and $Q(A)$ be density of $A$.
Then:
The discrimination value of the index term $k$ is:
$DV_{k}=Q(A)-Q(A_{k})$
How to compute
Given an occurrency matrix: $A$ and one keyword: $k$
• Find the global document centroid: $C$ (this is just the average document vector)
• Find the average euclidean distance from every document vector, $D_{i}$ to $C$
• Find the average euclidean distance from every document vector, $D_{i}$ to $C$ IGNORING $k$
• The difference between the two values in the above step is the discrimination value for keyword $K$
A higher value is better because including the keyword will result in better information retrieval.
Qualitative Observations
Keywords that are sparse should be poor discriminators because they have poor recall, whereas keywords that are frequent should be poor discriminators because they have poor precision.
References
• G. Salton, A. Wong, and C. S. Yang (1975), "A Vector Space Model for Automatic Indexing," Communications of the ACM, vol. 18, nr. 11, pages 613–620. (The article in which the vector space model was first presented)
• Can, F., Ozkarahan, E. A (1987), "Computation of term/document discrimination values by use of the cover coefficient concept." Journal of the American Society for Information Science, vol. 38, nr. 3, pages 171-183.
| Wikipedia |
Terminal digit preference
Terminal digit preference, terminal digit bias, or end-digit preference is a commonly-observed statistical phenomenon whereby humans recording numbers have a bias or preference for a specific final digit in a number. In medical science, this is often seen when recording measurements such as blood pressure by hand, where those taking measurements will round to the nearest 5 or 0.[1] The phenomenon has been blamed for misdiagnoses.[2] Terminal digit bias has been used to identify errors in research,[3][4][5][6] and is one method used in the identification of scientific fraud.[7] Severe terminal digit bias has been found in datasets for scientific papers that were later retracted [8][9]
See also
• Benford's law
References
1. Thavarajah (1 December 2003). "Terminal digit bias in a specialty hypertension faculty practice". Nature. 17 (12): 819–822. doi:10.1038/sj.jhh.1001625. PMID 14704725.
2. Nietert, Paul J.; Wessell, Andrea; Feifer, Chris; Ornstein, Steven (2006). "Effect of Terminal Digit Preference on Blood Pressure Measurement and Treatment in Primary Care". American Journal of Hypertension. 19 (2): 147–152. doi:10.1016/j.amjhyper.2005.08.016. PMID 16448884. S2CID 25597886.
3. Thavarajah (1 December 2003). "Terminal digit bias in a specialty hypertension faculty practice". Nature. 17 (12): 819–822. doi:10.1038/sj.jhh.1001625. PMID 14704725.
4. Hla, Khin (1986). "Observer Error in Systolic Blood Pressure Measurement in the Elderly". Arch Intern Med. 146 (12): 2373. doi:10.1001/archinte.1986.00360240099017. Retrieved 29 September 2021.
5. Hayes (2008). "Terminal digit preference occurs in pathology reporting irrespective of patient management implication". Journal of Clinical Pathology. 61 (9): 1071–1072. doi:10.1136/jcp.2008.059543. PMID 18755731. S2CID 10737432. Retrieved 29 September 2021.
6. Lusignan (23 March 2004). "End-digit preference in blood pressure recordings of patients with ischaemic heart disease in primary care". Nature. 18 (4): 261–265. doi:10.1038/sj.jhh.1001663. PMID 15037875. S2CID 430764. Retrieved 29 September 2021.
7. Lawrence, Jack (22 September 2021). "The lesson of ivermectin: meta-analyses based on summary data alone are inherently unreliable". Nature. 27 (11): 1853–1854. doi:10.1038/s41591-021-01535-y. PMID 34552263.
8. Brown, Nick. "Dr". More problematic articles from the Food and Brand Lab. Nick Brown. Retrieved 30 November 2021.
9. Retraction Watch (19 September 2018). "JAMA journals retract six papers by food marketing researcher Brian Wansink". Retraction Watch. Retrieved 30 November 2021.
| Wikipedia |
Initial and terminal objects
In category theory, a branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X.
"Zero object" redirects here. For zero object in an algebraic structure, see zero object (algebra).
The dual notion is that of a terminal object (also called terminal element): T is terminal if for every object X in C there exists exactly one morphism X → T. Initial objects are also called coterminal or universal, and terminal objects are also called final.
If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object.
A strict initial object I is one for which every morphism into I is an isomorphism.
Examples
• The empty set is the unique initial object in Set, the category of sets. Every one-element set (singleton) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category.
• In the category Rel of sets and relations, the empty set is the unique initial object, the unique terminal object, and hence the unique zero object.
• In the category of pointed sets (whose objects are non-empty sets together with a distinguished element; a morphism from (A, a) to (B, b) being a function f : A → B with f(a) = b), every singleton is a zero object. Similarly, in the category of pointed topological spaces, every singleton is a zero object.
• In Grp, the category of groups, any trivial group is a zero object. The trivial object is also a zero object in Ab, the category of abelian groups, Rng the category of pseudo-rings, R-Mod, the category of modules over a ring, and K-Vect, the category of vector spaces over a field. See Zero object (algebra) for details. This is the origin of the term "zero object".
• In Ring, the category of rings with unity and unity-preserving morphisms, the ring of integers Z is an initial object. The zero ring consisting only of a single element 0 = 1 is a terminal object.
• In Rig, the category of rigs with unity and unity-preserving morphisms, the rig of natural numbers N is an initial object. The zero rig, which is the zero ring, consisting only of a single element 0 = 1 is a terminal object.
• In Field, the category of fields, there are no initial or terminal objects. However, in the subcategory of fields of fixed characteristic, the prime field is an initial object.
• Any partially ordered set (P, ≤) can be interpreted as a category: the objects are the elements of P, and there is a single morphism from x to y if and only if x ≤ y. This category has an initial object if and only if P has a least element; it has a terminal object if and only if P has a greatest element.
• Cat, the category of small categories with functors as morphisms has the empty category, 0 (with no objects and no morphisms), as initial object and the terminal category, 1 (with a single object with a single identity morphism), as terminal object.
• In the category of schemes, Spec(Z), the prime spectrum of the ring of integers, is a terminal object. The empty scheme (equal to the prime spectrum of the zero ring) is an initial object.
• A limit of a diagram F may be characterised as a terminal object in the category of cones to F. Likewise, a colimit of F may be characterised as an initial object in the category of co-cones from F.
• In the category ChR of chain complexes over a commutative ring R, the zero complex is a zero object.
Properties
Existence and uniqueness
Initial and terminal objects are not required to exist in a given category. However, if they do exist, they are essentially unique. Specifically, if I1 and I2 are two different initial objects, then there is a unique isomorphism between them. Moreover, if I is an initial object then any object isomorphic to I is also an initial object. The same is true for terminal objects.
For complete categories there is an existence theorem for initial objects. Specifically, a (locally small) complete category C has an initial object if and only if there exist a set I (not a proper class) and an I-indexed family (Ki) of objects of C such that for any object X of C, there is at least one morphism Ki → X for some i ∈ I.
Equivalent formulations
Terminal objects in a category C may also be defined as limits of the unique empty diagram 0 → C. Since the empty category is vacuously a discrete category, a terminal object can be thought of as an empty product (a product is indeed the limit of the discrete diagram {Xi}, in general). Dually, an initial object is a colimit of the empty diagram 0 → C and can be thought of as an empty coproduct or categorical sum.
It follows that any functor which preserves limits will take terminal objects to terminal objects, and any functor which preserves colimits will take initial objects to initial objects. For example, the initial object in any concrete category with free objects will be the free object generated by the empty set (since the free functor, being left adjoint to the forgetful functor to Set, preserves colimits).
Initial and terminal objects may also be characterized in terms of universal properties and adjoint functors. Let 1 be the discrete category with a single object (denoted by •), and let U : C → 1 be the unique (constant) functor to 1. Then
• An initial object I in C is a universal morphism from • to U. The functor which sends • to I is left adjoint to U.
• A terminal object T in C is a universal morphism from U to •. The functor which sends • to T is right adjoint to U.
Relation to other categorical constructions
Many natural constructions in category theory can be formulated in terms of finding an initial or terminal object in a suitable category.
• A universal morphism from an object X to a functor U can be defined as an initial object in the comma category (X ↓ U). Dually, a universal morphism from U to X is a terminal object in (U ↓ X).
• The limit of a diagram F is a terminal object in Cone(F), the category of cones to F. Dually, a colimit of F is an initial object in the category of cones from F.
• A representation of a functor F to Set is an initial object in the category of elements of F.
• The notion of final functor (respectively, initial functor) is a generalization of the notion of final object (respectively, initial object).
Other properties
• The endomorphism monoid of an initial or terminal object I is trivial: End(I) = Hom(I, I) = { idI }.
• If a category C has a zero object 0, then for any pair of objects X and Y in C, the unique composition X → 0 → Y is a zero morphism from X to Y.
References
• Adámek, Jiří; Herrlich, Horst; Strecker, George E. (1990). Abstract and Concrete Categories. The joy of cats (PDF). John Wiley & Sons. ISBN 0-471-60922-6. Zbl 0695.18001. Archived from the original (PDF) on 2015-04-21. Retrieved 2008-01-15.
• Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004). Categorical foundations. Special topics in order, topology, algebra, and sheaf theory. Encyclopedia of Mathematics and Its Applications. Vol. 97. Cambridge: Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001.
• Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001.
• This article is based in part on PlanetMath's article on examples of initial and terminal objects.
Category theory
Key concepts
Key concepts
• Category
• Adjoint functors
• CCC
• Commutative diagram
• Concrete category
• End
• Exponential
• Functor
• Kan extension
• Morphism
• Natural transformation
• Universal property
Universal constructions
Limits
• Terminal objects
• Products
• Equalizers
• Kernels
• Pullbacks
• Inverse limit
Colimits
• Initial objects
• Coproducts
• Coequalizers
• Cokernels and quotients
• Pushout
• Direct limit
Algebraic categories
• Sets
• Relations
• Magmas
• Groups
• Abelian groups
• Rings (Fields)
• Modules (Vector spaces)
Constructions on categories
• Free category
• Functor category
• Kleisli category
• Opposite category
• Quotient category
• Product category
• Comma category
• Subcategory
Higher category theory
Key concepts
• Categorification
• Enriched category
• Higher-dimensional algebra
• Homotopy hypothesis
• Model category
• Simplex category
• String diagram
• Topos
n-categories
Weak n-categories
• Bicategory (pseudofunctor)
• Tricategory
• Tetracategory
• Kan complex
• ∞-groupoid
• ∞-topos
Strict n-categories
• 2-category (2-functor)
• 3-category
Categorified concepts
• 2-group
• 2-ring
• En-ring
• (Traced)(Symmetric) monoidal category
• n-group
• n-monoid
• Category
• Outline
• Glossary
| Wikipedia |
n-ary associativity
In algebra, n-ary associativity is a generalization of the associative law to n-ary operations.
An ternary operation is ternary associative if one has always
$(abc)de=a(bcd)e=ab(cde);$
that is, the operation gives the same result when any three adjacent elements are bracketed inside a sequence of five operands.
Similarly, an n-ary operation is n-ary associative if bracketing any n adjacent elements in a sequence of n + (n − 1) operands do not change the result.[1]
References
1. Dudek, W.A. (2001), "On some old problems in n-ary groups", Quasigroups and Related Systems, 8: 15–36, archived from the original on 2009-07-14.
| Wikipedia |
Ternary commutator
In mathematical physics, the ternary commutator is an additional ternary operation on a triple system defined by
$[a,b,c]=abc-acb-bac+bca+cab-cba.\,$
Also called the ternutator or alternating ternary sum, it is a special case of the n-commutator for n = 3, whereas the 2-commutator is the ordinary commutator.
Properties
• When one or more of a, b, c is equal to 0, [a, b, c] is also 0. This statement makes 0 the absorbing element of the ternary commutator.
• The same happens when a = b = c.
Further reading
• Bremner, Murray R. (15 August 1998), "Identities for the Ternary Commutator", Journal of Algebra, 206 (2): 615–623, doi:10.1006/jabr.1998.7433
• Bremner, Murray R.; Ortega, Juana Sánchez (25 October 2010), "The partially alternating ternary sum in an associative dialgebra", Journal of Physics A: Mathematical and Theoretical, 43 (56): 455215, arXiv:1008.2721, Bibcode:2010JPhA...43S5215B, doi:10.1088/1751-8113/43/45/455215, S2CID 6636902
• Bremner, Murray R.; Peresi, Luiz A. (1 April 2006), "Ternary analogues of Lie and Malcev algebras", Linear Algebra and Its Applications, 414 (1): 1–18, doi:10.1016/j.laa.2005.09.004
• Bremner, Murray R.; Peresi, Luiz A. (26 July 2012), "Higher identities for the ternary commutator", Journal of Physics A: Mathematical and General, 45 (50): 505201, arXiv:1207.6312, Bibcode:2012JPhA...45X5201B, doi:10.1088/1751-8113/45/50/505201, S2CID 17037773
• Devchand, Chandrashekar; Fairlie, David; Nuyts, Jean; Weingart, Gregor (6 November 2009), "Ternutator identities", Journal of Physics A: Mathematical and Theoretical, 42 (47): 475209, arXiv:0908.1738, Bibcode:2009JPhA...42U5209D, doi:10.1088/1751-8113/42/47/475209, S2CID 17246666
• Nambu, Yoichiro (1973), "Generalized Hamiltonian Dynamics", Physical Review D, 7 (8): 2405–2412, Bibcode:1973PhRvD...7.2405N, doi:10.1103/PhysRevD.7.2405
| Wikipedia |
n-ary group
In mathematics, and in particular universal algebra, the concept of an n-ary group (also called n-group or multiary group) is a generalization of the concept of a group to a set G with an n-ary operation instead of a binary operation.[1] By an n-ary operation is meant any map f: Gn → G from the n-th Cartesian power of G to G. The axioms for an n-ary group are defined in such a way that they reduce to those of a group in the case n = 2. The earliest work on these structures was done in 1904 by Kasner and in 1928 by Dörnte;[2] the first systematic account of (what were then called) polyadic groups was given in 1940 by Emil Leon Post in a famous 143-page paper in the Transactions of the American Mathematical Society.[3]
Axioms
Associativity
The easiest axiom to generalize is the associative law. Ternary associativity is the polynomial identity (abc)de = a(bcd)e = ab(cde), i.e. the equality of the three possible bracketings of the string abcde in which any three consecutive symbols are bracketed. (Here it is understood that the equations hold for all choices of elements a, b, c, d, e in G.) In general, n-ary associativity is the equality of the n possible bracketings of a string consisting of n + (n − 1) = 2n − 1 distinct symbols with any n consecutive symbols bracketed. A set G which is closed under an associative n-ary operation is called an n-ary semigroup. A set G which is closed under any (not necessarily associative) n-ary operation is called an n-ary groupoid.
Inverses / unique solutions
The inverse axiom is generalized as follows: in the case of binary operations the existence of an inverse means ax = b has a unique solution for x, and likewise xa = b has a unique solution. In the ternary case we generalize this to abx = c, axb = c and xab = c each having unique solutions, and the n-ary case follows a similar pattern of existence of unique solutions and we get an n-ary quasigroup.
Definition of n-ary group
An n-ary group is an n-ary semigroup which is also an n-ary quasigroup.
Identity / neutral elements
In the 2-ary case, there can be zero or one identity elements: the empty set is a 2-ary group, since the empty set is both a semigroup and a quasigroup, and every inhabited 2-ary group is a group. In n-ary groups for n ≥ 3 there can be zero, one, or many identity elements.
An n-ary groupoid (G, f) with f = (x1 ◦ x2 ◦ ⋯ ◦ xn), where (G, ◦) is a group is called reducible or derived from the group (G, ◦). In 1928 Dörnte [2] published the first main results: An n-ary groupoid which is reducible is an n-ary group, however for all n > 2 there exist inhabited n-ary groups which are not reducible. In some n-ary groups there exists an element e (called an n-ary identity or neutral element) such that any string of n-elements consisting of all e's, apart from one place, is mapped to the element at that place. E.g., in a quaternary group with identity e, eeae = a for every a.
An n-ary group containing a neutral element is reducible. Thus, an n-ary group that is not reducible does not contain such elements. There exist n-ary groups with more than one neutral element. If the set of all neutral elements of an n-ary group is non-empty it forms an n-ary subgroup.[4]
Some authors include an identity in the definition of an n-ary group but as mentioned above such n-ary operations are just repeated binary operations. Groups with intrinsically n-ary operations do not have an identity element.[5]
Weaker axioms
The axioms of associativity and unique solutions in the definition of an n-ary group are stronger than they need to be. Under the assumption of n-ary associativity it suffices to postulate the existence of the solution of equations with the unknown at the start or end of the string, or at one place other than the ends; e.g., in the 6-ary case, xabcde = f and abcdex = f, or an expression like abxcde = f. Then it can be proved that the equation has a unique solution for x in any place in the string.[3] The associativity axiom can also be given in a weaker form.[1]: 17
Example
The following is an example of a three element ternary group, one of four such groups[6]
${\begin{matrix}aaa=a&aab=b&aac=c&aba=c&abb=a&abc=b&aca=b&acb=c&acc=a\\baa=b&bab=c&bac=a&bba=a&bbb=b&bbc=c&bca=c&bcb=a&bcc=b\\caa=c&cab=a&cac=b&cba=b&cbb=c&cbc=a&cca=a&ccb=b&ccc=c\end{matrix}}$
(n,m)-group
The concept of an n-ary group can be further generalized to that of an (n,m)-group, also known as a vector valued group, which is a set G with a map f: Gn → Gm where n > m, subject to similar axioms as for an n-ary group except that the result of the map is a word consisting of m letters instead of a single letter. So an (n,1)-group is an n-ary group. (n,m)-groups were introduced by G Ĉupona in 1983.[7]
See also
• Universal algebra
References
1. Dudek, W.A. (2001), "On some old and new problems in n-ary groups" (PDF), Quasigroups and Related Systems, 8: 15–36.
2. W. Dörnte, Untersuchungen über einen verallgemeinerten Gruppenbegriff, Mathematische Zeitschrift, vol. 29 (1928), pp. 1-19.
3. E. L. Post, Polyadic groups, Transactions of the American Mathematical Society 48 (1940), 208–350.
4. Wiesław A. Dudek, Remarks to Głazek's results on n-ary groups, Discussiones Mathematicae. General Algebra and Applications 27 (2007), 199–233.
5. Wiesław A. Dudek and Kazimierz Głazek, Around the Hosszú-Gluskin theorem for n-ary groups, Discrete Mathematics 308 (2008), 486–4876.
6. http://tamivox.org/dave/math/tern_quasi/assoc12345.html
7. On (n, m)-groups, J Ušan - Mathematica Moravica, 2000
Further reading
• S. A. Rusakov: Some applications of n-ary group theory, (Russian), Belaruskaya navuka, Minsk 1998.
| Wikipedia |
Ternary operation
In mathematics, a ternary operation is an n-ary operation with n = 3. A ternary operation on a set A takes any given three elements of A and combines them to form a single element of A.
In computer science, a ternary operator is an operator that takes three arguments as input and returns one output.[1]
Examples
The function $T(a,b,c)=ab+c$ is an example of a ternary operation on the integers (or on any structure where $+$ and $\times $ are both defined). Properties of this ternary operation have been used to define planar ternary rings in the foundations of projective geometry.
In the Euclidean plane with points a, b, c referred to an origin, the ternary operation $[a,b,c]=a-b+c$ has been used to define free vectors.[2] Since (abc) = d implies a – b = c – d, these directed segments are equipollent and are associated with the same free vector. Any three points in the plane a, b, c thus determine a parallelogram with d at the fourth vertex.
In projective geometry, the process of finding a projective harmonic conjugate is a ternary operation on three points. In the diagram, points A, B and P determine point V, the harmonic conjugate of P with respect to A and B. Point R and the line through P can be selected arbitrarily, determining C and D. Drawing AC and BD produces the intersection Q, and RQ then yields V.
Suppose A and B are given sets and ${\mathcal {B}}(A,B)$ is the collection of binary relations between A and B. Composition of relations is always defined when A = B, but otherwise a ternary composition can be defined by $[p,q,r]=pq^{T}r$ where $q^{T}$ is the converse relation of q. Properties of this ternary relation have been used to set the axioms for a heap.[3]
In Boolean algebra, $T(A,B,C)=AC+(1-A)B$ defines the formula $(A\lor B)\land (\lnot A\lor C)$.
Computer science
In computer science, a ternary operator is an operator that takes three arguments (or operands).[1] The arguments and result can be of different types. Many programming languages that use C-like syntax[4] feature a ternary operator, ?:, which defines a conditional expression. In some languages, this operator is referred to as the conditional operator.
In Python, the ternary conditional operator reads x if C else y. Python also supports ternary operations called array slicing, e.g. a[b:c] return an array where the first element is a[b] and last element is a[c-1].[5] OCaml expressions provide ternary operations against records, arrays, and strings: a.[b]<-c would mean the string a where index b has value c.[6]
The multiply–accumulate operation is another ternary operator.
Another example of a ternary operator is between, as used in SQL.
The Icon programming language has a "to-by" ternary operator: the expression 1 to 10 by 2 generates the odd integers from 1 through 9.
In Excel formulae, the form is =if(C, x, y).
See also
• Median algebra
• Ternary conditional operator for a list of ternary operators in computer programming languages
References
1. MDN, nmve. "Conditional (ternary) Operator". Mozilla Developer Network. Retrieved 20 February 2017.
2. Jeremiah Certaine (1943) The ternary operation (abc) = a b−1c of a group, Bulletin of the American Mathematical Society 49: 868–77 MR0009953
3. Christopher Hollings (2014) Mathematics across the Iron Curtain: a history of the algebraic theory of semigroups, page 264, History of Mathematics 41, American Mathematical Society ISBN 978-1-4704-1493-1
4. Hoffer, Alex. "Ternary Operator". Cprogramming.com. Retrieved 20 February 2017.
5. "6. Expressions — Python 3.9.1 documentation". docs.python.org. Retrieved 2021-01-19.
6. "The OCaml Manual: Chapter 11 The OCaml language: (7) Expressions". ocaml.org. Retrieved 2023-05-03.
External links
• Media related to Ternary operations at Wikimedia Commons
| Wikipedia |
Ternary quartic
In mathematics, a ternary quartic form is a degree 4 homogeneous polynomial in three variables.
Hilbert's theorem
Hilbert (1888) showed that a positive semi-definite ternary quartic form over the reals can be written as a sum of three squares of quadratic forms.
Invariant theory
The ring of invariants is generated by 7 algebraically independent invariants of degrees 3, 6, 9, 12, 15, 18, 27 (discriminant) (Dixmier 1987), together with 6 more invariants of degrees 9, 12, 15, 18, 21, 21, as conjectured by Shioda (1967). Salmon (1879) discussed the invariants of order up to about 15.
The Salmon invariant is a degree 60 invariant vanishing on ternary quartics with an inflection bitangent. (Dolgachev 2012, 6.4)
Catalecticant
The catalecticant of a ternary quartic is the resultant of its 6 second partial derivatives. It vanishes when the ternary quartic can be written as a sum of five 4th powers of linear forms.
See also
• Ternary cubic
• Invariants of a binary form
References
• Cohen, Teresa (1919), "Investigations on the Plane Quartic", American Journal of Mathematics, 41 (3): 191–211, doi:10.2307/2370332, hdl:2027/mdp.39015079994953, ISSN 0002-9327, JSTOR 2370332
• Dixmier, Jacques (1987), "On the projective invariants of quartic plane curves", Advances in Mathematics, 64 (3): 279–304, doi:10.1016/0001-8708(87)90010-7, ISSN 0001-8708, MR 0888630
• Dolgachev, Igor (2012), Classical Algebraic Geometry : A Modern View, Cambridge University Press, ISBN 978-1-1070-1765-8
• Hilbert, David (1888), "Ueber die Darstellung definiter Formen als Summe von Formenquadraten", Mathematische Annalen, 32 (3): 342–350, doi:10.1007/BF01443605, ISSN 0025-5831
• Noether, Emmy (1908), "Über die Bildung des Formensystems der ternären biquadratischen Form (On Complete Systems of Invariants for Ternary Biquadratic Forms)", Journal für die reine und angewandte Mathematik, 134: 23–90 and two tables, archived from the original on 2013-03-08.
• Salmon, George (1879) [1852], A treatise on the higher plane curves, Hodges, Foster and Figgis, ISBN 978-1-4181-8252-6, MR 0115124
• Shioda, Tetsuji (1967), "On the graded ring of invariants of binary octavics", American Journal of Mathematics, 89 (4): 1022–1046, doi:10.2307/2373415, ISSN 0002-9327, JSTOR 2373415, MR 0220738
• Thomsen, H. Ivah (1916), "Some Invariants of the Ternary Quartic", American Journal of Mathematics, 38 (3): 249–258, doi:10.2307/2370450, ISSN 0002-9327, JSTOR 2370450
External links
• Invariants of the ternary quartic
| Wikipedia |
Terry Millar
Terrence Staples (Terry) Millar (September 18, 1948 – March 9, 2019) was professor emeritus of mathematics and former associate dean for physical sciences in the Graduate School and assistant to the provost at the University of Wisconsin–Madison. He joined the math faculty in 1976 after serving two years in the Marines and obtaining a Ph.D. from Cornell University.[1] Millar retired in 2015 and was considered to be one of the world's foremost researchers in computable model theory.[2]
Biography
He earned a bachelor’s degree in mathematics from Cornell University in 1970 and his Ph.D. in Mathematics from Cornell University in 1975.[3]
Millar died March 9, 2019, at the age of 70 due to pancreatic cancer.[1]
Career
Along with physics professors Sau Lan Wu and Wesley Smith (academic), he “was central to Wisconsin’s contribution to development of the Large Hadron Collider.”[4] [5] Working alongside Francis Halzen, he was “integral in launching the IceCube South Pole Neutrino Observatory.”[1]
References
1. "Millar was math professor, noted UW–Madison research administrator". Wisconsin News. March 15, 2019. Retrieved 25 March 2021.
2. "In Memoriam: Terry Millar". Department of Mathematics. University of Wisconsin Madison. Retrieved 25 March 2021.
3. "Millar, Terrence S." Madison.com. March 13, 2019. Retrieved 25 March 2021.
4. Seely, Ron (September 10, 2008). "UW has big role in giant particle collider". La Crosse Tribune. Retrieved 25 March 2021.
5. Sakai, Jill (July 4, 2012). "UW scientists play key role in discovery of a new particle consistent with Higgs boson". UWM News Reports. Retrieved 25 March 2021.
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Tertiary ideal
In mathematics, a tertiary ideal is a two-sided ideal in a perhaps noncommutative ring that cannot be expressed as a nontrivial intersection of a right fractional ideal with another ideal. Tertiary ideals generalize primary ideals to the case of noncommutative rings. Although primary decompositions do not exist in general for ideals in noncommutative rings, tertiary decompositions do, at least if the ring is Noetherian.
Every primary ideal is tertiary. Tertiary ideals and primary ideals coincide for commutative rings. To any (two-sided) ideal, a tertiary ideal can be associated called the tertiary radical, defined as
$t(I)=\{r\in R{\mbox{ }}|{\mbox{ }}\forall s\notin I,{\mbox{ }}\exists x\in (s){\mbox{ }}x\notin I{\text{ and }}(x)(r)\subset I\}.$
Then t(I) always contains I.
If R is a (not necessarily commutative) Noetherian ring and I a right ideal in R, then I has a unique irredundant decomposition into tertiary ideals
$I=T_{1}\cap \dots \cap T_{n}$.
See also
• Primary ideal
• Lasker–Noether theorem
References
• Riley, J.A. (1962), "Axiomatic primary and tertiary decomposition theory", Trans. Amer. Math. Soc., 105 (2): 177–201, doi:10.1090/s0002-9947-1962-0141683-4
• Tertiary ideal, Encyclopedia of Mathematics, Springer Online Reference Works.
• Behrens, Ernst-August (1972), Ring Theory, Verlag Academic Press, ISBN 9780080873572
• Kurata, Yoshiki (1965), "On an additive ideal theory in a non-associative ring", Mathematische Zeitschrift, 88 (2): 129–135, doi:10.1007/BF01112095, S2CID 119531162
| Wikipedia |
Teruhisa Matsusaka
Teruhisa Matsusaka (松阪 輝久, Matsusaka Teruhisa) (1926–2006) was a Japanese-born American mathematician, who specialized in algebraic geometry.
Teruhisa Matsusaka
Born(1926-04-05)5 April 1926
Died4 March 2006(2006-03-04) (aged 79)
NationalityJapanese/American
Alma materKyoto University
Known forMatsusaka's Criterion[1]
Matsusaka's Big Theorem[1]
Scientific career
FieldsAlgebraic geometry
InstitutionsBrandeis University
Doctoral advisorYasuo Akizuki[2]
Doctoral studentsJános Kollár[2]
Matsusaka received his Ph.D. in 1954 at Kyoto University;[2] he was a member of the Brandeis Mathematics Department from 1961 until his retirement in 1994, and was that department's chair from 1984–1986. He was invited to address the International Congress of Mathematicians held in Edinburgh in 1958 and was elected to the American Academy of Arts and Sciences in 1966.[3]
During the difficult years after the Second World War, Matsusaka worked on several problems connected with Weil's Foundations of Algebraic Geometry. This led to a correspondence and eventually Weil invited Matsusaka to the University of Chicago (1954–57) where they became life-long friends. After three years at Northwestern University and a year at the Institute for Advanced Study,[4] Princeton, he went to Brandeis University in 1961 where he stayed until 1994, helping to build the department to its current prominence.[1]
Matsusaka was awarded a Guggenheim Fellowship for the academic year 1959–1960.[5]
In 1972, Matsusaka introduced Matsusaka's big theorem, a key technical result on ample line bundles.
Selected publications
• Matsusaka, Teruhisa (1952). "On the algebraic construction of the Picard variety". Proc. Japan Acad. 28 (1): 5–8. doi:10.3792/pja/1195571116. MR 0048858.
• Matsusaka, Teruhisa (1956). "Polarized varieties, the field of moduli and generalized Kummer varieties of abelian varieties". Proc. Japan Acad. 32 (6): 367–372. doi:10.3792/pja/1195525333. MR 0079815.
• Matsusaka, Teruhisa (1959). "On a characterization of a Jacobian variety". Mem. College Sci. Univ. Kyoto Ser. A Math. 32 (1): 1–19. doi:10.1215/kjm/1250776695. MR 0108497.
• Theory of Q-varieties. Vol. 8. Mathematical society of Japan. 1964; 158 pp.{{cite book}}: CS1 maint: postscript (link)
• Matsusaka, T. (1972). "Polarized varieties with a given Hilbert polynomial". American Journal of Mathematics. 94 (4): 1027–1077. doi:10.2307/2373563. JSTOR 2373563.
• Matsusaka, T. (1974). "Global deformation of polarized varieties". Bull. Amer. Math. Soc. 80 (1): 1–7. doi:10.1090/s0002-9904-1974-13340-9. MR 0364246.
References
1. Kollár, János (August 2006). "Teruhisa Matsusaka (1926–2006)" (PDF). Notices of the American Mathematical Society. 53 (7): 766–768.
2. Teruhisa Matsusaka at the Mathematics Genealogy Project
3. "In Memory Of ... (2006-2007)". American Mathematical Society.
4. Matsusaka, Teruhisa — Institute for Advanced Study
5. John Simon Guggenheim Foundation, Teruhisa Matsusaka
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| Wikipedia |
Bicomplex number
In abstract algebra, a bicomplex number is a pair (w, z) of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate $(w,z)^{*}=(w,-z)$, and the product of two bicomplex numbers as
$(u,v)(w,z)=(uw-vz,uz+vw).$
"Tessarine" redirects here. For real tessarines, see Split-complex number.
Then the bicomplex norm is given by
$(w,z)^{*}(w,z)=(w,-z)(w,z)=(w^{2}+z^{2},0),$ a quadratic form in the first component.
The bicomplex numbers form a commutative algebra over C of dimension two, which is isomorphic to the direct sum of algebras C ⊕ C.
The product of two bicomplex numbers yields a quadratic form value that is the product of the individual quadratic forms of the numbers: a verification of this property of the quadratic form of a product refers to the Brahmagupta–Fibonacci identity. This property of the quadratic form of a bicomplex number indicates that these numbers form a composition algebra. In fact, bicomplex numbers arise at the binarion level of the Cayley–Dickson construction based on $\mathbb {C} $ with norm z2.
The general bicomplex number can be represented by the matrix ${\begin{pmatrix}w&iz\\iz&w\end{pmatrix}}$, which has determinant $w^{2}+z^{2}$. Thus, the composing property of the quadratic form concurs with the composing property of the determinant.
Bicomplex numbers feature two distinct imaginary units. Multiplication being associative and commutative, the product of these imaginary units must have positive one for its square. Such an element as this product has been called a hyperbolic unit.[1]
As a real algebra
Tessarine multiplication
× 1 i j k
1 1 i j k
i i −1 k −j
j j k 1 i
k k −j i −1
Bicomplex numbers form an algebra over C of dimension two, and since C is of dimension two over R, the bicomplex numbers are an algebra over R of dimension four. In fact the real algebra is older than the complex one; it was labelled tessarines in 1848 while the complex algebra was not introduced until 1892.
A basis for the tessarine 4-algebra over R specifies z = 1 and z = −i, giving the matrices $k={\begin{pmatrix}0&i\\i&0\end{pmatrix}},\quad \ j={\begin{pmatrix}0&1\\1&0\end{pmatrix}}$, which multiply according to the table given. When the identity matrix is identified with 1, then a tessarine t = w + z j .
History
The subject of multiple imaginary units was examined in the 1840s. In a long series "On quaternions, or on a new system of imaginaries in algebra" beginning in 1844 in Philosophical Magazine, William Rowan Hamilton communicated a system multiplying according to the quaternion group. In 1848 Thomas Kirkman reported on his correspondence with Arthur Cayley regarding equations on the units determining a system of hypercomplex numbers.[2]
Tessarines
In 1848 James Cockle introduced the tessarines in a series of articles in Philosophical Magazine.[3]
A tessarine is a hypercomplex number of the form
$t=w+xi+yj+zk,\quad w,x,y,z\in \mathbb {R} $
where $ij=ji=k,\quad i^{2}=-1,\quad j^{2}=+1.$ Cockle used tessarines to isolate the hyperbolic cosine series and the hyperbolic sine series in the exponential series. He also showed how zero divisors arise in tessarines, inspiring him to use the term "impossibles". The tessarines are now best known for their subalgebra of real tessarines $t=w+yj\ $, also called split-complex numbers, which express the parametrization of the unit hyperbola.
Bicomplex numbers
In a 1892 Mathematische Annalen paper, Corrado Segre introduced bicomplex numbers,[4] which form an algebra isomorphic to the tessarines.[5]
Segre read W. R. Hamilton's Lectures on Quaternions (1853) and the works of W. K. Clifford. Segre used some of Hamilton's notation to develop his system of bicomplex numbers: Let h and i be elements that square to −1 and that commute. Then, presuming associativity of multiplication, the product hi must square to +1. The algebra constructed on the basis { 1, h, i, hi } is then the same as James Cockle's tessarines, represented using a different basis. Segre noted that elements
$g=(1-hi)/2,\quad g'=(1+hi)/2$ are idempotents.
When bicomplex numbers are expressed in terms of the basis { 1, h, i, −hi }, their equivalence with tessarines is apparent. Looking at the linear representation of these isomorphic algebras shows agreement in the fourth dimension when the negative sign is used; consider the sample product given above under linear representation.
Bibinarions
The modern theory of composition algebras positions the algebra as a binarion construction based on another binarion construction, hence the bibinarions.[6] The unarion level in the Cayley-Dickson process must be a field, and starting with the real field, the usual complex numbers arises as division binarions, another field. Thus the process can begin again to form bibinarions. Kevin McCrimmon noted the simplification of nomenclature provided by the term binarion in his text A Taste of Jordan Algebras (2004).
Polynomial roots
Write 2C = C ⊕ C and represent elements of it by ordered pairs (u,v) of complex numbers. Since the algebra of tessarines T is isomorphic to 2C, the rings of polynomials T[X] and 2C[X] are also isomorphic, however polynomials in the latter algebra split:
$\sum _{k=1}^{n}(a_{k},b_{k})(u,v)^{k}\quad =\quad \left({\sum _{k=1}^{n}a_{i}u^{k}},\quad \sum _{k=1}^{n}b_{k}v^{k}\right).$
In consequence, when a polynomial equation $f(u,v)=(0,0)$ in this algebra is set, it reduces to two polynomial equations on C. If the degree is n, then there are n roots for each equation: $u_{1},u_{2},\dots ,u_{n},\ v_{1},v_{2},\dots ,v_{n}.$ Any ordered pair $(u_{i},v_{j})\!$ from this set of roots will satisfy the original equation in 2C[X], so it has n2 roots.[7]
Due to the isomorphism with T[X], there is a correspondence of polynomials and a correspondence of their roots. Hence the tessarine polynomials of degree n also have n2 roots, counting multiplicity of roots.
Applications
Bicomplex number appears as the center of CAPS (complexified algebra of physical space), which is Clifford algebra $Cl(3,\mathbb {C} )$.[8] Since the linear space of CAPS can be viewed as the four dimensional space span {$1,e_{1},e_{2},e_{3}$} over {$1,i,k,j$}.
Tessarines have been applied in digital signal processing.[9][10][11]
Bicomplex numbers are employed in fluid mechanics. The use of bicomplex algebra reconciles two distinct applications of complex numbers: the representation of two-dimensional potential flows in the complex plane and the complex exponential function.[12]
References
1. M.E. Luna-Elizarrarás, M. Shapiro, D.C. Struppa (2013) Bicomplex Holomorphic Functions: the algebra, geometry and analysis of bicomplex numbers, page 6, Birkhauser ISBN 978-3-319-24868-4
2. Thomas Kirkman (1848) "On Pluquaternions and Homoid Products of n Squares", London and Edinburgh Philosophical Magazine 1848, p 447 Google books link
3. James Cockle in London-Dublin-Edinburgh Philosophical Magazine, series 3
• 1848 On Certain Functions Resembling Quaternions and on a New Imaginary in Algebra, 33:435–9.
• 1849 On a New Imaginary in Algebra 34:37–47.
• 1849 On the Symbols of Algebra and on the Theory of Tessarines 34:406–10.
• 1850 On the True Amplitude of a Tessarine 36:290-2.
• 1850 On Impossible Equations, on Impossible Quantities and on Tessarines 37:281–3.
Links from Biodiversity Heritage Library.
4. Segre, Corrado (1892), "Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici" [The real representation of complex elements and hyperalgebraic entities], Mathematische Annalen, 40 (3): 413–467, doi:10.1007/bf01443559, S2CID 121807474. (see especially pages 455–67)
5. Abstract Algebra/Polynomial Rings at Wikibooks
6. Associative Composition Algebra/Bibinarions at Wikibooks
7. Poodiack, Robert D. & Kevin J. LeClair (2009) "Fundamental theorems of algebra for the perplexes", The College Mathematics Journal 40(5):322–35.
8. Baylis, W.E.; Kiselica, J.D. (2012). The Complex Algebra of Physical Space: A Framework for Relativity. Adv. Appl. Clifford Algebras. Vol. 22. SpringerLink. pp. 537–561.
9. Pei, Soo-Chang; Chang, Ja-Han; Ding, Jian-Jiun (21 June 2004). "Commutative reduced biquaternions and their Fourier transform for signal and image processing" (PDF). IEEE Transactions on Signal Processing. IEEE. 52 (7): 2012–2031. doi:10.1109/TSP.2004.828901. ISSN 1941-0476. S2CID 13907861.
10. Alfsmann, Daniel (4–8 September 2006). On families of 2N dimensional hypercomplex algebras suitable for digital signal processing (PDF). 14th European Signal Processing Conference, Florence, Italy: EURASIP.{{cite conference}}: CS1 maint: location (link)
11. Alfsmann, Daniel; Göckler, Heinz G. (2007). On Hyperbolic Complex LTI Digital Systems (PDF). EURASIP.
12. Kleine, Vitor G.; Hanifi, Ardeshir; Henningson, Dan S. (2022). "Stability of two-dimensional potential flows using bicomplex numbers". Proc. R. Soc. A. 478 (20220165). doi:10.1098/rspa.2022.0165. PMC 9185835. PMID 35702595.
Further reading
• G. Baley Price (1991) An Introduction to Multicomplex Spaces and Functions Marcel Dekker ISBN 0-8247-8345-X
• F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti. (2008) The Mathematics of Minkowski Space-Time with an Introduction to Commutative Hypercomplex Numbers, Birkhäuser Verlag, Basel ISBN 978-3-7643-8613-9
• Alpay D, Luna-Elizarrarás ME, Shapiro M, Struppa DC. (2014) Basics of functional analysis with bicomplex scalars, and bicomplex Schur analysis, Cham, Switzerland: Springer Science & BusinessMedia
• Luna-Elizarrarás ME, Shapiro M, Struppa DC, Vajiac A. (2015) Bicomplex holomorphic functions:the algebra, geometry and analysis of bicomplex numbers, Cham, Switzerland: Birkhäuser
• Rochon, Dominic, and Michael Shapiro (2004). "On algebraic properties of bicomplex and hyperbolic numbers." Anal. Univ. Oradea, fasc. math 11, no. 71: 110. http://3dfractals.com/docs/Article01_bicomplex.pdf
Number systems
Sets of definable numbers
• Natural numbers ($\mathbb {N} $)
• Integers ($\mathbb {Z} $)
• Rational numbers ($\mathbb {Q} $)
• Constructible numbers
• Algebraic numbers ($\mathbb {A} $)
• Closed-form numbers
• Periods
• Computable numbers
• Arithmetical numbers
• Set-theoretically definable numbers
• Gaussian integers
Composition algebras
• Division algebras: Real numbers ($\mathbb {R} $)
• Complex numbers ($\mathbb {C} $)
• Quaternions ($\mathbb {H} $)
• Octonions ($\mathbb {O} $)
Split
types
• Over $\mathbb {R} $:
• Split-complex numbers
• Split-quaternions
• Split-octonions
Over $\mathbb {C} $:
• Bicomplex numbers
• Biquaternions
• Bioctonions
Other hypercomplex
• Dual numbers
• Dual quaternions
• Dual-complex numbers
• Hyperbolic quaternions
• Sedenions ($\mathbb {S} $)
• Split-biquaternions
• Multicomplex numbers
• Geometric algebra/Clifford algebra
• Algebra of physical space
• Spacetime algebra
Other types
• Cardinal numbers
• Extended natural numbers
• Irrational numbers
• Fuzzy numbers
• Hyperreal numbers
• Levi-Civita field
• Surreal numbers
• Transcendental numbers
• Ordinal numbers
• p-adic numbers (p-adic solenoids)
• Supernatural numbers
• Profinite integers
• Superreal numbers
• Normal numbers
• Classification
• List
| Wikipedia |
Tesseractic honeycomb
In four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol {4,3,3,4}, and constructed by a 4-dimensional packing of tesseract facets.
Tesseractic honeycomb
Perspective projection of a 3x3x3x3 red-blue chessboard.
TypeRegular 4-space honeycomb
Uniform 4-honeycomb
FamilyHypercubic honeycomb
Schläfli symbols{4,3,3,4}
t0,4{4,3,3,4}
{4,3,31,1}
{4,4}(2)
{4,3,4}×{∞}
{4,4}×{∞}(2)
{∞}(4)
Coxeter-Dynkin diagrams
4-face type{4,3,3}
Cell type{4,3}
Face type{4}
Edge figure{3,4}
(octahedron)
Vertex figure{3,3,4}
(16-cell)
Coxeter groups${\tilde {C}}_{4}$, [4,3,3,4]
${\tilde {B}}_{4}$, [4,3,31,1]
Dualself-dual
Propertiesvertex-transitive, edge-transitive, face-transitive, cell-transitive, 4-face-transitive
Its vertex figure is a 16-cell. Two tesseracts meet at each cubic cell, four meet at each square face, eight meet on each edge, and sixteen meet at each vertex.
It is an analog of the square tiling, {4,4}, of the plane and the cubic honeycomb, {4,3,4}, of 3-space. These are all part of the hypercubic honeycomb family of tessellations of the form {4,3,...,3,4}. Tessellations in this family are Self-dual.
Coordinates
Vertices of this honeycomb can be positioned in 4-space in all integer coordinates (i,j,k,l).
Sphere packing
Like all regular hypercubic honeycombs, the tesseractic honeycomb corresponds to a sphere packing of edge-length-diameter spheres centered on each vertex, or (dually) inscribed in each cell instead. In the hypercubic honeycomb of 4 dimensions, vertex-centered 3-spheres and cell-inscribed 3-spheres will both fit at once, forming the unique regular body-centered cubic lattice of equal-sized spheres (in any number of dimensions). Since the tesseract is radially equilateral, there is exactly enough space in the hole between the 16 vertex-centered 3-spheres for another edge-length-diameter 3-sphere. (This 4-dimensional body centered cubic lattice is actually the union of two tesseractic honeycombs, in dual positions.)
This is the same densest known regular 3-sphere packing, with kissing number 24, that is also seen in the other two regular tessellations of 4-space, the 16-cell honeycomb and the 24-cell-honeycomb. Each tesseract-inscribed 3-sphere kisses a surrounding shell of 24 3-spheres, 16 at the vertices of the tesseract and 8 inscribed in the adjacent tesseracts. These 24 kissing points are the vertices of a 24-cell of radius (and edge length) 1/2.
Constructions
There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,3,3,4}. Another form has two alternating tesseract facets (like a checkerboard) with Schläfli symbol {4,3,31,1}. The lowest symmetry Wythoff construction has 16 types of facets around each vertex and a prismatic product Schläfli symbol {∞}4. One can be made by stericating another.
Related polytopes and tessellations
The [4,3,3,4], , Coxeter group generates 31 permutations of uniform tessellations, 21 with distinct symmetry and 20 with distinct geometry. The expanded tesseractic honeycomb (also known as the stericated tesseractic honeycomb) is geometrically identical to the tesseractic honeycomb. Three of the symmetric honeycombs are shared in the [3,4,3,3] family. Two alternations (13) and (17), and the quarter tesseractic (2) are repeated in other families.
C4 honeycombs
Extended
symmetry
Extended
diagram
Order Honeycombs
[4,3,3,4]: ×1
1, 2, 3, 4,
5, 6, 7, 8,
9, 10, 11, 12,
13
[[4,3,3,4]] ×2 (1), (2), (13), 18
(6), 19, 20
[(3,3)[1+,4,3,3,4,1+]]
↔ [(3,3)[31,1,1,1]]
↔ [3,4,3,3]
↔
↔
×6
14, 15, 16, 17
The [4,3,31,1], , Coxeter group generates 31 permutations of uniform tessellations, 23 with distinct symmetry and 4 with distinct geometry. There are two alternated forms: the alternations (19) and (24) have the same geometry as the 16-cell honeycomb and snub 24-cell honeycomb respectively.
B4 honeycombs
Extended
symmetry
Extended
diagram
Order Honeycombs
[4,3,31,1]: ×1
5, 6, 7, 8
<[4,3,31,1]>:
↔[4,3,3,4]
↔
×2
9, 10, 11, 12, 13, 14,
(10), 15, 16, (13), 17, 18, 19
[3[1+,4,3,31,1]]
↔ [3[3,31,1,1]]
↔ [3,3,4,3]
↔
↔
×3
1, 2, 3, 4
[(3,3)[1+,4,3,31,1]]
↔ [(3,3)[31,1,1,1]]
↔ [3,4,3,3]
↔
↔
×12
20, 21, 22, 23
The 24-cell honeycomb is similar, but in addition to the vertices at integers (i,j,k,l), it has vertices at half integers (i+1/2,j+1/2,k+1/2,l+1/2) of odd integers only. It is a half-filled body centered cubic (a checkerboard in which the red 4-cubes have a central vertex but the black 4-cubes do not).
The tesseract can make a regular tessellation of the 4-sphere, with three tesseracts per face, with Schläfli symbol {4,3,3,3}, called an order-3 tesseractic honeycomb. It is topologically equivalent to the regular polytope penteract in 5-space.
The tesseract can make a regular tessellation of 4-dimensional hyperbolic space, with 5 tesseracts around each face, with Schläfli symbol {4,3,3,5}, called an order-5 tesseractic honeycomb.
Birectified tesseractic honeycomb
A birectified tesseractic honeycomb, , contains all rectified 16-cell (24-cell) facets and is the Voronoi tessellation of the D4* lattice. Facets can be identically colored from a doubled ${\tilde {C}}_{4}$×2, [[4,3,3,4]] symmetry, alternately colored from ${\tilde {C}}_{4}$, [4,3,3,4] symmetry, three colors from ${\tilde {B}}_{4}$, [4,3,31,1] symmetry, and 4 colors from ${\tilde {D}}_{4}$, [31,1,1,1] symmetry.
See also
Regular and uniform honeycombs in 4-space:
• 16-cell honeycomb
• 24-cell honeycomb
• 5-cell honeycomb
• Truncated 5-cell honeycomb
• Omnitruncated 5-cell honeycomb
References
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) - Model 1
• Klitzing, Richard. "4D Euclidean tesselations". x∞o x∞o x∞o x∞o, x∞x x∞o x∞o x∞o, x∞x x∞x x∞o x∞o, x∞x x∞x x∞x x∞o,x∞x x∞x x∞x x∞x, x∞o x∞o x4o4o, x∞o x∞o o4x4o, x∞x x∞o x4o4o, x∞x x∞o o4x4o, x∞o x∞o x4o4x, x∞x x∞x x4o4o, x∞x x∞x o4x4o, x∞x x∞o x4o4x, x∞x x∞x x4o4x, x4o4x x4o4x, x4o4x o4x4o, x4o4x x4o4o, o4x4o o4x4o, x4o4o o4x4o, x4o4o x4o4o, x∞x o3o3o *d4x, x∞o o3o3o *d4x, x∞x x4o3o4x, x∞o x4o3o4x, x∞x x4o3o4o, x∞o x4o3o4o, o3o3o *b3o4x, x4o3o3o4x, x4o3o3o4o - test - O1
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
| Wikipedia |
Randomness test
A randomness test (or test for randomness), in data evaluation, is a test used to analyze the distribution of a set of data to see whether it can be described as random (patternless). In stochastic modeling, as in some computer simulations, the hoped-for randomness of potential input data can be verified, by a formal test for randomness, to show that the data are valid for use in simulation runs. In some cases, data reveals an obvious non-random pattern, as with so-called "runs in the data" (such as expecting random 0–9 but finding "4 3 2 1 0 4 3 2 1..." and rarely going above 4). If a selected set of data fails the tests, then parameters can be changed or other randomized data can be used which does pass the tests for randomness.
Background
The issue of randomness is an important philosophical and theoretical question. Tests for randomness can be used to determine whether a data set has a recognisable pattern, which would indicate that the process that generated it is significantly non-random. For the most part, statistical analysis has, in practice, been much more concerned with finding regularities in data as opposed to testing for randomness. Many "random number generators" in use today are defined by algorithms, and so are actually pseudo-random number generators. The sequences they produce are called pseudo-random sequences. These generators do not always generate sequences which are sufficiently random, but instead can produce sequences which contain patterns. For example, the infamous RANDU routine fails many randomness tests dramatically, including the spectral test.
Stephen Wolfram used randomness tests on the output of Rule 30 to examine its potential for generating random numbers,[1] though it was shown to have an effective key size far smaller than its actual size[2] and to perform poorly on a chi-squared test.[3] The use of an ill-conceived random number generator can put the validity of an experiment in doubt by violating statistical assumptions. Though there are commonly used statistical testing techniques such as NIST standards, Yongge Wang showed that NIST standards are not sufficient. Furthermore, Yongge Wang[4] designed statistical–distance–based and law–of–the–iterated–logarithm–based testing techniques. Using this technique, Yongge Wang and Tony Nicol[5] detected the weakness in commonly used pseudorandom generators such as the well known Debian version of OpenSSL pseudorandom generator which was fixed in 2008.
Specific tests for randomness
There have been a fairly small number of different types of (pseudo-)random number generators used in practice. They can be found in the list of random number generators, and have included:
• Linear congruential generator and Linear-feedback shift register
• Generalized Fibonacci generator
• Cryptographic generators
• Quadratic congruential generator
• Cellular automaton generators
• Pseudorandom binary sequence
These different generators have varying degrees of success in passing the accepted test suites. Several widely used generators fail the tests more or less badly, while other 'better' and prior generators (in the sense that they passed all current batteries of tests and they already existed) have been largely ignored.
There are many practical measures of randomness for a binary sequence. These include measures based on statistical tests, transforms, and complexity or a mixture of these. A well-known and widely used collection of tests was the Diehard Battery of Tests, introduced by Marsaglia; this was extended to the TestU01 suite by L'Ecuyer and Simard. The use of Hadamard transform to measure randomness was proposed by S. Kak and developed further by Phillips, Yuen, Hopkins, Beth and Dai, Mund, and Marsaglia and Zaman.[6]
Several of these tests, which are of linear complexity, provide spectral measures of randomness. T. Beth and Z-D. Dai purported to show that Kolmogorov complexity and linear complexity are practically the same,[7] although Y. Wang later showed their claims are incorrect.[8] Nevertheless, Wang also demonstrated that for Martin-Löf random sequences, the Kolmogorov complexity is essentially the same as linear complexity.
These practical tests make it possible to compare the randomness of strings. On probabilistic grounds, all strings of a given length have the same randomness. However different strings have a different Kolmogorov complexity. For example, consider the following two strings.
String 1: 0101010101010101010101010101010101010101010101010101010101010101
String 2: 1100100001100001110111101110110011111010010000100101011110010110
String 1 admits a short linguistic description: "32 repetitions of '01'". This description has 22 characters, and it can be efficiently constructed out of some basis sequences. String 2 has no obvious simple description other than writing down the string itself, which has 64 characters, and it has no comparably efficient basis function representation. Using linear Hadamard spectral tests (see Hadamard transform), the first of these sequences will be found to be of much less randomness than the second one, which agrees with intuition.
Notable software implementations
• Diehard tests
• TestU01
• ENT utility from Fourmilab[9]
• NIST Statistical Test Suite[10][11]
See also
• Randomness
• Statistical randomness
• Algorithmically random sequence
• Seven states of randomness
• Wald–Wolfowitz runs test
Notes
1. Wolfram, Stephen (2002). A New Kind of Science. Wolfram Media, Inc. pp. 975–976. ISBN 978-1-57955-008-0.
2. Willi Meier; Othmar Staffelbach (1991). "Analysis of Pseudo Random Sequences Generated by Cellular Automata". Advances in Cryptology — EUROCRYPT '91. Lecture Notes in Computer Science. Vol. 547. pp. 186–199. doi:10.1007/3-540-46416-6_17. ISBN 978-3-540-54620-7.
3. Moshe Sipper; Marco Tomassini (1996), "Generating parallel random number generators by cellular programming", International Journal of Modern Physics C, 7 (2): 181–190, Bibcode:1996IJMPC...7..181S, CiteSeerX 10.1.1.21.870, doi:10.1142/S012918319600017X.
4. Yongge Wang. On the Design of LIL Tests for (Pseudo) Random Generators and Some Experimental Results, http://webpages.uncc.edu/yonwang/, 2014
5. Yongge Wang; Tony Nicol (2014), "Statistical Properties of Pseudo Random Sequences and Experiments with PHP and Debian OpenSSL", Esorics 2014, LNCS 8712: 454–471
6. Terry Ritter, "Randomness tests: a literature survey", webpage: CBR-rand.
7. Beth, T. and Z-D. Dai. 1989. On the Complexity of Pseudo-Random Sequences -- or: If You Can Describe a Sequence It Can't be Random. Advances in Cryptology – EUROCRYPT '89. 533-543. Springer-Verlag
8. Yongge Wang 1999. Linear complexity versus pseudorandomness: on Beth and Dai's result. In: Proc. Asiacrypt 99, pages 288--298. LNCS 1716, Springer Verlag
9. ENT: A Pseudorandom Number Sequence Test Program, Fourmilab, 2008.
10. A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications, Special Publication 800-22 Revision 1a, National Institute of Standards and Technology, 2010.
11. Implementation of the NIST Statistical Test Suite
External links
• Randomness tests included in the Cryptographic Toolkit from NIST
• George Marsaglia, Wai Wan Tsang (2002), "Some Difficult-to-pass Tests of Randomness", Journal of Statistical Software, Volume 7, Issue 3
• DieHarder: A Random Number Test Suite by Robert G. Brown, Duke University
• Online Random Number Generator Analysis from CAcert.org
| Wikipedia |
Test functions for optimization
In applied mathematics, test functions, known as artificial landscapes, are useful to evaluate characteristics of optimization algorithms, such as:
• Convergence rate.
• Precision.
• Robustness.
• General performance.
Here some test functions are presented with the aim of giving an idea about the different situations that optimization algorithms have to face when coping with these kinds of problems. In the first part, some objective functions for single-objective optimization cases are presented. In the second part, test functions with their respective Pareto fronts for multi-objective optimization problems (MOP) are given.
The artificial landscapes presented herein for single-objective optimization problems are taken from Bäck,[1] Haupt et al.[2] and from Rody Oldenhuis software.[3] Given the number of problems (55 in total), just a few are presented here.
The test functions used to evaluate the algorithms for MOP were taken from Deb,[4] Binh et al.[5] and Binh.[6] The software developed by Deb can be downloaded,[7] which implements the NSGA-II procedure with GAs, or the program posted on Internet,[8] which implements the NSGA-II procedure with ES.
Just a general form of the equation, a plot of the objective function, boundaries of the object variables and the coordinates of global minima are given herein.
Test functions for single-objective optimization
NamePlotFormulaGlobal minimumSearch domain
Rastrigin function $f(\mathbf {x} )=An+\sum _{i=1}^{n}\left[x_{i}^{2}-A\cos(2\pi x_{i})\right]$
${\text{where: }}A=10$
$f(0,\dots ,0)=0$ $-5.12\leq x_{i}\leq 5.12$
Ackley function $f(x,y)=-20\exp \left[-0.2{\sqrt {0.5\left(x^{2}+y^{2}\right)}}\right]$
$-\exp \left[0.5\left(\cos 2\pi x+\cos 2\pi y\right)\right]+e+20$
$f(0,0)=0$ $-5\leq x,y\leq 5$
Sphere function $f({\boldsymbol {x}})=\sum _{i=1}^{n}x_{i}^{2}$ $f(x_{1},\dots ,x_{n})=f(0,\dots ,0)=0$ $-\infty \leq x_{i}\leq \infty $, $1\leq i\leq n$
Rosenbrock function $f({\boldsymbol {x}})=\sum _{i=1}^{n-1}\left[100\left(x_{i+1}-x_{i}^{2}\right)^{2}+\left(1-x_{i}\right)^{2}\right]$ ${\text{Min}}={\begin{cases}n=2&\rightarrow \quad f(1,1)=0,\\n=3&\rightarrow \quad f(1,1,1)=0,\\n>3&\rightarrow \quad f(\underbrace {1,\dots ,1} _{n{\text{ times}}})=0\\\end{cases}}$ $-\infty \leq x_{i}\leq \infty $, $1\leq i\leq n$
Beale function $f(x,y)=\left(1.5-x+xy\right)^{2}+\left(2.25-x+xy^{2}\right)^{2}$
$+\left(2.625-x+xy^{3}\right)^{2}$
$f(3,0.5)=0$ $-4.5\leq x,y\leq 4.5$
Goldstein–Price function $f(x,y)=\left[1+\left(x+y+1\right)^{2}\left(19-14x+3x^{2}-14y+6xy+3y^{2}\right)\right]$
$\left[30+\left(2x-3y\right)^{2}\left(18-32x+12x^{2}+48y-36xy+27y^{2}\right)\right]$
$f(0,-1)=3$ $-2\leq x,y\leq 2$
Booth function $f(x,y)=\left(x+2y-7\right)^{2}+\left(2x+y-5\right)^{2}$ $f(1,3)=0$ $-10\leq x,y\leq 10$
Bukin function N.6 $f(x,y)=100{\sqrt {\left|y-0.01x^{2}\right|}}+0.01\left|x+10\right|.\quad $ $f(-10,1)=0$ $-15\leq x\leq -5$, $-3\leq y\leq 3$
Matyas function $f(x,y)=0.26\left(x^{2}+y^{2}\right)-0.48xy$ $f(0,0)=0$ $-10\leq x,y\leq 10$
Lévi function N.13 $f(x,y)=\sin ^{2}3\pi x+\left(x-1\right)^{2}\left(1+\sin ^{2}3\pi y\right)$
$+\left(y-1\right)^{2}\left(1+\sin ^{2}2\pi y\right)$
$f(1,1)=0$ $-10\leq x,y\leq 10$
Himmelblau's function $f(x,y)=(x^{2}+y-11)^{2}+(x+y^{2}-7)^{2}.\quad $ ${\text{Min}}={\begin{cases}f\left(3.0,2.0\right)&=0.0\\f\left(-2.805118,3.131312\right)&=0.0\\f\left(-3.779310,-3.283186\right)&=0.0\\f\left(3.584428,-1.848126\right)&=0.0\\\end{cases}}$ $-5\leq x,y\leq 5$
Three-hump camel function $f(x,y)=2x^{2}-1.05x^{4}+{\frac {x^{6}}{6}}+xy+y^{2}$ $f(0,0)=0$ $-5\leq x,y\leq 5$
Easom function $f(x,y)=-\cos \left(x\right)\cos \left(y\right)\exp \left(-\left(\left(x-\pi \right)^{2}+\left(y-\pi \right)^{2}\right)\right)$ $f(\pi ,\pi )=-1$ $-100\leq x,y\leq 100$
Cross-in-tray function $f(x,y)=-0.0001\left[\left|\sin x\sin y\exp \left(\left|100-{\frac {\sqrt {x^{2}+y^{2}}}{\pi }}\right|\right)\right|+1\right]^{0.1}$ ${\text{Min}}={\begin{cases}f\left(1.34941,-1.34941\right)&=-2.06261\\f\left(1.34941,1.34941\right)&=-2.06261\\f\left(-1.34941,1.34941\right)&=-2.06261\\f\left(-1.34941,-1.34941\right)&=-2.06261\\\end{cases}}$ $-10\leq x,y\leq 10$
Eggholder function[9][10] $f(x,y)=-\left(y+47\right)\sin {\sqrt {\left|{\frac {x}{2}}+\left(y+47\right)\right|}}-x\sin {\sqrt {\left|x-\left(y+47\right)\right|}}$ $f(512,404.2319)=-959.6407$ $-512\leq x,y\leq 512$
Hölder table function $f(x,y)=-\left|\sin x\cos y\exp \left(\left|1-{\frac {\sqrt {x^{2}+y^{2}}}{\pi }}\right|\right)\right|$ ${\text{Min}}={\begin{cases}f\left(8.05502,9.66459\right)&=-19.2085\\f\left(-8.05502,9.66459\right)&=-19.2085\\f\left(8.05502,-9.66459\right)&=-19.2085\\f\left(-8.05502,-9.66459\right)&=-19.2085\end{cases}}$ $-10\leq x,y\leq 10$
McCormick function $f(x,y)=\sin \left(x+y\right)+\left(x-y\right)^{2}-1.5x+2.5y+1$ $f(-0.54719,-1.54719)=-1.9133$ $-1.5\leq x\leq 4$, $-3\leq y\leq 4$
Schaffer function N. 2 $f(x,y)=0.5+{\frac {\sin ^{2}\left(x^{2}-y^{2}\right)-0.5}{\left[1+0.001\left(x^{2}+y^{2}\right)\right]^{2}}}$ $f(0,0)=0$ $-100\leq x,y\leq 100$
Schaffer function N. 4 $f(x,y)=0.5+{\frac {\cos ^{2}\left[\sin \left(\left|x^{2}-y^{2}\right|\right)\right]-0.5}{\left[1+0.001\left(x^{2}+y^{2}\right)\right]^{2}}}$ ${\text{Min}}={\begin{cases}f\left(0,1.25313\right)&=0.292579\\f\left(0,-1.25313\right)&=0.292579\\f\left(1.25313,0\right)&=0.292579\\f\left(-1.25313,0\right)&=0.292579\end{cases}}$ $-100\leq x,y\leq 100$
Styblinski–Tang function $f({\boldsymbol {x}})={\frac {\sum _{i=1}^{n}x_{i}^{4}-16x_{i}^{2}+5x_{i}}{2}}$ $-39.16617n<f(\underbrace {-2.903534,\ldots ,-2.903534} _{n{\text{ times}}})<-39.16616n$ $-5\leq x_{i}\leq 5$, $1\leq i\leq n$..
Test functions for constrained optimization
NamePlotFormulaGlobal minimumSearch domain
Rosenbrock function constrained with a cubic and a line[11] $f(x,y)=(1-x)^{2}+100(y-x^{2})^{2}$,
subjected to: $(x-1)^{3}-y+1\leq 0{\text{ and }}x+y-2\leq 0$
$f(1.0,1.0)=0$ $-1.5\leq x\leq 1.5$, $-0.5\leq y\leq 2.5$
Rosenbrock function constrained to a disk[12] $f(x,y)=(1-x)^{2}+100(y-x^{2})^{2}$,
subjected to: $x^{2}+y^{2}\leq 2$
$f(1.0,1.0)=0$ $-1.5\leq x\leq 1.5$, $-1.5\leq y\leq 1.5$
Mishra's Bird function - constrained[13][14] $f(x,y)=\sin(y)e^{\left[(1-\cos x)^{2}\right]}+\cos(x)e^{\left[(1-\sin y)^{2}\right]}+(x-y)^{2}$,
subjected to: $(x+5)^{2}+(y+5)^{2}<25$
$f(-3.1302468,-1.5821422)=-106.7645367$ $-10\leq x\leq 0$, $-6.5\leq y\leq 0$
Townsend function (modified)[15] $f(x,y)=-[\cos((x-0.1)y)]^{2}-x\sin(3x+y)$,
subjected to:$x^{2}+y^{2}<\left[2\cos t-{\frac {1}{2}}\cos 2t-{\frac {1}{4}}\cos 3t-{\frac {1}{8}}\cos 4t\right]^{2}+[2\sin t]^{2}$ where: t = Atan2(x,y)
$f(2.0052938,1.1944509)=-2.0239884$ $-2.25\leq x\leq 2.25$, $-2.5\leq y\leq 1.75$
Gomez and Levy function (modified)[16] $f(x,y)=4x^{2}-2.1x^{4}+{\frac {1}{3}}x^{6}+xy-4y^{2}+4y^{4}$,
subjected to:$-\sin(4\pi x)+2\sin ^{2}(2\pi y)\leq 1.5$
$f(0.08984201,-0.7126564)=-1.031628453$ $-1\leq x\leq 0.75$, $-1\leq y\leq 1$
Simionescu function[17] $f(x,y)=0.1xy$,
subjected to: $x^{2}+y^{2}\leq \left[r_{T}+r_{S}\cos \left(n\arctan {\frac {x}{y}}\right)\right]^{2}$ ${\text{where: }}r_{T}=1,r_{S}=0.2{\text{ and }}n=8$
$f(\pm 0.84852813,\mp 0.84852813)=-0.072$ $-1.25\leq x,y\leq 1.25$
Test functions for multi-objective optimization
NamePlotFunctionsConstraintsSearch domain
Binh and Korn function:[5] ${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)=4x^{2}+4y^{2}\\f_{2}\left(x,y\right)=\left(x-5\right)^{2}+\left(y-5\right)^{2}\\\end{cases}}$ ${\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)=\left(x-5\right)^{2}+y^{2}\leq 25\\g_{2}\left(x,y\right)=\left(x-8\right)^{2}+\left(y+3\right)^{2}\geq 7.7\\\end{cases}}$ $0\leq x\leq 5$, $0\leq y\leq 3$
Chankong and Haimes function:[18] ${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)=2+\left(x-2\right)^{2}+\left(y-1\right)^{2}\\f_{2}\left(x,y\right)=9x-\left(y-1\right)^{2}\\\end{cases}}$ ${\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)=x^{2}+y^{2}\leq 225\\g_{2}\left(x,y\right)=x-3y+10\leq 0\\\end{cases}}$ $-20\leq x,y\leq 20$
Fonseca–Fleming function:[19] ${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=1-\exp \left[-\sum _{i=1}^{n}\left(x_{i}-{\frac {1}{\sqrt {n}}}\right)^{2}\right]\\f_{2}\left({\boldsymbol {x}}\right)=1-\exp \left[-\sum _{i=1}^{n}\left(x_{i}+{\frac {1}{\sqrt {n}}}\right)^{2}\right]\\\end{cases}}$ $-4\leq x_{i}\leq 4$, $1\leq i\leq n$
Test function 4:[6] ${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)=x^{2}-y\\f_{2}\left(x,y\right)=-0.5x-y-1\\\end{cases}}$ ${\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)=6.5-{\frac {x}{6}}-y\geq 0\\g_{2}\left(x,y\right)=7.5-0.5x-y\geq 0\\g_{3}\left(x,y\right)=30-5x-y\geq 0\\\end{cases}}$ $-7\leq x,y\leq 4$
Kursawe function:[20] ${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=\sum _{i=1}^{2}\left[-10\exp \left(-0.2{\sqrt {x_{i}^{2}+x_{i+1}^{2}}}\right)\right]\\&\\f_{2}\left({\boldsymbol {x}}\right)=\sum _{i=1}^{3}\left[\left|x_{i}\right|^{0.8}+5\sin \left(x_{i}^{3}\right)\right]\\\end{cases}}$ $-5\leq x_{i}\leq 5$, $1\leq i\leq 3$.
Schaffer function N. 1:[21] ${\text{Minimize}}={\begin{cases}f_{1}\left(x\right)=x^{2}\\f_{2}\left(x\right)=\left(x-2\right)^{2}\\\end{cases}}$ $-A\leq x\leq A$. Values of $A$ from $10$ to $10^{5}$ have been used successfully. Higher values of $A$ increase the difficulty of the problem.
Schaffer function N. 2: ${\text{Minimize}}={\begin{cases}f_{1}\left(x\right)={\begin{cases}-x,&{\text{if }}x\leq 1\\x-2,&{\text{if }}1<x\leq 3\\4-x,&{\text{if }}3<x\leq 4\\x-4,&{\text{if }}x>4\\\end{cases}}\\f_{2}\left(x\right)=\left(x-5\right)^{2}\\\end{cases}}$ $-5\leq x\leq 10$.
Poloni's two objective function: ${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)=\left[1+\left(A_{1}-B_{1}\left(x,y\right)\right)^{2}+\left(A_{2}-B_{2}\left(x,y\right)\right)^{2}\right]\\f_{2}\left(x,y\right)=\left(x+3\right)^{2}+\left(y+1\right)^{2}\\\end{cases}}$
${\text{where}}={\begin{cases}A_{1}=0.5\sin \left(1\right)-2\cos \left(1\right)+\sin \left(2\right)-1.5\cos \left(2\right)\\A_{2}=1.5\sin \left(1\right)-\cos \left(1\right)+2\sin \left(2\right)-0.5\cos \left(2\right)\\B_{1}\left(x,y\right)=0.5\sin \left(x\right)-2\cos \left(x\right)+\sin \left(y\right)-1.5\cos \left(y\right)\\B_{2}\left(x,y\right)=1.5\sin \left(x\right)-\cos \left(x\right)+2\sin \left(y\right)-0.5\cos \left(y\right)\end{cases}}$
$-\pi \leq x,y\leq \pi $
Zitzler–Deb–Thiele's function N. 1:[22] ${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=x_{1}\\f_{2}\left({\boldsymbol {x}}\right)=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)=1+{\frac {9}{29}}\sum _{i=2}^{30}x_{i}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)=1-{\sqrt {\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}}\\\end{cases}}$ $0\leq x_{i}\leq 1$, $1\leq i\leq 30$.
Zitzler–Deb–Thiele's function N. 2:[22] ${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=x_{1}\\f_{2}\left({\boldsymbol {x}}\right)=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)=1+{\frac {9}{29}}\sum _{i=2}^{30}x_{i}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)=1-\left({\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}\right)^{2}\\\end{cases}}$ $0\leq x_{i}\leq 1$, $1\leq i\leq 30$.
Zitzler–Deb–Thiele's function N. 3:[22] ${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=x_{1}\\f_{2}\left({\boldsymbol {x}}\right)=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)=1+{\frac {9}{29}}\sum _{i=2}^{30}x_{i}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)=1-{\sqrt {\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}}-\left({\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}\right)\sin \left(10\pi f_{1}\left({\boldsymbol {x}}\right)\right)\end{cases}}$ $0\leq x_{i}\leq 1$, $1\leq i\leq 30$.
Zitzler–Deb–Thiele's function N. 4:[22] ${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=x_{1}\\f_{2}\left({\boldsymbol {x}}\right)=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)=91+\sum _{i=2}^{10}\left(x_{i}^{2}-10\cos \left(4\pi x_{i}\right)\right)\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)=1-{\sqrt {\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}}\end{cases}}$ $0\leq x_{1}\leq 1$, $-5\leq x_{i}\leq 5$, $2\leq i\leq 10$
Zitzler–Deb–Thiele's function N. 6:[22] ${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=1-\exp \left(-4x_{1}\right)\sin ^{6}\left(6\pi x_{1}\right)\\f_{2}\left({\boldsymbol {x}}\right)=g\left({\boldsymbol {x}}\right)h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)\\g\left({\boldsymbol {x}}\right)=1+9\left[{\frac {\sum _{i=2}^{10}x_{i}}{9}}\right]^{0.25}\\h\left(f_{1}\left({\boldsymbol {x}}\right),g\left({\boldsymbol {x}}\right)\right)=1-\left({\frac {f_{1}\left({\boldsymbol {x}}\right)}{g\left({\boldsymbol {x}}\right)}}\right)^{2}\\\end{cases}}$ $0\leq x_{i}\leq 1$, $1\leq i\leq 10$.
Osyczka and Kundu function:[23] ${\text{Minimize}}={\begin{cases}f_{1}\left({\boldsymbol {x}}\right)=-25\left(x_{1}-2\right)^{2}-\left(x_{2}-2\right)^{2}-\left(x_{3}-1\right)^{2}-\left(x_{4}-4\right)^{2}-\left(x_{5}-1\right)^{2}\\f_{2}\left({\boldsymbol {x}}\right)=\sum _{i=1}^{6}x_{i}^{2}\\\end{cases}}$ ${\text{s.t.}}={\begin{cases}g_{1}\left({\boldsymbol {x}}\right)=x_{1}+x_{2}-2\geq 0\\g_{2}\left({\boldsymbol {x}}\right)=6-x_{1}-x_{2}\geq 0\\g_{3}\left({\boldsymbol {x}}\right)=2-x_{2}+x_{1}\geq 0\\g_{4}\left({\boldsymbol {x}}\right)=2-x_{1}+3x_{2}\geq 0\\g_{5}\left({\boldsymbol {x}}\right)=4-\left(x_{3}-3\right)^{2}-x_{4}\geq 0\\g_{6}\left({\boldsymbol {x}}\right)=\left(x_{5}-3\right)^{2}+x_{6}-4\geq 0\end{cases}}$ $0\leq x_{1},x_{2},x_{6}\leq 10$, $1\leq x_{3},x_{5}\leq 5$, $0\leq x_{4}\leq 6$.
CTP1 function (2 variables):[4][24] ${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)=x\\f_{2}\left(x,y\right)=\left(1+y\right)\exp \left(-{\frac {x}{1+y}}\right)\end{cases}}$ ${\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)={\frac {f_{2}\left(x,y\right)}{0.858\exp \left(-0.541f_{1}\left(x,y\right)\right)}}\geq 1\\g_{2}\left(x,y\right)={\frac {f_{2}\left(x,y\right)}{0.728\exp \left(-0.295f_{1}\left(x,y\right)\right)}}\geq 1\end{cases}}$ $0\leq x,y\leq 1$.
Constr-Ex problem:[4] ${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)=x\\f_{2}\left(x,y\right)={\frac {1+y}{x}}\\\end{cases}}$ ${\text{s.t.}}={\begin{cases}g_{1}\left(x,y\right)=y+9x\geq 6\\g_{2}\left(x,y\right)=-y+9x\geq 1\\\end{cases}}$ $0.1\leq x\leq 1$, $0\leq y\leq 5$
Viennet function: ${\text{Minimize}}={\begin{cases}f_{1}\left(x,y\right)=0.5\left(x^{2}+y^{2}\right)+\sin \left(x^{2}+y^{2}\right)\\f_{2}\left(x,y\right)={\frac {\left(3x-2y+4\right)^{2}}{8}}+{\frac {\left(x-y+1\right)^{2}}{27}}+15\\f_{3}\left(x,y\right)={\frac {1}{x^{2}+y^{2}+1}}-1.1\exp \left(-\left(x^{2}+y^{2}\right)\right)\\\end{cases}}$ $-3\leq x,y\leq 3$.
See also
Wikimedia Commons has media related to Test functions (mathematical optimization).
• Ackley function
• Himmelblau's function
• Rastrigin function
• Rosenbrock function
• Shekel function
• Binh function
References
1. Bäck, Thomas (1995). Evolutionary algorithms in theory and practice : evolution strategies, evolutionary programming, genetic algorithms. Oxford: Oxford University Press. p. 328. ISBN 978-0-19-509971-3.
2. Haupt, Randy L. Haupt, Sue Ellen (2004). Practical genetic algorithms with CD-Rom (2nd ed.). New York: J. Wiley. ISBN 978-0-471-45565-3.{{cite book}}: CS1 maint: multiple names: authors list (link)
3. Oldenhuis, Rody. "Many test functions for global optimizers". Mathworks. Retrieved 1 November 2012.
4. Deb, Kalyanmoy (2002) Multiobjective optimization using evolutionary algorithms (Repr. ed.). Chichester [u.a.]: Wiley. ISBN 0-471-87339-X.
5. Binh T. and Korn U. (1997) MOBES: A Multiobjective Evolution Strategy for Constrained Optimization Problems. In: Proceedings of the Third International Conference on Genetic Algorithms. Czech Republic. pp. 176–182
6. Binh T. (1999) A multiobjective evolutionary algorithm. The study cases. Technical report. Institute for Automation and Communication. Barleben, Germany
7. Deb K. (2011) Software for multi-objective NSGA-II code in C. Available at URL: https://www.iitk.ac.in/kangal/codes.shtml
8. Ortiz, Gilberto A. "Multi-objective optimization using ES as Evolutionary Algorithm". Mathworks. Retrieved 1 November 2012.
9. Whitley, Darrell; Rana, Soraya; Dzubera, John; Mathias, Keith E. (1996). "Evaluating evolutionary algorithms". Artificial Intelligence. Elsevier BV. 85 (1–2): 264. doi:10.1016/0004-3702(95)00124-7. ISSN 0004-3702.
10. Vanaret C. (2015) Hybridization of interval methods and evolutionary algorithms for solving difficult optimization problems. PhD thesis. Ecole Nationale de l'Aviation Civile. Institut National Polytechnique de Toulouse, France.
11. Simionescu, P.A.; Beale, D. (September 29 – October 2, 2002). New Concepts in Graphic Visualization of Objective Functions (PDF). ASME 2002 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Montreal, Canada. pp. 891–897. Retrieved 7 January 2017.
12. "Solve a Constrained Nonlinear Problem - MATLAB & Simulink". www.mathworks.com. Retrieved 2017-08-29.
13. "Bird Problem (Constrained) | Phoenix Integration". Archived from the original on 2016-12-29. Retrieved 2017-08-29.{{cite web}}: CS1 maint: bot: original URL status unknown (link)
14. Mishra, Sudhanshu (2006). "Some new test functions for global optimization and performance of repulsive particle swarm method". MPRA Paper.
15. Townsend, Alex (January 2014). "Constrained optimization in Chebfun". chebfun.org. Retrieved 2017-08-29.
16. Simionescu, P.A. (2020). "A collection of bivariate nonlinear optimisation test problems with graphical representations". International Journal of Mathematical Modelling and Numerical Optimisation. 10 (4): 365–398. doi:10.1504/IJMMNO.2020.110704.
17. Simionescu, P.A. (2014). Computer Aided Graphing and Simulation Tools for AutoCAD Users (1st ed.). Boca Raton, FL: CRC Press. ISBN 978-1-4822-5290-3.
18. Chankong, Vira; Haimes, Yacov Y. (1983). Multiobjective decision making. Theory and methodology. ISBN 0-444-00710-5.
19. Fonseca, C. M.; Fleming, P. J. (1995). "An Overview of Evolutionary Algorithms in Multiobjective Optimization". Evol Comput. 3 (1): 1–16. CiteSeerX 10.1.1.50.7779. doi:10.1162/evco.1995.3.1.1. S2CID 8530790.
20. F. Kursawe, “A variant of evolution strategies for vector optimization,” in PPSN I, Vol 496 Lect Notes in Comput Sc. Springer-Verlag, 1991, pp. 193–197.
21. Schaffer, J. David (1984). "Multiple Objective Optimization with Vector Evaluated Genetic Algorithms". In G.J.E Grefensette; J.J. Lawrence Erlbraum (eds.). Proceedings of the First International Conference on Genetic Algorithms. OCLC 20004572.
22. Deb, Kalyan; Thiele, L.; Laumanns, Marco; Zitzler, Eckart (2002). "Scalable multi-objective optimization test problems". Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600). Vol. 1. pp. 825–830. doi:10.1109/CEC.2002.1007032. ISBN 0-7803-7282-4. S2CID 61001583.
23. Osyczka, A.; Kundu, S. (1 October 1995). "A new method to solve generalized multicriteria optimization problems using the simple genetic algorithm". Structural Optimization. 10 (2): 94–99. doi:10.1007/BF01743536. ISSN 1615-1488. S2CID 123433499.
24. Jimenez, F.; Gomez-Skarmeta, A. F.; Sanchez, G.; Deb, K. (May 2002). "An evolutionary algorithm for constrained multi-objective optimization". Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600). Vol. 2. pp. 1133–1138. doi:10.1109/CEC.2002.1004402. ISBN 0-7803-7282-4. S2CID 56563996.
| Wikipedia |
Test ideal
A test ideal is a positive characteristic analog of a multiplier ideal in, say, the field of complex numbers.[1] Test ideals are used in the study of singularities in algebraic geometry in positive characteristic.[2]
References
1. Henriques, Inês B.; Varbaro, M. (2014). "Test, multiplier and invariant ideals". arXiv:1407.4324 [math.AC].
2. Hassett, Brendan; McKernan, James; Starr, Jason; Vakil, Ravi (September 11, 2013). A Celebration of Algebraic Geometry. American Mathematical Society. ISBN 9780821889831. Retrieved 3 March 2017.
| Wikipedia |
Test of Mathematics for University Admission
The Test of Mathematics for University Admission is a paper-based test sometimes used in the United Kingdom and other countries in Africa and the United States all assess the mathematical thinking and reasoning skills needed for undergraduate mathematics courses or courses featuring mathematics. A number of universities across the world accept the test as an optional part of their application process for mathematics-based courses.[1]
History
The test was developed by Cambridge Assessment Admissions Testing and launched in 2016. It was designed to assess the key skills that students need to succeed on demanding university-level mathematics courses, and assist university mathematics tutors in making admissions decisions.[2]
Durham University[3] and Lancaster University[4] began using the test in 2016, with the University of Warwick,[5] the University of Sheffield[6] and the University of Southampton recognising the test in 2017, and the London School of Economics and Political Science (LSE)[7] and Cardiff University in 2018 [8]
Research indicates that the test has good predictive validity, with good correlation between candidates' scores in the test and their performance in their exams at the end of first year university study. There is also correlation between A-level Further Maths performance and performance in the test.[2]
Test format and specification
The Test of Mathematics for University Admission is a paper-based 2 hour and 30 minute long test. It has two papers which are taken consecutively.
Paper 1: Mathematical Thinking
Paper 1 has 20 multiple-choice questions, with 75 minutes allowed to complete the paper. This paper assesses a candidate’s ability to apply their knowledge of mathematics in new situations. It comprises a core set of ideas from Pure Mathematics. These ideas reflect those that would be met early on in a typical A Level Mathematics course: algebra, basic functions, sequences and series, coordinate geometry, trigonometry, exponentials and logarithms, differentiation, integration, graphs of functions. In addition, knowledge of the GCSE curriculum is assumed.[9]
Paper 2: Mathematical Reasoning
Paper 2 has 20 multiple-choice questions, with 75 minutes allowed to complete the paper. The second paper assesses a candidate’s ability to justify and interpret mathematical arguments and conjectures, and deal with elementary concepts from logic. It assumes knowledge of the Paper 1 specification and, in addition, requires students to have some knowledge of the structure of proof and basic logical concepts.
Calculators or dictionaries are not allowed to be used in the test.
Scoring
There is no pass/fail for the test. Candidates’ scores are determined by the number of correct answers given in both papers. Each question has the same weighting, and no penalties are given for incorrect answers. Raw scores are converted to a scale of 1.0 to 9.0 (with 9.0 being the highest). A score is also reported for each of the two papers (also reported on the 1.0 to 9.0 scale), but these are for candidate information only and do not form part of the formal test result.[10]
Timing and results
The test is made available once a year in late October or early November. Candidates can sit the test at their school or at test centres around the world. Entry for the test typically opens in September and candidates must be registered by early October. Results are released in late November. Candidates can access their results online and share them with their chosen institutions.
Preparation
Students generally spend several weeks preparing for the TMUA exam. There are various different preparation materials available for students wanting to get ready for the exam such as textbooks, courses and online materials.[11] However, past papers are the most valuable resource as they are directly from the exam administrators themselves. The past papers are freely available from the exam administrator, and various other sources. Answer keys are also released alongside TMUA past papers.[12]
References
1. http://www.admissionstesting.org/images/302050-courses-accepting-test-of-mathematics-for-undergraduate-admissions.pdf retrieved 20 October 2018
2. Cambridge Assessment Admissions Testing. (2019) Test of Mathematics of University Admission trial – Durham University 2015. Cambridge, UK: Cambridge Assessment. retrieved 14 April 2019
3. How to Apply, University of Durham website. Retrieved 19 April 2019
4. Our Offers, Lancaster University website. Retrieved 19 April 2019
5. MAT, TMUA and STEP, the University of Warwick website. Retrieved 19 April 2019
6. Further Mathematics A Level, TMUA and STEP, The University of Sheffield website. Retrieved 19 April 2019
7. Admissions Information, LSE website. Retrieved April 19, 2019.
8. Qualifications accepted by Cardiff University, Cardiff University website. Retrieved 19 April 2019
9. Test of Mathematics for University Admission - Specification for October 2018 retrieved 22 April 2019
10. "TMUA Scoring: What Is A Good TMUA Score?". UniAdmissions: The Oxbridge Experts. Retrieved 4 November 2022.
11. "TMUA Tuition Programme". UniAdmissions: The Oxbridge Experts. Retrieved 10 October 2022.
12. "TMUA Past Papers Collection (2016 - 2021)". Exams Ninja. Retrieved 7 November 2022.
External links
• http://www.admissionstesting.org Cambridge Assessment Admissions Testing
University admissions tests in the United Kingdom
English Literature
• English Literature Admissions Test (ELAT)
History
• History Aptitude Test (HAT)
Languages
• Modern and Medieval Languages Test
Law
• National Admissions Test for Law (LNAT)
Mathematics
• Sixth Term Examination Paper (STEP)
• Mathematics Admissions Test (MAT)
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• BioMedical Admissions Test (BMAT)
• Graduate Medical School Admissions Test (GAMSAT)
• Health Professions Admissions Test (HPAT)
• University Clinical Aptitude Test (UCAT)
Science
• Physics Aptitude Test (PAT)
General academic study
• Thinking Skills Assessment (TSA)
| Wikipedia |
Testimator
A testimator is an estimator whose value depends on the result of a test for statistical significance. In the simplest case the value of the final estimator is that of the basic estimator if the test result is significant, and otherwise the value is zero. However more general testimators are possible.[1]
History
An early use of the term "testimator" way made by Brewster & Zidek (1974).[2]
References
1. Adke, S.R., Waikar, V.B. & Schurmann, F.J. (1987). "A two stage shrinkage testimator for the mean of an exponential distribution". Communications in Statistics - Theory and Methods, 16 (6), 1821-1834. Retrieved April 16, 2009, from http://www.informaworld.com/10.1080/03610928708829474 (restricted access)
2. Brewster, J.F., Zidek, J.V.(1974) "Improving on Equivariant Estimators". Annals of Statistics, 2 (1), 21–38
Further reading
• S. M. Kanbur, C. Ngeow, A. Nanthakumar and R. Stevens (2007). "Investigations of the Nonlinear LMC Cepheid Period‐Luminosity Relation with Testimator and Schwarz Information Criterion Methods", Publications of the Astronomical Society of the Pacific 119, 512–522 online
• Mezbahur Rahman, Gokhale, D.V. (1996) "Testimation in Regression Parameter Estimation", Biometrical Journal, 38 (7), 809–817 online
• Abramovich, F., Grinshtein, V., Pensky, M. (2007) "On optimality of Bayesian testimation in the normal means problem", Annals of Statistics, 35, 2261–2286 online
| Wikipedia |
Testing in binary response index models
Denote a binary response index model as: $P[Y_{i}=1\mid X_{i}]=G(X_{i}\beta )$, $[Y_{i}=0\mid X_{i}]=1-G(X_{i}'\beta )$ where $X_{i}\in R^{N}$.
Description
This type of model is applied in many economic contexts, especially in modelling the choice-making behavior. For instance, $Y_{i}$ here denotes whether consumer $i$ chooses to purchase a certain kind of chocolate, and $X_{i}$ includes many variables characterizing the features of consumer $i$ . Through function $G(\cdot )$ , the probability of choosing to purchase is determined.[1]
Now, suppose its maximum likelihood estimator (MLE) ${\hat {\beta }}_{u}$ has an asymptotic distribution as ${\sqrt {n}}({\hat {\beta }}_{u}-\beta ){\xrightarrow {d}}N(0,V)$ and there is a feasible consistent estimator for the asymptotic variance $V$ denoted as ${\hat {V}}$ . Usually, there are two different types of hypothesis needed to be tested in binary response index model.
The first type is testing the multiple exclusion restrictions, namely, testing $\beta _{2}=0with=[\beta _{1};\beta _{2}]where\beta _{2}\in R^{Q}$. If the unrestricted MLE can be easily computed, it is convenient to use the Wald test[2] whose test statistic is constructed as:
$(D{\hat {\beta }}_{u})^{T}(D{\hat {V}}D^{T}/n)^{-1}(D{\hat {\beta }}_{u}){\xrightarrow {d}}X_{Q}^{2}$
Where D is a diagonal matrix with the last Q diagonal entries as 0 and others as 1. If the restricted MLE can be easily computed, it is more convenient to use the Score test (LM test). Denote the maximum likelihood estimator under the restricted model as $({\hat {\beta }}_{r})$ and define ${\hat {u}}_{i}\equiv Y_{i}-G(X_{i}'{\hat {\beta }}_{r}),{\hat {G}}_{i}\equiv G(X_{i}'{\hat {\beta }}_{r})$ and ${\hat {g}}_{i}\equiv g(X_{i}'{\hat {\beta }}_{r}$, where $g(\cdot )=G'(\cdot )$. Then run the OLS regression ${\frac {{\hat {u}}_{i}}{\sqrt {({\hat {G}}_{i}(1-{\hat {G}}_{i})}}}$ on ${\frac {{\hat {g}}_{i}}{\sqrt {({\hat {G}}_{i}(1-{\hat {G}}_{i})}}}X_{1i}',{\frac {{\hat {g}}_{i}}{\sqrt {({\hat {G}}_{i}(1-{\hat {G}}_{i})}}}X_{2i}'$, where $X_{i}=[X_{1i};X_{2i}]$ and $X_{2i}\varepsilon R^{Q}$. The LM statistic is equal to the explained sum of squares from this regression [3] and it is asymptotically distributed as $X_{Q}^{2}$. If the MLE can be computed easily under both of the restricted and unrestricted models, Likelihood-ratio test is also a choice: let $L_{u}$ denote the value of the log-likelihood function under the unrestricted model and let $L_{r}$ denote the value under the restricted model, then $2(L_{u}-L_{r})$ has an asymptotic $X_{Q}^{2}$ distribution.
The second type is testing a nonlinear hypothesis about $\beta $, which can be represented as $H_{0}:c(\beta )=0$ where $c(\beta )$ is a Q×1 vector of possibly nonlinear functions satisfying the differentiability and rank requirements. In most of the cases, it is not easy or even feasible to compute the MLE under the restricted model when $c(\beta )$ include some complicated nonlinear functions. Hence, Wald test is usually used to deal with this problem. The test statistic is constructed as:
$c({\hat {\beta }}_{u}')[\nabla _{\beta }c({\hat {\beta }}_{u}){\hat {V}}\nabla _{\beta }c({\hat {\beta }}_{u})']^{-1}c({\hat {\beta }}_{u}){\xrightarrow {d}}X_{Q}^{2}$
where $\nabla _{\beta }c({\hat {\beta }}_{u})$ is the Q×N Jacobian of $c(\beta )$ evaluated at ${\hat {\beta }}_{u}$.
For the tests with very general and complicated alternatives, the formula of the test statistics might not have the exactly same representation as above. But we can still derive the formulas as well as its asymptotic distribution by Delta method[4] and implement Wald test, Score test or Likelihood-ratio test.[5] Which test should be used is determined by the relative computation difficulty of the MLE under restricted and unrestricted models.
References
1. For an application example, refer to: Rayton, B. A. (2006): “Examining the Interconnection of Job Satisfaction and Organizational Commitment: an Application of the Bivariate Probit Model”,The International Journal of Human Resource Management, Vol. 17, Iss. 1.
2. Greene, W. H. (2003), Econometric Analysis , Prentice Hall , Upper Saddle River, NJ .
3. Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass.
4. Casella, G., and Berger, R. L. (2002). Statistical inference. Duxbury Press.
5. Engle, Robert F. (1983). "Wald, Likelihood Ratio, and Lagrange Multiplier Tests in Econometrics". In Intriligator, M. D.; and Griliches, Z. Handbook of Econometrics II. Elsevier. pp. 796–801. ISBN 978-0-444-86185-6.
Statistics
• Outline
• Index
Descriptive statistics
Continuous data
Center
• Mean
• Arithmetic
• Arithmetic-Geometric
• Cubic
• Generalized/power
• Geometric
• Harmonic
• Heronian
• Heinz
• Lehmer
• Median
• Mode
Dispersion
• Average absolute deviation
• Coefficient of variation
• Interquartile range
• Percentile
• Range
• Standard deviation
• Variance
Shape
• Central limit theorem
• Moments
• Kurtosis
• L-moments
• Skewness
Count data
• Index of dispersion
Summary tables
• Contingency table
• Frequency distribution
• Grouped data
Dependence
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| Wikipedia |
Numeral prefix
Numeral or number prefixes are prefixes derived from numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example:
• unicycle, bicycle, tricycle (1-cycle, 2-cycle, 3-cycle)
• dyad, triad (2 parts, 3 parts)
• biped, quadruped (2 legs, 4 legs)
• September, October, November, December (month 7, month 8, month 9, month 10)[upper-alpha 1]
• decimal, hexadecimal (base-10, base-16)
• septuagenarian, octogenarian (70-79 years old, 80-89 years old)
• centipede, millipede (around 100 legs, around 1000 legs)
In many European languages there are two principal systems, taken from Latin and Greek, each with several subsystems; in addition, Sanskrit occupies a marginal position.[upper-alpha 2] There is also an international set of metric prefixes, which are used in the metric system and which for the most part are either distorted from the forms below or not based on actual number words.
Table of number prefixes in English
In the following prefixes, a final vowel is normally dropped before a root that begins with a vowel, with the exceptions of bi-, which is bis- before a vowel, and of the other monosyllables, du-, di-, dvi-, tri-, which are invariable.
The cardinal series are derived from cardinal numbers, such as the English one, two, three. The multiple series are based on adverbial numbers like the English once, twice, thrice. The distributive series originally meant one each, two each or one by one, two by two, etc., though that meaning is now frequently lost. The ordinal series are based on ordinal numbers such as the English first, second, third (for numbers higher than 2, the ordinal forms are also used for fractions; only the fraction 1⁄2 has special forms).
For the hundreds, there are competing forms: those in -gent-, from the original Latin, and those in -cent-, derived from centi-, etc. plus the prefixes for 1–9.
Many of the items in the following tables are not in general use, but may rather be regarded as coinages by individuals. In scientific contexts, either scientific notation or SI prefixes are used to express very large or very small numbers, and not unwieldy prefixes.
The same suffix may be used with more than one series:
Ordinal numbers primarysecondarytertiaryquartary
quintarysextaryseptimaryoctavarynonarydecimary
Distributive numbers singulary
binaryternary, trinaryquaternaryquinarysenaryseptenaryoctonarynovenarydenary
Number prefixes in English
Number Latin Greek[upper-alpha 3] Sanskrit[upper-alpha 2]
CardinalMultipleDistributiveOrdinal CardinalMultiple,
proportional, or
quantitative
Ordinal
0nulli-nullesim-miden-, ouden- -medeproto-, oudeproto-shūnya-
1⁄12unci-[1] –For fractions, Greek uses ordinals (i.e. dodecato-) –
1⁄8octant- – – –As above; ogdoö– –
1⁄6sextant- – – –As above; hecto- –
1⁄5quintant- – – –As above; Pempto– –
1⁄4quadrant- – – –As above; tetarto– –
1⁄3trient- – – –As above; trito– –
1⁄2semi- –demi-[upper-alpha 4] –hemi- ("half")[lower-alpha 1] – – –
3⁄4dodrant- – – – – –
1uni-[lower-alpha 2]
sol-[upper-alpha 5][lower-alpha 3]
sim-[upper-alpha 6]singul-prim-mono- ("one", "alone")[lower-alpha 4]
holo- ("entire", "full")[lower-alpha 5]
hen-[upper-alpha 7] rare
mono- ("one, alone")
hapax- ("once")
haplo-[upper-alpha 7] ("single")
monad- ("one of a kind", "unique", "unit")
prot-[2][lower-alpha 6]
protaio- ("[every] first day")
eka- [3]
1+1⁄4 –quasqui-[lower-alpha 7] – – – – – –
1+1⁄2 –sesqui-[lower-alpha 8] – – – – – –
2du-bi-, bis-[lower-alpha 9]bin-second-di-, dy-,[4] duo-, dyo-dis-[5] ("twice") common
dyakis- ("twice") rare
diplo- ("double")
dyad- ("two of a
kind")
deuter-[6][lower-alpha 10]
deuteraio- ("[every] second day")
dvi-[7]
2+1⁄2 –semiquin-[lower-alpha 11] – – – – – –
3tri-[lower-alpha 12]ter-tern-, trin-terti-tri-[lower-alpha 13]tris-[8] ("thrice") common
triakis- ("thrice") rare
triplo- ("triple")
triad- ("three of a kind")
trit-[9] ("third")[lower-alpha 14]
tritaio- ("[every] third day")
tri-[10]
4quadri-, quadru-[upper-alpha 8]quater-[11]quatern-[12]quart-[13]tetra-, tessara-tetrakis- ("four times")
tetraplo- ("quadruple")
tetrad- ("four of a kind")[lower-alpha 15]
tetarto- ("fourth")
tetartaio-
("[every] fourth day")
catur-[14]
5quinque-[15] –quin-[16]quint-[17]penta-pentakis-
pentaplo-
pentad-[lower-alpha 16]
pempt-[18]
pemptaio-
pañca-[19]
6sexa-[upper-alpha 9] –sen-[20]sext-[21]hexa-[22]hexakis-
hexaplo-
hexad-[lower-alpha 17]
hect-[23]
hectaio-
ṣaṭ-[24]
7septem-, septi-[lower-alpha 18]septen-[25]septim-hepta-[26][lower-alpha 19]heptakis-
heptaplo-
heptad-
hebdomo- ("seventh")
hebdomaio- ("seventh day")[lower-alpha 20]
sapta-[27]
8octo-[lower-alpha 21] –octon-[28]octav-[29]octo-[lower-alpha 22]octakis-
octaplo-
octad-[lower-alpha 23]
ogdoö-
ogdoaio-
aṣṭa-
9novem-[lower-alpha 24]noven-nona-ennea-[30]enneakis-
enneaplo-
ennead-
enat-[31]
enataio-
nava-
10decem-, dec-[lower-alpha 25]den-[32]decim-[33]deca-[34][lower-alpha 26]decakis-
decaplo-
decad-
decat-[35]
decataio-
dasha-
11undec-unden-[36]undecim-[37]hendeca-[38]hendeca/kis/plo/d-hendecat-[39]/o/aio-ekadasha-
12duodec-duoden-[lower-alpha 27]duodecim-dodeca-[40][lower-alpha 28]dodeca/kis/plo/d-dodecat-[41]/o/aio-dvadasha-
13tredec-treden-tredecim-tria(kai)deca-, decatria-[lower-alpha 29]tris(kai)decakis-,
decatria/kis/plo/d-
decatotrito-
etc.
trayodasha-
14quattuordec-quattuorden-quattuordecim-
quartadecim-
tessara(kai)deca-, decatettara-, decatessara-tetra(kai)decakis-,
decatetra/kis/plo/d-[lower-alpha 30]
decatotetarto-chaturdasha-
15quinquadec-, quindec-[42]quinden-[43]quindecim-[44]
quintadecim-
pente(kai)deca-, decapente-penta(kai)decakis-,
decapentakis- etc.
decatopempto-panchadasha-
16sedec-,[45] sexdec-
(but hybrid hexadecimal)
seden-sedecim-
sextadecim-
hexa(kai)deca-,
hekkaideca-,
decahex-
hexa(kai)decakis-,
decahexakis- etc.
decatohecto-shodasha-
17septendec-septenden-septendecim-
septimadecim-
hepta(kai)deca-,
decahepta-
hepta(kai)decakis-,
decaheptakis- etc.
decatohebdomo-saptadasha-
18octodec-octoden-octodecim-
duodevicesim-
octo(kai)deca-,
decaocto-
octa(kai)decakis-,
decaoctakis- etc.
decatoogdoö-ashtadasha-
19novemdec-, novendec-
undeviginti-
novemden-
novenden-
novemdecim-
novendecim-
undevisim-
ennea(kai)deca-, decaennea-ennea(kai)decakis-,
decaenneakis- etc.
decatoenato-navadasha-
20[upper-alpha 10]viginti-vicen-, vigen-vigesim-(e)icosi-eicosa/kis/plo/d-[lower-alpha 31]eicosto-vimshati-
22duovigint-(e)icosidyo-, dyo(e)icosi- rare[lower-alpha 32](e)icosidyakis-
(e)icosidiplo-
(e)icosidyad-
eicostodeutero- –
24quattuorvigint-(e)icositettara-, (e)icosikaitettara-
rare
(e)icositetrakis-
(e)icositetraplo-
(e)icositetrad-[lower-alpha 33]
eicostotetarto-chaturvimshati-
25quinvigint-(e)icosipente-[lower-alpha 34](e)icosipentakis-
(e)icosipentaplo-
(e)icosipentad-
eicostopempto- –
30triginti-tricen-trigesim-triaconta-triacontakis- etc.[lower-alpha 35]triacosto-trimshat-
31untriginti-triacontahen-triacontahenakis-
triacontahenaplo-
triacontahenad-
triacostoproto-
triacostoprotaio-
–
40quadraginti-quadragen-quadragesim-tettaraconta-,
tessaraconta-
tettaracontakis-,
tessaracontakis- etc.
tessaracosto-chatvarimshat-
50quinquaginti-[46]quinquagen-[47]quinquagesim-[48]penteconta-[lower-alpha 36]pentecontakis- etc.pentecosto-[lower-alpha 37]panchashat-
60sexaginti-sexagen-sexagesim-hexeconta-hexecontakis- etc.hexecosto-shasti-
70septuaginti-[lower-alpha 38]septuagen-septuagesim-[49]hebdomeconta-hebdomecontakis- etc.hebdomecosto-saptati-
80octogint-octogen-octogesim-ogdoëconta-ogdoëcontakis- etc.ogdoëcosto-ashiti-
90nonagint-nonagen-nonagesim-eneneconta-enenecontakis- etc.enenecosto-navati-
100centi-centen-centesim-hecato(n)-hecatontakis-
hundred times
hecatontaplo-
hundred-multiple
hecatontad-
hundred of a kind
also abbreviated in
hec[aton]tad-
hecatosto-
hundredth
hecatostaio-
the hundredth day
shata–
120viginticenti- – –hecaton(e)icosi-hecaton(e)icosakis- etc.hecatostoeicosto- –
150 –sesquicenten-[lower-alpha 39] – – – – –
200ducenti-ducen-, bicenten-ducentesim-diacosia-diacosakis- etc.diacosiosto- –
250 –semiquincenten-[lower-alpha 40] – – – – –
300trecenti-trecen-, tercenten-, tricenten-trecentesim-triacosia- etc.triacosakis-
triacosaplo-
triacosad-
triacosiosto- –
400quadringenti-quadringen-, quatercenten-, quadricenten-quadringentesim-tetracosia-tetracosakis- etc.tetracosiosto- –
500quingent-,[50] quincent-[51]quingen-,[52] quingenten-, quincenten-quingentesim-[53]pentacosia-pentacosakis- etc.pentacosiosto- –
600sescenti-, sexcenti-sescen-, sexcenten-sescentesim-hexacosia-hexacosakis- etc.hexacosiosto- –
700septingenti-septingen-, septingenten-, septcenten-septingentesim-heptacosia-heptacosakis- etc.heptacosiosto- –
800octingenti-octingen-, octingenten-, octocenten-octingentesim-octacosia-octacosakis- etc.octacosiosto- –
900nongenti-nongen-nongentesim-ennacosi-[54]
derived from
en(n)iacosia-,
a pejoration of
enneacosia-
enneacosakis- etc.enacosiost-,[55]
alt. spelling
en(n)iacosiost(o)-
a pejoration of
enneacosiosto-
–
1000milli-millen-millesim-chili-,[56] kilo-chiliakis-
chiliaplo-
chiliad-
chiliost-[57]sahasra–
2000duomilli – –dischili-[58]dischiliakis- etc.dischiliosto- –
3000tremilli-trischili-[59] –trischiliost-[60] –
5000quinmilli–pentacischili-[61] – – –
10000decamilli–myria-,[62][lower-alpha 41] decakischilia-myriakis-
myriaplo-
myriad-
decakischiliakis- etc.
myriast-,[63]
decakischiliosto-
ayuta–
80000octogintmilli–octacismyri-[64] – – –
105centimilli–decakismyria-, hecatontakischilia-decakismyriakis-,
hecatontakischiliakis-
etc.laksha–
106million- –hecatommyria-
(see also Mega-)
hecatommyriakis-
("a million times")
hecatommyriaplo-
(million-multiple)
hecatommyriad-
(a million of a kind)
hecatommyriosto-
(ranked millionth;
also one piece of
a million [fraction] see above in
fractions)
hecatommyriostaio-
("the millionth day")
–
109billion- –dis
hecatommyria-
dis
hecatommyriakis-
etc. –
1012trillion- –tris
hecatommyria-
tris
hecatommyriakis-
etc. –
1015quadrillion- –tetrakis
hecatommyria-
tetrakis
hecatommyriakis-
etc. –
1018quintillion- –pentakis
hecatommyria-
pentakis
hecatommyriakis-
etc. –
1020vingtillion- –eicosakis
hecatommyria-
eicosakis
hecatommyriakis-
etc. –
1021sextillion- –hexakis
hecatommyria-
hexakis
hecatommyriakis-
etc. –
1024septillion- –heptakis
hecatommyria-
heptakis
hecatommyriakis-
etc. –
1027octillion- –octakis
hecatommyria-
octakis
hecatommyriakis-
etc. –
1030nonillion- –enneakis
hecatommyria-
enneakis
hecatommyriakis-
etc. –
1033decillion- –decakis
hecatommyria-
decakis
hecatommyriakis-
etc. –
1036undecillion- –hendecakis
hecatommyria-
hendecakis
hecatommyriakis-
etc. –
1039duodecillion- –dodecakis
hecatommyria-
dodecakis
hecatommyriakis-
etc. –
1042tredecillion- –triskaidecakis
hecatommyria-
triskaidecakis
hecatommyriakis-
etc. –
1045quattuordecillion- –tetrakaidecakis
hecatommyria-
tetrakaidecakis
hecatommyriakis-
etc. –
1048quindecillion- –pentakaidecakis
hecatommyria-
pentakaidecakis
hecatommyriakis-
etc. –
1051sexdecillion- –hexakaidecakis
hecatommyriakis-
hexakaidecakis
hecatommyriakis-
etc. –
1054septendecillion- –heptakaidecakis
hecatommyria-
heptakaidecakis
hecatommyriakis-
etc. –
1057octodecillion- –octakaidecakis
hecatommyria-
octakaidecakis
hecatommyriakis-
etc. –
1060novemdecillion- –enneakaidecakis
hecatommyria-
enneakaidecakis
hecatommyriakis-
etc. –
1063vigintillion- -Icosakis
hectotommyria-
Icosakis
hectotommyriakis-
etc. -
10303centillion- –hecatontakis
hecatommyria-
hecatontakis
hecatommyriakis-
etc. –
103003millinilion- / milliatillion- –chiliakis
hecatommyria-
chiliakis
hecatommyriakis-
etc. –
1030003decillinillion- / decmilliatillion- –myriakis
hecatommyria-
myriakis
hecatommyriakis-
etc.
10300003centillinillion- / centimilliatillion- –decakismyriakis
hecatommyria- and so on -
virtually endless
decakismyriakis
hecatommyriakis-
etc.
103000003micrillion-[65] -hectotomyriakis hectotommyria- Hectotommyrakis hectatommyria-
∞ infini- apeiro-
Few
(1–20)
pauci-[lower-alpha 42] –oligo-[lower-alpha 43] – – –
Many
(> 1)
multi-, pluri-[lower-alpha 44] –poly-[lower-alpha 45]pollakis- (many times)
pollaplo- (multiple)
plethos- (many of a kind)[lower-alpha 46]
pollosto-
(rank/order of many [manieth])
bahut–
Examples
1. e.g. hemisphere
2. e.g. universe, unilateral
3. e.g. solo, soliloquy
4. e.g. monogamy
5. e.g. holocaust, holography
6. e.g. proton, protozoa
7. e.g. quasquicentennial
8. e.g. sesquicentennial, sesquipedalian
9. e.g. bireme, bilingual, bipolar, bipartisan
10. e.g. Deuteron/ium, Deuteronomy
11. e.g. semiquincentennial
12. e.g. trireme
13. e.g. triathlon, Tripolis
14. e.g. Triton/ium
15. e.g. tetrahedron
16. e.g. pentahedron
17. e.g. hexahedron
18. e.g. September
19. e.g. heptathlon
20. e.g. hebdomas
21. e.g. October
22. e.g. octopus
23. e.g. octahedron
24. e.g. November
25. e.g. December
26. e.g. decathlon, decahedron, decagon
27. e.g. duodenum
28. e.g. dodecahedron
29. e.g. triskaidekaphobia
30. e.g. tetradecahedron/
decatetrahedron
31. e.g. (e)icosahedron
32. e.g. docosa-hexaenoic acid (a pejoration of dyoicosa-hexanoic)
33. e.g. (e)icositetragon
34. e.g. eicosapenta-enoic acid
35. e.g. triacontahedron
36. e.g. penteconter
37. e.g. pentecost
38. e.g. Septuagint
39. e.g. sesquicentennial
40. e.g. semiquincentennial
41. e.g. myriapoda
42. e.g. pauciparous
43. e.g. oligopoly, oligarchy, oligomer, oligonucleotide, oligopeptide, oligosaccharide
44. e.g. multilingual, multiple, pluripotent, pluricentric
45. e.g. polyhedra, polygamy, polypod, polyglot, polymath, polymer
46. e.g. plethora
Occurrences
• Numerical prefixes occur in 19th-, 20th-, and 21st-century coinages, mainly the terms that are used in relation to or that are the names of technological innovations, such as hexadecimal and bicycle. Also used in medals that commemorate an anniversary, such as sesquicentennial (150 years), centennial (100 years), or bicentennial (200 years).
• They occur in constructed words such as systematic names. Systematic names use numerical prefixes derived from Greek, with one principal exception, nona-.
• They occur as prefixes to units of measure in the SI system. See SI prefix.
• They occur as prefixes to units of computer data. See binary prefixes.
• They occur in words in the same languages as the original number word, and their respective derivatives. (Strictly speaking, some of the common citations of these occurrences are not in fact occurrences of the prefixes. For example: millennium is not formed from milli-, but is in fact derived from the same shared Latin root – mille.)
Because of the common inheritance of Greek and Latin roots across the Romance languages, the import of much of that derived vocabulary into non-Romance languages (such as into English via Norman French), and the borrowing of 19th and 20th century coinages into many languages, the same numerical prefixes occur in many languages.
Numerical prefixes are not restricted to denoting integers. Some of the SI prefixes denote negative powers of 10, i.e. division by a multiple of 10 rather than multiplication by it. Several common-use numerical prefixes denote vulgar fractions.
Words containing non-technical numerical prefixes are usually not hyphenated. This is not an absolute rule, however, and there are exceptions (for example: quarter-deck occurs in addition to quarterdeck). There are no exceptions for words comprising technical numerical prefixes, though. Systematic names and words comprising SI prefixes and binary prefixes are not hyphenated, by definition.
Nonetheless, for clarity, dictionaries list numerical prefixes in hyphenated form, to distinguish the prefixes from words with the same spellings (such as duo- and duo).
Several technical numerical prefixes are not derived from words for numbers. (mega- is not derived from a number word, for example.) Similarly, some are only derived from words for numbers inasmuch as they are word play. (Peta- is word play on penta-, for example. See its etymology for details.)
The root language of a numerical prefix need not be related to the root language of the word that it prefixes. Some words comprising numerical prefixes are hybrid words.
In certain classes of systematic names, there are a few other exceptions to the rule of using Greek-derived numerical prefixes. The IUPAC nomenclature of organic chemistry, for example, uses the numerical prefixes derived from Greek, except for the prefix for 9 (as mentioned) and the prefixes from 1 to 4 (meth-, eth-, prop-, and but-), which are not derived from words for numbers. These prefixes were invented by the IUPAC, deriving them from the pre-existing names for several compounds that it was intended to preserve in the new system: methane (via methyl, which is in turn from the Greek word for wine), ethane (from ethyl coined by Justus von Liebig in 1834), propane (from propionic, which is in turn from pro- and the Greek word for fat), and butane (from butyl, which is in turn from butyric, which is in turn from the Latin word for butter).
Cardinal Latin series
• unicycle, bicycle, tricycle, quadricycle
• uniped, biped, triped, quadruped,[upper-alpha 8] centipede, millipede
Distributive Latin series
• unary, binary, trinary, quaternary, quinary, senary, … vicenary … centenary …
• denarian, vicenarian, tricenarian, quadragenarian, quinquagenarian, sexagenarian, septuagenarian, octogenarian, nonagenarian, centenarian, … millenarian
Greek series
• monad, dyad, triad, tetrad, pentad, hexad, heptad, ogdoad, ennead, decad, ... triacontad, ... hecatontad, chiliad, myriad
• digon, trigon, tetragon, pentagon, hexagon, heptagon, octagon, enneagon, decagon, hendecagon, dodecagon, ... enneadecagon, icosagon, triacontagon, ... chiliagon, myriagon
• trilogy, tetralogy, pentalogy, hexalogy, heptalogy
• monopod, dipod, tripod, tetrapod, hexapod, octopod, decapod
Mixed language series
• pentane, hexane, heptane, octane, nonane,[upper-alpha 11] decane, undecane, ... icosane
• binary, ternary, quaternary, quinary, senary, septenary, octal, nonary, decimal, duodecimal, hexadecimal, vigesimal, quadrovigesimal, duotrigesimal, sexagesimal, octogesimal
See also
• IUPAC numerical multiplier
• List of numbers
• List of numeral systems
• List of commonly used taxonomic affixes
• Numerals in English and other languages
• Names for tuples of specific lengths
Notes
1. These months' prefixes originated with the early Roman 10-month calendar. See Roman Calendar.
2. See Mendeleev's predicted elements for the most common use of Sanskrit numerical prefixes.
3. The numbering adjectives in Greek are inflectional for grammatical gender (i.e. there is monos [masculine for single/alone], mone [feminine for single/alone] and monon [neuter for single/alone]), grammatical case (i.e. nominative, genitive, etc.) and grammatical number (singular/plural). The prefixes are produced from the default grammatical type (masculine/nominative/singular).
4. Demi- is French, from Latin dimidium.
5. sol (sōlus) is more appropriately a Latin root for ("only", "oneself").
6. sim- (sin-) is found in the words simplex, simple.
7. The Greek prefix for 'one' is normally mono- 'alone'. Hen- 'one' is only used in compound numbers (hendeka- 11) and a few words like henad (= monad). Haplo- 'single' is found is a few technical words such as haploid.
8. The forms related to quattuor "four", like the previous three integers, are irregular in Latin and other Indo-European languages, and the details, while presumably a form of assimilation, are unclear. Andrew Sihler, New comparative grammar of Greek and Latin, p. 412, and Carl Darling Buck, Comparative grammar of Greek and Latin. In particular, quadri- has the alternate form quadru- before p in some Latin words, such as quadruple.
9. Sometimes Greek hexa- is used in Latin compounds, such as hexadecimal, due to taboo avoidance with the English word sex.
10. For Latinate 21, 22, etc., the pattern for the teens is followed: unvigint-, duovigint-, etc. For higher numbers, the reverse order may be found: 36 is trigintisex-. For Greek, the word kai ("and") is used: icosikaihena-, icosikaidi-, pentacontakaipenta-, etc. In these and in the tens, the kai is frequently omitted, though not in triskaidekaphobia. (The inconsistency of triskaidekaphobia with the table above is explained by the fact that the Greek letter kappa can be transliterated either "c" or "k".) In chemical nomenclature, 11 is generally mixed Latin-Greek undeca-, and the 20s are based on -cos-, for example tricos- for 23.
11. In organic chemistry, most prefixes are Greek but the prefixes for 9 and 11 are Latin.
References
1. uncia. Charlton T. Lewis and Charles Short. A Latin Dictionary on Perseus Project.
2. πρῶτος. Liddell, Henry George; Scott, Robert; A Greek–English Lexicon at the Perseus Project
3. Monier-Williams, Monier (1899). "एक-". A Sanskrit-English Dictionary: Etymologically and Philologically Arranged with Special Reference to Cognate Indo-European Languages. Oxford: Clarendon Press. OCLC 685239912.
4. δύο
5. δίς
6. δεύτερος
7. Monier-Williams, Monier (1899). "द्वि-". A Sanskrit-English Dictionary: Etymologically and Philologically Arranged with Special Reference to Cognate Indo-European Languages. Oxford: Clarendon Press. OCLC 685239912.
8. τρίς
9. τρίτος
10. Monier-Williams, Monier (1899). "त्रि-". A Sanskrit-English Dictionary: Etymologically and Philologically Arranged with Special Reference to Cognate Indo-European Languages. Oxford: Clarendon Press. OCLC 685239912.
11. "quăter". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
12. "quăterni". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
13. "quartus". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
14. Monier-Williams, Monier (1899). "चतुर्-". A Sanskrit-English Dictionary: Etymologically and Philologically Arranged with Special Reference to Cognate Indo-European Languages. Oxford: Clarendon Press. OCLC 685239912.
15. "quinque". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
16. "quīni". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
17. "quintus". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
18. πέμπτος
19. Monier-Williams, Monier (1899). "पञ्च-". A Sanskrit-English Dictionary: Etymologically and Philologically Arranged with Special Reference to Cognate Indo-European Languages. Oxford: Clarendon Press. OCLC 685239912.
20. "sēni". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
21. "sextus". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
22. "Henry George Liddell, Robert Scott, A Greek-English Lexicon, ἕξ". www.perseus.tufts.edu. Archived from the original on 23 December 2021. Retrieved 2022-02-24.
23. ἕκτος
24. Monier-Williams, Monier (1899). "षट्-". A Sanskrit-English Dictionary: Etymologically and Philologically Arranged with Special Reference to Cognate Indo-European Languages. Oxford: Clarendon Press. OCLC 685239912.
25. "septēni". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
26. ἑπτά
27. Monier-Williams, Monier (1899). "सप्त-". A Sanskrit-English Dictionary: Etymologically and Philologically Arranged with Special Reference to Cognate Indo-European Languages. Oxford: Clarendon Press. OCLC 685239912.
28. "octōni". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
29. "octāvus". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
30. ἐννέα
31. ἔνατος
32. "dēni". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
33. "dĕcĭmus". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
34. δέκα
35. δέκατος
36. "undēni". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
37. "undĕcĭmus". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
38. ἕνδεκα
39. ἑνδέκατος
40. δώδεκα
41. δωδέκατος
42. "quindĕcim". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
43. "quindēni". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
44. "quindĕcĭmus". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
45. "sēdĕcim". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
46. "quinquāginta". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
47. "quinquāgēni". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
48. "quinquāgēsĭmus". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
49. "Charlton T. Lewis, Charles Short, A Latin Dictionary, S , septĭfārĭam , septŭāgēsimus". www.perseus.tufts.edu. Archived from the original on 26 July 2020. Retrieved 26 February 2019.
50. "quingenti". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
51. "quincenti". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
52. "quingēni". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
53. "quingentēsĭmus". Archived from the original on 2012-10-14. Retrieved 2011-05-14.
54. ἐννακόσιοι
55. ἐνακοσιοστός
56. χίλιοι
57. χιλιοστός
58. δισχίλιοι
59. τρισχίλιοι
60. τρισχιλιοστός
61. πεντακισχίλιοι
62. μυρίος
63. μυριαστός
64. ὀκτακισμύριοι
65. Wallard, Andrew. "Micrillion". BIPM. Retrieved January 13, 2010.
Bibliography
• Bauer-Ramazani, Christine (April 2008). "Prefixes—Amount, Relationship, Judgment, Other Prefixes". Archived from the original on 2021-04-30. Retrieved 2022-02-24.
• Buck, Carl Darling. Comparative Grammar of Greek and Latin.
• Chrisomalis, Stephen. "Numerical Adjectives, Greek and Latin Number Prefixes". The Phrontistery. Archived from the original on 2022-01-29. Retrieved 2022-02-24.
• Sihler, Andrew L. (1995). New Comparative Grammar of Greek and Latin. Oxford University Press. ISBN 0195083458.
• Oxford English Dictionary (2 ed.).
| Wikipedia |
Edge-matching puzzle
An edge-matching puzzle is a type of tiling puzzle involving tiling an area with (typically regular) polygons whose edges are distinguished with colours or patterns, in such a way that the edges of adjacent tiles match.
Edge-matching puzzles are known to be NP-complete, and capable of conversion to and from equivalent jigsaw puzzles and polyomino packing puzzle.[1]
The first edge-matching puzzles were patented in the U.S. by E. L. Thurston in 1892.[2] Current examples of commercial edge-matching puzzles include the Eternity II puzzle, Tantrix, Kadon Enterprises' range of edge-matching puzzles, and the Edge Match Puzzles iPhone app.
Notable variations
MacMahon Squares
MacMahon Squares is the name given to a recreational math puzzle suggested by British mathematician Percy MacMahon, who published a treatise on edge-colouring of a variety of shapes in 1921.[4] This particular puzzle uses 24 tiles consisting of all permutations of 3 colors for the edges of a square. The tiles must be arranged into a 6×4 rectangular area such that all edges match and, furthermore, only one color is used for the outside edge of the rectangle.[5]
This puzzle can be extended to tiles with permutations of 4 colors, arranged in 10×7.[6] In either case, the squares are a subset of the Wang tiles, reducing tiles that are similar under rotation. Solutions number well into the thousands.[7]
MacMahon Squares, along with variations on the idea, was commercialized as Multimatch.
TetraVex
TetraVex is a computer game that presents the player with a square grid and a collection of tiles, by default nine square tiles for a 3×3 grid. Each tile has four single-digit numbers, one on each edge. The objective of the game is to place the tiles into the grid in the proper position, completing this puzzle as quickly as possible. The tiles cannot be rotated, and two can be placed next to each other only if the numbers on adjacent edges match.[8][9]
TetraVex was inspired by "the problem of tiling the plane" as described by Donald Knuth on page 382 of Volume 1: Fundamental Algorithms, the first book in his series The Art of Computer Programming. It was named by Scott Ferguson, the development lead and an architect of the first version of Visual Basic, who wrote it for Windows Entertainment Pack 3.[10]
TetraVex is also available as an open source game in the GNOME Games collection.[11]
The possible number of TetraVex can be counted. On a $n\times {}n$ board there are $n(n-1)$ horizontal and vertical pairs that must match and $4n$ numbers along the edges that can be chosen arbitrarily. Hence there are $2n(n-1)+4n=2n(n+1)$ choices of 10 digits, i.e. $10^{2n(n+1)}$ possible boards.
Deciding if a TetraVex puzzle has a solution is in general NP-complete.[12] Its computational approach involves the Douglas-Rachford algorithm.[13][14]
Hexagons
Serpentiles are the hexagonal tiles used in various abstract strategy games such as Psyche-Paths, Kaliko, and Tantrix. Within each serpentile, the edges are paired, thus restricting the set of tiles in such a way that no edge color occurs an odd number of times within the hexagon.
Three dimensions
Mathematically, edge-matching puzzles are two-dimensional. A 3D edge-matching puzzle is such a puzzle that is not flat in Euclidean space, so involves tiling a three-dimensional area such as the surface of a regular polyhedron. As before, polygonal pieces have distinguished edges to require that the edges of adjacent pieces match.
3D edge-matching puzzles are not currently under direct U.S. patent protection, since the 1892 patent by E. L. Thurston has expired.[2] Current examples of commercial puzzles include the Dodek Duo, The Enigma, Mental Misery,[15] and Kadon Enterprises' range of three-dimensional edge-matching puzzles.[16]
Incorporation of edge matching
The Carcassonne board game employs edge matching to constrain where its square tiles may be placed. The original game has three types of edges: fields, roads and cities.
See also
• Domino tiling
• Wang dominoes
References
1. Erik D. Demaine, Martin L. Demaine. "Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity" (PDF). Retrieved 2007-08-12.
2. "Rob's puzzle page: Edge Matching". Archived from the original on 2007-10-22. Retrieved 2007-08-12.
3. Gardner, Martin (2009). Sphere Packing, Lewis Caroll and Reversi. Cambridge University Press.
4. MacMahon, Percy Alexander (1921). New mathematical pastimes. Gerstein - University of Toronto. Cambridge, University Press.
5. Steckles, Katie. Blackboard Bold: MacMahon Squares. Retrieved 10 March 2021.
6. Guy. Cube Root of 31. Wang Tiles. Retrieved 12 April 2021.
7. Wade Philpott (credited). Kadon Enterprises. Multimatch. Retrieved 12 April 2021.
8. Whittum, Christopher (2013). Energize Education Through Open Source. pg 32.
9. Gagné, Marcel (2006). Moving to Ubuntu Linux.
10. "The Birth of Visual Basic". Forestmoon.com. Retrieved 2010-05-11.
11. "License - README". gnome-games. gnome.org. 2011. Retrieved 2012-10-02.
12. Takenaga, Yasuhiko; Walsh, Toby (15 September 2006). "TetraVex is NP-complete". Information Processing Letters. Information Processing Letters, Volume 99, Issue 5, Pages 171–174. 99 (5): 171–174. doi:10.1016/j.ipl.2006.04.010. S2CID 7228681.
13. Bansal, Pulkit(2010). "Code for solving Tetravex using Douglas–Rachford algorithm". Retrieved 10 March 2021.
14. Linstrom, Scott B.; Sims, Brailey (2020). Survey: Sixty years of Douglas Rachford. Cambridge University Press.
15. "Rob's puzzle page: Pattern Puzzles". Retrieved 2009-06-22.
16. "Kadon Enterprises, More About Edgematching". Retrieved 2009-06-22.
External links
• Erich's Matching Puzzles Collection
• Color- and Edge-Matching Polygons by Peter Esser
• Rob's puzzle page by Rob Stegmann
• Edge matching squares
| Wikipedia |
Polyabolo
In recreational mathematics, a polyabolo (also known as a polytan) is a shape formed by gluing isosceles right triangles edge-to-edge, making a polyform with the isosceles right triangle as the base form. Polyaboloes were introduced by Martin Gardner in his June 1967 "Mathematical Games column" in Scientific American.[1]
Nomenclature
The name polyabolo is a back formation from the juggling object 'diabolo', although the shape formed by joining two triangles at just one vertex is not a proper polyabolo. By false analogy, treating the di- in diabolo as meaning "two", polyaboloes with from 1 to 10 cells are called respectively monaboloes, diaboloes, triaboloes, tetraboloes, pentaboloes, hexaboloes, heptaboloes, octaboloes, enneaboloes, and decaboloes. The name polytan is derived from Henri Picciotto's name tetratan and alludes to the ancient Chinese amusement of tangrams.
Combinatorial enumeration
There are two ways in which a square in a polyabolo can consist of two isosceles right triangles, but polyaboloes are considered equivalent if they have the same boundaries. The number of nonequivalent polyaboloes composed of 1, 2, 3, … triangles is 1, 3, 4, 14, 30, 107, 318, 1116, 3743, … (sequence A006074 in the OEIS).
Polyaboloes that are confined strictly to the plane and cannot be turned over may be termed one-sided. The number of one-sided polyaboloes composed of 1, 2, 3, … triangles is 1, 4, 6, 22, 56, 198, 624, 2182, 7448, … (sequence A151519 in the OEIS).
As for a polyomino, a polyabolo that can be neither turned over nor rotated may be termed fixed. A polyabolo with no symmetries (rotation or reflection) corresponds to 8 distinct fixed polyaboloes.
A non-simply connected polyabolo is one that has one or more holes in it. The smallest value of n for which an n-abolo is non-simply connected is 7.
Tiling rectangles with copies of a single polyabolo
In 1968, David A. Klarner defined the order of a polyomino. Similarly, the order of a polyabolo P can be defined as the minimum number of congruent copies of P that can be assembled (allowing translation, rotation, and reflection) to form a rectangle.
A polyabolo has order 1 if and only if it is itself a rectangle. Polyaboloes of order 2 are also easily recognisable. Solomon W. Golomb found polyaboloes, including a triabolo, of order 8.[2] Michael Reid found a heptabolo of order 6.[3] Higher orders are possible.
There are interesting tessellations of the Euclidean plane involving polyaboloes. One such is the tetrakis square tiling, a monohedral tessellation that fills the entire Euclidean plane with 45–45–90 triangles.
Tiling a common figure with various polyaboloes
The Compatibility Problem is to take two or more polyaboloes and find a figure that can be tiled with each. This problem has been studied far less than the Compatibility Problem for polyominoes. Systematic results first appeared in 2004 at Erich Friedman's website Math Magic.[4]
References
1. Gardner, Martin (June 1967). "The polyhex and the polyabolo, polygonal jigsaw puzzle pieces". Scientific American. 216 (6): 124–132.
2. Golomb, Solomon W. (1994). Polyominoes (2nd ed.). Princeton University Press. p. 101. ISBN 0-691-02444-8.
3. Goodman, Jacob E.; O'Rourke, Joseph, eds. (2004). Handbook of Discrete and Computational Geometry (2nd ed.). Chapman & Hall/CRC. p. 349. ISBN 1-58488-301-4.
4. Friedman, Erich. "Polypolyforms". Math Magic.
External links
Wikimedia Commons has media related to Polyabolo.
• Weisstein, Eric W. "Polyabolo". MathWorld.
• Weisstein, Eric W. "Triabolo". MathWorld.
Polyforms
Polyominoes
• Domino
• Tromino
• Tetromino
• Pentomino
• Hexomino
• Heptomino
• Octomino
• Nonomino
• Decomino
Higher dimensions
• Polyominoid
• Polycube
Others
• Polyabolo
• Polydrafter
• Polyhex
• Polyiamond
• Pseudo-polyomino
• Polystick
Games and puzzles
• Blokus
• Soma cube
• Snake cube
• Tangram
• Hexastix
• Tantrix
• Tetris
WikiProject Portal
| Wikipedia |
Tetracategory
In category theory, a tetracategory is a weakened definition of a 4-category.
See also
• Weak n-category
• infinity category
External links
• Notes on tetracategories by Todd Trimble.
Category theory
Key concepts
Key concepts
• Category
• Adjoint functors
• CCC
• Commutative diagram
• Concrete category
• End
• Exponential
• Functor
• Kan extension
• Morphism
• Natural transformation
• Universal property
Universal constructions
Limits
• Terminal objects
• Products
• Equalizers
• Kernels
• Pullbacks
• Inverse limit
Colimits
• Initial objects
• Coproducts
• Coequalizers
• Cokernels and quotients
• Pushout
• Direct limit
Algebraic categories
• Sets
• Relations
• Magmas
• Groups
• Abelian groups
• Rings (Fields)
• Modules (Vector spaces)
Constructions on categories
• Free category
• Functor category
• Kleisli category
• Opposite category
• Quotient category
• Product category
• Comma category
• Subcategory
Higher category theory
Key concepts
• Categorification
• Enriched category
• Higher-dimensional algebra
• Homotopy hypothesis
• Model category
• Simplex category
• String diagram
• Topos
n-categories
Weak n-categories
• Bicategory (pseudofunctor)
• Tricategory
• Tetracategory
• Kan complex
• ∞-groupoid
• ∞-topos
Strict n-categories
• 2-category (2-functor)
• 3-category
Categorified concepts
• 2-group
• 2-ring
• En-ring
• (Traced)(Symmetric) monoidal category
• n-group
• n-monoid
• Category
• Outline
• Glossary
| Wikipedia |
Astroid
In mathematics, an astroid is a particular type of roulette curve: a hypocycloid with four cusps. Specifically, it is the locus of a point on a circle as it rolls inside a fixed circle with four times the radius.[1] By double generation, it is also the locus of a point on a circle as it rolls inside a fixed circle with 4/3 times the radius. It can also be defined as the envelope of a line segment of fixed length that moves while keeping an end point on each of the axes. It is therefore the envelope of the moving bar in the Trammel of Archimedes.
Its modern name comes from the Greek word for "star". It was proposed, originally in the form of "Astrois", by Joseph Johann von Littrow in 1838.[2][3] The curve had a variety of names, including tetracuspid (still used), cubocycloid, and paracycle. It is nearly identical in form to the evolute of an ellipse.
Equations
If the radius of the fixed circle is a then the equation is given by[4]
$x^{2/3}+y^{2/3}=a^{2/3}.$
This implies that an astroid is also a superellipse.
Parametric equations are
${\begin{aligned}x=a\cos ^{3}t&={\frac {a}{4}}\left(3\cos \left(t\right)+\cos \left(3t\right)\right),\\[2ex]y=a\sin ^{3}t&={\frac {a}{4}}\left(3\sin \left(t\right)-\sin \left(3t\right)\right).\end{aligned}}$
The pedal equation with respect to the origin is
$r^{2}=a^{2}-3p^{2},$
the Whewell equation is
$s={3a \over 4}\cos 2\varphi ,$
and the Cesàro equation is
$R^{2}+4s^{2}={\frac {9a^{2}}{4}}.$
The polar equation is[5]
$r={\frac {a}{\left(\cos ^{2/3}\theta +\sin ^{2/3}\theta \right)^{3/2}}}.$
The astroid is a real locus of a plane algebraic curve of genus zero. It has the equation[6]
$\left(x^{2}+y^{2}-a^{2}\right)^{3}+27a^{2}x^{2}y^{2}=0.$
The astroid is, therefore, a real algebraic curve of degree six.
Derivation of the polynomial equation
The polynomial equation may be derived from Leibniz's equation by elementary algebra:
$x^{2/3}+y^{2/3}=a^{2/3}.$
Cube both sides:
${\begin{aligned}x^{6/3}+3x^{4/3}y^{2/3}+3x^{2/3}y^{4/3}+y^{6/3}&=a^{6/3}\\[1.5ex]x^{2}+3x^{2/3}y^{2/3}\left(x^{2/3}+y^{2/3}\right)+y^{2}&=a^{2}\\[1ex]x^{2}+y^{2}-a^{2}&=-3x^{2/3}y^{2/3}\left(x^{2/3}+y^{2/3}\right)\end{aligned}}$
Cube both sides again:
$\left(x^{2}+y^{2}-a^{2}\right)^{3}=-27x^{2}y^{2}\left(x^{2/3}+y^{2/3}\right)^{3}$
But since:
$x^{2/3}+y^{2/3}=a^{2/3}\,$
It follows that
$\left(x^{2/3}+y^{2/3}\right)^{3}=a^{2}.$
Therefore:
$\left(x^{2}+y^{2}-a^{2}\right)^{3}=-27x^{2}y^{2}a^{2}$
or
$\left(x^{2}+y^{2}-a^{2}\right)^{3}+27x^{2}y^{2}a^{2}=0.$
Metric properties
Area enclosed[7]
${\frac {3}{8}}\pi a^{2}$
Length of curve
$6a$
Volume of the surface of revolution of the enclose area about the x-axis.
${\frac {32}{105}}\pi a^{3}$
Area of surface of revolution about the x-axis
${\frac {12}{5}}\pi a^{2}$
Properties
The astroid has four cusp singularities in the real plane, the points on the star. It has two more complex cusp singularities at infinity, and four complex double points, for a total of ten singularities.
The dual curve to the astroid is the cruciform curve with equation $ x^{2}y^{2}=x^{2}+y^{2}.$ The evolute of an astroid is an astroid twice as large.
The astroid has only one tangent line in each oriented direction, making it an example of a hedgehog.[8]
See also
• Cardioid (epicycloid with one cusp)
• Nephroid (epicycloid with two cusps)
• Deltoid (hypocycloid with three cusps)
• Stoner–Wohlfarth astroid a use of this curve in magnetics.
• Spirograph
References
1. Yates
2. J. J. v. Littrow (1838). "§99. Die Astrois". Kurze Anleitung zur gesammten Mathematik. Wien. p. 299.
3. Loria, Gino (1902). Spezielle algebraische und transscendente ebene kurven. Theorie und Geschichte. Leipzig. pp. 224.{{cite book}}: CS1 maint: location missing publisher (link)
4. Yates, for section
5. Weisstein, Eric W. "Astroid". MathWorld.
6. A derivation of this equation is given on p. 3 of http://xahlee.info/SpecialPlaneCurves_dir/Astroid_dir/astroid.pdf
7. Yates, for section
8. Nishimura, Takashi; Sakemi, Yu (2011). "View from inside". Hokkaido Mathematical Journal. 40 (3): 361–373. doi:10.14492/hokmj/1319595861. MR 2883496.
• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 4–5, 34–35, 173–174. ISBN 0-486-60288-5.
• Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 10–11. ISBN 0-14-011813-6.
• R.C. Yates (1952). "Astroid". A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards. pp. 1 ff.
External links
Wikimedia Commons has media related to Astroid.
• "Astroid", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Astroid" at The MacTutor History of Mathematics archive
• "Astroid" at The Encyclopedia of Remarkable Mathematical Forms
• Article on 2dcurves.com
• Visual Dictionary Of Special Plane Curves, Xah Lee
• Bars of an Astroid by Sándor Kabai, The Wolfram Demonstrations Project.
| Wikipedia |
Tetrad (geometry puzzle)
In geometry, a tetrad is a set of four simply connected disjoint planar regions in the plane, each pair sharing a finite portion of common boundary. It was named by Michael R. W. Buckley in 1975 in the Journal of Recreational Mathematics. A further question was proposed that became a puzzle, whether the 4 regions could be congruent, with or without holes, other enclosed regions.[1]
Tetrad with one central region and 3 surrounding ones
Tetrad with a hole
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Fewest sides and vertices
The solutions with four congruent tiles include some with five sides.[2] However, their placement surrounds an uncovered hole in the plane. Among solutions without holes, the ones with the fewest possible sides are given by a hexagon identified by Scott Kim as a student at Stanford University.[1] It is not known whether five-sided solutions without holes are possible.[2]
Kim's solution has 16 vertices, while some of the pentagon solutions have as few as 11 vertices. It is not known whether fewer vertices are possible.[2]
Congruent polyform solutions
Gardner offered a number of polyform (polyomino, polyiamond, and polyhex) solutions, with no holes.[1]
• 11 squares
• 12 squares
• 10 triangles
• 22 triangles
• 26 triangles
• 4 hexagons
References
1. Martin Gardner, Penrose Tiles to Trapdoor ciphers, 1989, p.121-123
2. Further Questions about Tetrads by Walter Trump
External links
• Polyform Tetrads and Polyomino and Polynar Tetrads
• A Tetrad Puzzle 7 April 2020
• Application of IT in Mathematical Proofs and in Checking of Results of Pupils’ Research
• Tetrads and their Counting Juris ČERŅENOKS, Andrejs CIBULIS
| Wikipedia |
Tetradecagon
In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon.
Regular tetradecagon
A regular tetradecagon
TypeRegular polygon
Edges and vertices14
Schläfli symbol{14}, t{7}
Coxeter–Dynkin diagrams
Symmetry groupDihedral (D14), order 2×14
Internal angle (degrees)154+2/7°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf
Regular tetradecagon
A regular tetradecagon has Schläfli symbol {14} and can be constructed as a quasiregular truncated heptagon, t{7}, which alternates two types of edges.
The area of a regular tetradecagon of side length a is given by
$A={\frac {14}{4}}a^{2}\cot {\frac {\pi }{14}}\approx 15.3345a^{2}$
Construction
As 14 = 2 × 7, a regular tetradecagon cannot be constructed using a compass and straightedge.[1] However, it is constructible using neusis with use of the angle trisector,[2] or with a marked ruler,[3] as shown in the following two examples.
Symmetry
The regular tetradecagon has Dih14 symmetry, order 28. There are 3 subgroup dihedral symmetries: Dih7, Dih2, and Dih1, and 4 cyclic group symmetries: Z14, Z7, Z2, and Z1.
These 8 symmetries can be seen in 10 distinct symmetries on the tetradecagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order.[4] Full symmetry of the regular form is r28 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g14 subgroup has no degrees of freedom but can seen as directed edges.
The highest symmetry irregular tetradecagons are d14, an isogonal tetradecagon constructed by seven mirrors which can alternate long and short edges, and p14, an isotoxal tetradecagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular tetradecagon.
Dissection
14-cube projection
84 rhomb dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.[5] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular tetradecagon, m=7, and it can be divided into 21: 3 sets of 7 rhombs. This decomposition is based on a Petrie polygon projection of a 7-cube, with 21 of 672 faces. The list OEIS: A006245 defines the number of solutions as 24698, including up to 14-fold rotations and chiral forms in reflection.
Dissection into 21 rhombs
Numismatic use
The regular tetradecagon is used as the shape of some commemorative gold and silver Malaysian coins, the number of sides representing the 14 states of the Malaysian Federation.[6]
Related figures
A tetradecagram is a 14-sided star polygon, represented by symbol {14/n}. There are two regular star polygons: {14/3} and {14/5}, using the same vertices, but connecting every third or fifth points. There are also three compounds: {14/2} is reduced to 2{7} as two heptagons, while {14/4} and {14/6} are reduced to 2{7/2} and 2{7/3} as two different heptagrams, and finally {14/7} is reduced to seven digons.
A notable application of a fourteen-pointed star is in the flag of Malaysia, which incorporates a yellow {14/6} tetradecagram in the top-right corner, representing the unity of the thirteen states with the federal government.
Compounds and star polygons
n1234567
Form Regular Compound Star polygon Compound Star polygon Compound
Image
{14/1} = {14}
{14/2} = 2{7}
{14/3}
{14/4} = 2{7/2}
{14/5}
{14/6} = 2{7/3}
{14/7} or 7{2}
Internal angle ≈154.286° ≈128.571° ≈102.857° ≈77.1429° ≈51.4286° ≈25.7143° 0°
Deeper truncations of the regular heptagon and heptagrams can produce isogonal (vertex-transitive) intermediate tetradecagram forms with equally spaced vertices and two edge lengths. Other truncations can form double covering polygons 2{p/q}, namely: t{7/6}={14/6}=2{7/3}, t{7/4}={14/4}=2{7/2}, and t{7/2}={14/2}=2{7}.[7]
Isogonal truncations of heptagon and heptagrams
Quasiregular Isogonal Quasiregular
Double covering
t{7}={14}
{7/6}={14/6}
=2{7/3}
t{7/3}={14/3}
t{7/4}={14/4}
=2{7/2}
t{7/5}={14/5}
t{7/2}={14/2}
=2{7}
Isotoxal forms
An isotoxal polygon can be labeled as {pα} with outer most internal angle α, and a star polygon {(p/q)α}, with q is a winding number, and gcd(p,q)=1, q<p. Isotoxal tetradecagons have p=7, and since 7 is prime all solutions, q=1..6, are polygons.
{7α}
{(7/2)α}
{(7/3)α}
{(7/4)α}
{(7/5)α}
{(7/6)α}
Petrie polygons
Regular skew tetradecagons exist as Petrie polygon for many higher-dimensional polytopes, shown in these skew orthogonal projections, including:
Petrie polygons
B7 2I2(7) (4D)
7-orthoplex
7-cube
7-7 duopyramid
7-7 duoprism
A13 D8 E8
13-simplex
511
151
421
241
References
1. Wantzel, Pierre (1837). "Recherches sur les moyens de Reconnaître si un Problème de géométrie peau se résoudre avec la règle et le compas" (PDF). Journal de Mathématiques: 366–372.
2. Gleason, Andrew Mattei (March 1988). "Angle trisection, the heptagon, p. 186 (Fig.1) –187" (PDF). The American Mathematical Monthly. 95 (3): 185–194. doi:10.2307/2323624. Archived from the original (PDF) on 2016-02-02.
3. Weisstein, Eric W. "Heptagon." From MathWorld, A Wolfram Web Resource.
4. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
5. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
6. The Numismatist, Volume 96, Issues 7-12, Page 1409, American Numismatic Association, 1983.
7. The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum
External links
• Weisstein, Eric W. "Tetradecagon". MathWorld.
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| Wikipedia |
Tetradecahedron
A tetradecahedron is a polyhedron with 14 faces. There are numerous topologically distinct forms of a tetradecahedron, with many constructible entirely with regular polygon faces.
A tetradecahedron is sometimes called a tetrakaidecahedron.[1][2] No difference in meaning is ascribed.[3][4] The Greek word kai means 'and'. There is evidence that mammalian epidermal cells are shaped like flattened tetrakaidecahedra, an idea first suggested by Lord Kelvin.[5] The polyhedron can also be found in soap bubbles and in sintered ceramics, due to its ability to tesselate in 3D space.[6][7]
Convex
There are 1,496,225,352 topologically distinct convex tetradecahedra, excluding mirror images, having at least 9 vertices.[8] (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)
Examples
An incomplete list of forms includes:
Tetradecahedra having all regular polygonal faces (all exist in irregular-faced forms as well):
• Archimedean solids:
• Cuboctahedron (8 triangles, 6 squares)
• Truncated cube (8 triangles, 6 octagons)
• Truncated octahedron (6 squares, 8 hexagons)
• Prisms and antiprisms:
• Dodecagonal prism (12 squares, 2 dodecagons)
• Hexagonal antiprism (12 triangles, 2 hexagons)
• Johnson solids:
• J18: Elongated triangular cupola (4 triangles, 9 squares, 1 hexagon)
• J27: Triangular orthobicupola (8 triangles, 6 squares)
• J51: Triaugmented triangular prism (14 triangles)
• J55: Parabiaugmented hexagonal prism (8 triangles, 4 squares, 2 hexagons)
• J56: Metabiaugmented hexagonal prism (8 triangles, 4 squares, 2 hexagons)
• J65: Augmented truncated tetrahedron (8 triangles, 3 squares, 3 hexagons)
• J86: Sphenocorona (12 triangles, 2 squares)
• J91: Bilunabirotunda (8 triangles, 2 squares, 4 pentagons)
Tetradecahedra having at least one irregular face:
• Heptagonal bipyramid (14 triangles) (see Dipyramid)
• Heptagonal trapezohedron (14 kites) (see Trapezohedron)
• Tridecagonal pyramid (13 triangles, 1 regular tridecagon) (see Pyramid (geometry))
• Dissected regular icosahedron (the vertex figure of the grand antiprism) (12 equilateral triangles and 2 trapezoids)
• Hexagonal truncated trapezohedron: (12 pentagons, 2 hexagons)
Includes an optimal space-filling shape in foams (see Weaire–Phelan structure) and in the crystal structure of Clathrate hydrate (see illustration, next to label 51262)
• Hexagonal bifrustum (12 trapezoids, 2 hexagons)
• The British £1 coin in circulation from 2017 - with twelve edges and two faces - is an irregular dodecagonal prism, when one disregards the edging and relief features.[9]
See also
• Császár polyhedron – A nonconvex tetradecahedron of all triangle faces
• Steffen's polyhedron – A flexible tetradecahedron
• Permutohedron – A polyhedron that can be defined in any dimension and equals the truncated octahedron in three dimensions
References
1. Weisstein, Eric W. "Tetradecahedron". MathWorld. Retrieved 16 August 2023.
2. Tetradecahedron at the Wayback Machine (archived 18 July 2011)
3. Weisstein, Eric W. "Tetrakaidecahedron". MathWorld. Retrieved 16 August 2023.
4. Tetrakaidecahedron at the Wayback Machine (archived 28 September 2011(Date mismatch))
5. Yokouchi, Mariko; Atsugi, Toru; Logtestijn, Mark van; Tanaka, Reiko J.; Kajimura, Mayumi; Suematsu, Makoto; Furuse, Mikio; Amagai, Masayuki; Kubo, Akiharu (2016). "Epidermal cell turnover across tight junctions based on Kelvin's tetrakaidecahedron cell shape". eLife. 5. doi:10.7554/eLife.19593. PMC 5127639. PMID 27894419.
6. "MOST SPACE FILLING STRUCTURE IN THE WORLD! – TETRADECAHEDRON". Ardent Metallurgist. 2020-07-26. Retrieved 2022-11-15.
7. Wey, Ming-Yen; Tseng, Hui-Hsin; Chiang, Chian-kai (2014-03-01). "Improving the mechanical strength and gas separation performance of CMS membranes by simply sintering treatment of α-Al2O3 support". Journal of Membrane Science. 453: 603–613. doi:10.1016/j.memsci.2013.11.039. ISSN 0376-7388.
8. Counting polyhedra
9. "New Pound Coin | the Royal Mint".
• "What Are Polyhedra?" at the Wayback Machine (archived 12 February 2005), with Greek Numerical Prefixes
External links
• Weisstein, Eric W. "Tetradecahedron". MathWorld.
• Self-dual tetradecahedra
Polyhedra
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elemental things
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• vertex
• uniform polyhedron (two infinite groups and 75)
• regular polyhedron (9)
• quasiregular polyhedron (7)
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• Platonic solid (5)
• Archimedean solid (13)
• Catalan solid (13)
• Johnson solid (92)
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• Kepler–Poinsot polyhedron (4)
• Star polyhedron (infinite)
• Uniform star polyhedron (57)
prismatoids
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| Wikipedia |
Flexagon
In geometry, flexagons are flat models, usually constructed by folding strips of paper, that can be flexed or folded in certain ways to reveal faces besides the two that were originally on the back and front.
Flexagons are usually square or rectangular (tetraflexagons) or hexagonal (hexaflexagons). A prefix can be added to the name to indicate the number of faces that the model can display, including the two faces (back and front) that are visible before flexing. For example, a hexaflexagon with a total of six faces is called a hexahexaflexagon.
In hexaflexagon theory (that is, concerning flexagons with six sides), flexagons are usually defined in terms of pats.[1][2]
Two flexagons are equivalent if one can be transformed to the other by a series of pinches and rotations. Flexagon equivalence is an equivalence relation.[1]
History
Discovery and introduction
The discovery of the first flexagon, a trihexaflexagon, is credited to the British mathematician Arthur H. Stone, while a student at Princeton University in the United States in 1939. His new American paper would not fit in his English binder so he cut off the ends of the paper and began folding them into different shapes.[3] One of these formed a trihexaflexagon. Stone's colleagues Bryant Tuckerman, Richard Feynman, and John Tukey became interested in the idea and formed the Princeton Flexagon Committee. Tuckerman worked out a topological method, called the Tuckerman traverse, for revealing all the faces of a flexagon.[4] Tuckerman traverses are shown as a diagram.
Flexagons were introduced to the general public by Martin Gardner in the December 1956 issue of Scientific American in an article so well-received that it launched Gardner's "Mathematical Games" column which then ran in that magazine for the next twenty-five years.[3][5] In 1974, the magician Doug Henning included a construct-your-own hexaflexagon with the original cast recording of his Broadway show The Magic Show.
Attempted commercial development
In 1955, Russell Rogers and Leonard D'Andrea of Homestead Park, Pennsylvania applied for a patent, and in 1959 they were granted U.S. Patent number 2,883,195 for the hexahexaflexagon, under the title "Changeable Amusement Devices and the Like."
Their patent imagined possible applications of the device "as a toy, as an advertising display device, or as an educational geometric device."[6] A few such novelties were produced by the Herbick & Held Printing Company, the printing company in Pittsburgh where Rogers worked, but the device, marketed as the "Hexmo", failed to catch on.
Varieties
Tritetraflexagon
The tritetraflexagon is the simplest tetraflexagon (flexagon with square sides). The "tri" in the name means it has three faces, two of which are visible at any given time if the flexagon is pressed flat. The construction of the tritetraflexagon is similar to the mechanism used in the traditional Jacob's Ladder children's toy, in Rubik's Magic and in the magic wallet trick or the Himber wallet.
The tritetraflexagon has two dead ends, where you cannot flex forward. To get to another face you must either flex backwards or flip the flexagon over.
Hexatetraflexagon
A more complicated cyclic hexatetraflexagon requires no gluing. A cyclic hexatetraflexagon does not have any "dead ends", but the person making it can keep folding it until they reach the starting position. If the sides are colored in the process, the states can be seen more clearly.
Contrary to the tritetraflexagon, the hexatetraflexagon has no dead ends, and does not ever need to be flexed backwards.
Hexaflexagons
Hexaflexagons come in great variety, distinguished by the number of faces that can be achieved by flexing the assembled figure. (Note that the word hexaflexagons [with no prefixes] can sometimes refer to an ordinary hexahexaflexagon, with six sides instead of other numbers.)
Trihexaflexagon
A hexaflexagon with three faces is the simplest of the hexaflexagons to make and to manage, and is made from a single strip of paper, divided into nine equilateral triangles. (Some patterns provide ten triangles, two of which are glued together in the final assembly.)
To assemble, the strip is folded every third triangle, connecting back to itself after three inversions in the manner of the international recycling symbol. This makes a Möbius strip whose single edge forms a trefoil knot.
Hexahexaflexagon
This hexaflexagon has six faces. It is made up of nineteen triangles folded from a strip of paper.
Once folded, faces 1, 2, and 3 are easier to find than faces 4, 5, and 6.
An easy way to expose all six faces is using the Tuckerman traverse, named after Bryant Tuckerman, one of the first to investigate the properties of hexaflexagons. The Tuckerman traverse involves the repeated flexing by pinching one corner and flex from exactly the same corner every time. If the corner refuses to open, move to an adjacent corner and keep flexing. This procedure brings you to a 12-face cycle. During this procedure, however, 1, 2, and 3 show up three times as frequently as 4, 5, and 6. The cycle proceeds as follows:
1 → 3 → 6 → 1 → 3 → 2 → 4 → 3 → 2 → 1 → 5 → 2
And then back to 1 again.
Each color/face can also be exposed in more than one way. In figure 6, for example, each blue triangle has at the center its corner decorated with a wedge, but it is also possible, for example, to make the ones decorated with Y's come to the center. There are 18 such possible configurations for triangles with different colors, and they can be seen by flexing the hexahexaflexagon in all possible ways in theory, but only 15 can be flexed by the ordinary hexahexaflexagon. The 3 extra configurations are impossible due to the arrangement of the 4, 5, and 6 tiles at the back flap. (The 60-degree angles in the rhombi formed by the adjacent 4, 5, or 6 tiles will only appear on the sides and never will appear at the center because it would require one to cut the strip, which is topologically forbidden.)
Hexahexaflexagons can be constructed from different shaped nets of eighteen equilateral triangles. One hexahexaflexagon, constructed from an irregular paper strip, is almost identical to the one shown above, except that all 18 configurations can be flexed on this version.
Other hexaflexagons
While the most commonly seen hexaflexagons have either three or six faces, variations exist with any number of faces. Straight strips produce hexaflexagons with a multiple of three number of faces. Other numbers are obtained from nonstraight strips, that are just straight strips with some joints folded, eliminating some faces. Many strips can be folded in different ways, producing different hexaflexagons, with different folding maps.
Right octaflexagon and right dodecaflexagon
In these more recently discovered flexagons, each square or equilateral triangular face of a conventional flexagon is further divided into two right triangles, permitting additional flexing modes.[7] The division of the square faces of tetraflexagons into right isosceles triangles yields the octaflexagons,[8] and the division of the triangular faces of the hexaflexagons into 30-60-90 right triangles yields the dodecaflexagons.[9]
Pentaflexagon and right decaflexagon
In its flat state, the pentaflexagon looks much like the Chrysler logo: a regular pentagon divided from the center into five isosceles triangles, with angles 72-54-54. Because of its fivefold symmetry, the pentaflexagon cannot be folded in half. However, a complex series of flexes results in its transformation from displaying sides one and two on the front and back, to displaying its previously hidden sides three and four.[10]
By further dividing the 72-54-54 triangles of the pentaflexagon into 36-54-90 right triangles produces one variation of the 10-sided decaflexagon.[11]
Generalized isosceles n-flexagon
The pentaflexagon is one of an infinite sequence of flexagons based on dividing a regular n-gon into n isosceles triangles. Other flexagons include the heptaflexagon,[12] the isosceles octaflexagon,[13] the enneaflexagon,[14] and others.
Nonplanar pentaflexagon and nonplanar heptaflexagon
Harold V. McIntosh also describes "nonplanar" flexagons (i.e., ones which cannot be flexed so they lie flat); ones folded from pentagons called pentaflexagons,[15] and from heptagons called heptaflexagons.[16] These should be distinguished from the "ordinary" pentaflexagons and heptaflexagons described above, which are made out of isosceles triangles, and they can be made to lie flat.
In popular culture
Flexagons are also a popular book structure used by artist's book creators such as Julie Chen (Life Cycle) and Edward H. Hutchins (Album and Voces de México). Instructions for making tetra-tetra-flexagon and cross-flexagons are included in Making Handmade Books: 100+ Bindings, Structures and Forms by Alisa Golden.[17]
A high-order hexaflexagon was used as a plot element in Piers Anthony's novel 0X, in which a flex was analogous to the travel between alternate universes.[18]
See also
• Cayley tree
• Geometric group theory
• Kaleidocycle
References
1. Oakley, C. O.; Wisner, R. J. (March 1957). "Flexagons". The American Mathematical Monthly. Mathematical Association of America. 64 (3): 143–154. doi:10.2307/2310544. JSTOR 2310544.
2. Anderson, Thomas; McLean, T. Bruce; Pajoohesh, Homeira; Smith, Chasen (January 2010). "The combinatorics of all regular flexagons". European Journal of Combinatorics. 31 (1): 72–80. doi:10.1016/j.ejc.2009.01.005.
3. Gardner, Martin (December 1956). "Flexagons". Scientific American. Vol. 195, no. 6. pp. 162–168. doi:10.1038/scientificamerican1256-162. JSTOR 24941843. OCLC 4657622161.
4. Gardner, Martin (1988). Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Puzzles and Games. University of Chicago Press. ISBN 0-226-28254-6.
5. Mulcahy, Colm (October 21, 2014). "The Top 10 Martin Gardner Scientific American Articles". Scientific American.
6. Rogers, Russell E.; Andrea, Leonard D. L. (April 21, 1959). "Changeable amusement devices and the like" (PDF). Freepatentsonline.com. U.S. Patent 2883195. Retrieved January 13, 2011.
7. Schwartz, Ann (2005). "Flexagon Discovery: The Shape-Shifting 12-Gon". Eighthsquare.com. Retrieved October 26, 2012.
8. Sherman, Scott (2007). "Octaflexagon". Loki3.com. Retrieved October 26, 2012.
9. Sherman, Scott (2007). "Dodecaflexagon". Loki3.com. Retrieved October 26, 2012.
10. Sherman, Scott (2007). "Pentaflexagon". Loki3.com. Retrieved October 26, 2012.
11. Sherman, Scott (2007). "Decaflexagon". Loki3.com. Retrieved October 26, 2012.
12. Sherman, Scott (2007). "Heptaflexagon". Loki3.com. Retrieved October 26, 2012.
13. Sherman, Scott (2007). "Octaflexagon: Isosceles Octaflexagon". Loki3.com. Retrieved October 26, 2012.
14. Sherman, Scott (2007). "Enneaflexagon: Isosceles Enneaflexagon". Loki3.com. Retrieved October 26, 2012.
15. McIntosh, Harold V. (August 24, 2000). "Pentagonal Flexagons". Cinvestav.mx. Universidad Autónoma de Puebla. Retrieved October 26, 2012.
16. McIntosh, Harold V. (March 11, 2000). "Heptagonal Flexagons". Cinvestav.mx. Universidad Autónoma de Puebla. Retrieved October 26, 2012.
17. Golden, Alisa J. (2011). Making Handmade Books: 100+ Bindings, Structures & Forms. Lark Crafts. pp. 130, 132–133. ISBN 978-1-60059-587-5.
18. Collings, Michael R. (1984). Piers Anthony. Starmont Reader's Guide #20. Borgo Press. pp. 47–48. ISBN 0-89370-058-4.
Bibliography
• Martin Gardner wrote an excellent introduction to hexaflexagons in the December 1956 Mathematical Games column in Scientific American. It also appears in:
• The "Scientific American" Book of Mathematical Puzzles and Diversions. Simon & Schuster. 1959.
• Hexaflexagons and Other Mathematical Diversions: The First "Scientific American" Book of Puzzles and Games. University of Chicago Press. 1988. ISBN 0-226-28254-6.
• The Colossal Book of Mathematics. W. W. Norton & Co. 2001. ISBN 0-393-02023-1.
• Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi: Martin Gardner's First Book of Mathematical Puzzles and Games. Cambridge University Press. 2008. ISBN 978-0-521-73525-4.
• Gardner, Martin (January 2012). "Hexaflexagons". The College Mathematics Journal. 43 (1): 2–5. doi:10.4169/college.math.j.43.1.002. JSTOR 10.4169/college.math.j.43.1.002. S2CID 218544330. The issue also contains another article by Pook, and one by Iacob, McLean, and Hua.
• Jones, Madeline (1966). The Mysterious Flexagons: An Introduction to a Fascinating New Concept in Paper Folding. Crown Publishers.
• Mitchell, David (2000). The Magic of Flexagons – Paper curiosities to cut out and make. Tarquin. ISBN 1-899618-28-7.
• Pook, Les (2006). Flexagons Inside Out. Cambridge University Press. ISBN 0-521-81970-9.
• Pook, Les (2009). Serious Fun with Flexagons, A Compendium and Guide. Springer. ISBN 978-90-481-2502-9.
External links
Wikimedia Commons has media related to Flexagons.
• My Flexagon Experiences by Harold V. McIntosh – contains historical information and theory
• The Flexagon Portal – Robin Moseley's site has patterns for a large variety of flexagons.
• Flexagons
• Flexagons – Scott Sherman's site, with variety of flexagons of different shapes.
• MathWorld's page on tetraflexagons, including three nets
• Flexagons – 1962 paper by Antony S. Conrad and Daniel K. Hartline (RIAS)
• MathWorld entry on Hexaflexagons
• Yutaka Nishiyama (2010). "General Solution for Multiple Foldings of Hexaflexagons" IJPAM, Vol. 58, No. 1, 113-124. "19 faces of Flexagons"
• Vi Hart's video on Hexaflexagons part 1 part 2
Mathematics of paper folding
Flat folding
• Big-little-big lemma
• Crease pattern
• Huzita–Hatori axioms
• Kawasaki's theorem
• Maekawa's theorem
• Map folding
• Napkin folding problem
• Pureland origami
• Yoshizawa–Randlett system
Strip folding
• Dragon curve
• Flexagon
• Möbius strip
• Regular paperfolding sequence
3d structures
• Miura fold
• Modular origami
• Paper bag problem
• Rigid origami
• Schwarz lantern
• Sonobe
• Yoshimura buckling
Polyhedra
• Alexandrov's uniqueness theorem
• Blooming
• Flexible polyhedron (Bricard octahedron, Steffen's polyhedron)
• Net
• Source unfolding
• Star unfolding
Miscellaneous
• Fold-and-cut theorem
• Lill's method
Publications
• Geometric Exercises in Paper Folding
• Geometric Folding Algorithms
• Geometric Origami
• A History of Folding in Mathematics
• Origami Polyhedra Design
• Origamics
People
• Roger C. Alperin
• Margherita Piazzola Beloch
• Robert Connelly
• Erik Demaine
• Martin Demaine
• Rona Gurkewitz
• David A. Huffman
• Tom Hull
• Kôdi Husimi
• Humiaki Huzita
• Toshikazu Kawasaki
• Robert J. Lang
• Anna Lubiw
• Jun Maekawa
• Kōryō Miura
• Joseph O'Rourke
• Tomohiro Tachi
• Eve Torrence
| Wikipedia |
Deltoidal hexecontahedron
In geometry, a deltoidal hexecontahedron (also sometimes called a trapezoidal hexecontahedron, a strombic hexecontahedron, or a tetragonal hexacontahedron[1]) is a Catalan solid which is the dual polyhedron of the rhombicosidodecahedron, an Archimedean solid. It is one of six Catalan solids to not have a Hamiltonian path among its vertices.[2]
Deltoidal hexecontahedron
(Click here for rotating model)
TypeCatalan
Conway notationoD or deD
Coxeter diagram
Face polygon
kite
Faces60
Edges120
Vertices62 = 12 + 20 + 30
Face configurationV3.4.5.4
Symmetry groupIh, H3, [5,3], (*532)
Rotation groupI, [5,3]+, (532)
Dihedral angle154° 7′ 17′′ arccos(-19-8√5/41)
Propertiesconvex, face-transitive
rhombicosidodecahedron
(dual polyhedron)
Net
It is topologically identical to the nonconvex rhombic hexecontahedron.
Lengths and angles
The 60 faces are deltoids or kites. The short and long edges of each kite are in the ratio 1:7 + √5/6 ≈ 1:1.539344663...
The angle between two short edges in a single face is arccos(-5-2√5/20)≈118.2686774705°. The opposite angle, between long edges, is arccos(-5+9√5/40)≈67.783011547435° . The other two angles of each face, between a short and a long edge each, are both equal to arccos(5-2√5/10)≈86.97415549104°.
The dihedral angle between any pair of adjacent faces is arccos(-19-8√5/41)≈154.12136312578°.
Topology
Topologically, the deltoidal hexecontahedron is identical to the nonconvex rhombic hexecontahedron. The deltoidal hexecontahedron can be derived from a dodecahedron (or icosahedron) by pushing the face centers, edge centers and vertices out to different radii from the body center. The radii are chosen so that the resulting shape has planar kite faces each such that vertices go to degree-3 corners, faces to degree-five corners, and edge centers to degree-four points.
Cartesian coordinates
The 62 vertices of the disdyakis triacontahedron fall in three sets centered on the origin:
• Twelve vertices are of the form of a unit circumradius regular icosahedron.
• Twenty vertices are of the form of a ${\frac {3}{11}}{\sqrt {15-{\frac {6}{\sqrt {5}}}}}\approx 0.9571$ scaled regular dodecahedron.
• Thirty vertices are of the form of a $3{\sqrt {1-{\frac {2}{\sqrt {5}}}}}\approx 0.9748$ scaled Icosidodecahedron.
These hulls are visualized in the figure below:
Orthogonal projections
The deltoidal hexecontahedron has 3 symmetry positions located on the 3 types of vertices:
Orthogonal projections
Projective
symmetry
[2] [2] [2] [2] [6] [10]
Image
Dual
image
Variations
The deltoidal hexecontahedron can be constructed from either the regular icosahedron or regular dodecahedron by adding vertices mid-edge, and mid-face, and creating new edges from each edge center to the face centers. Conway polyhedron notation would give these as oI, and oD, ortho-icosahedron, and ortho-dodecahedron. These geometric variations exist as a continuum along one degree of freedom.
Related polyhedra and tilings
Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
{5,3} t{5,3} r{5,3} t{3,5} {3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
V5.5.5 V3.10.10 V3.5.3.5 V5.6.6 V3.3.3.3.3 V3.4.5.4 V4.6.10 V3.3.3.3.5
When projected onto a sphere (see right), it can be seen that the edges make up the edges of an icosahedron and dodecahedron arranged in their dual positions.
This tiling is topologically related as a part of sequence of deltoidal polyhedra with face figure (V3.4.n.4), and continues as tilings of the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.
*n32 symmetry mutation of dual expanded tilings: V3.4.n.4
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco.
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Figure
Config.
V3.4.2.4
V3.4.3.4
V3.4.4.4
V3.4.5.4
V3.4.6.4
V3.4.7.4
V3.4.8.4
V3.4.∞.4
See also
• Deltoidal icositetrahedron
References
1. Conway, Symmetries of things, p.284-286
2. "Archimedean Dual Graph".
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal hexecontahedron)
• http://mathworld.wolfram.com/ArchimedeanDualGraph.html
External links
• Eric W. Weisstein, DeltoidalHexecontahedron and Hamiltonian path (Catalan solid) at MathWorld.
• Deltoidal Hexecontahedron (Trapezoidal Hexecontrahedron)—Interactive Polyhedron Model
• Example in real life—A ball almost 4 meters in diameter, from ripstop nylon, and inflated by the wind. It bounces around on the ground so that kids can play with it at kite festivals.
Catalan solids
Tetrahedron
(Dual)
Tetrahedron
(Seed)
Octahedron
(Dual)
Cube
(Seed)
Icosahedron
(Dual)
Dodecahedron
(Seed)
Triakis tetrahedron
(Needle)
Triakis tetrahedron
(Kis)
Triakis octahedron
(Needle)
Tetrakis hexahedron
(Kis)
Triakis icosahedron
(Needle)
Pentakis dodecahedron
(Kis)
Rhombic hexahedron
(Join)
Rhombic dodecahedron
(Join)
Rhombic triacontahedron
(Join)
Deltoidal dodecahedron
(Ortho)
Disdyakis hexahedron
(Meta)
Deltoidal icositetrahedron
(Ortho)
Disdyakis dodecahedron
(Meta)
Deltoidal hexecontahedron
(Ortho)
Disdyakis triacontahedron
(Meta)
Pentagonal dodecahedron
(Gyro)
Pentagonal icositetrahedron
(Gyro)
Pentagonal hexecontahedron
(Gyro)
Archimedean duals
Tetrahedron
(Seed)
Tetrahedron
(Dual)
Cube
(Seed)
Octahedron
(Dual)
Dodecahedron
(Seed)
Icosahedron
(Dual)
Truncated tetrahedron
(Truncate)
Truncated tetrahedron
(Zip)
Truncated cube
(Truncate)
Truncated octahedron
(Zip)
Truncated dodecahedron
(Truncate)
Truncated icosahedron
(Zip)
Tetratetrahedron
(Ambo)
Cuboctahedron
(Ambo)
Icosidodecahedron
(Ambo)
Rhombitetratetrahedron
(Expand)
Truncated tetratetrahedron
(Bevel)
Rhombicuboctahedron
(Expand)
Truncated cuboctahedron
(Bevel)
Rhombicosidodecahedron
(Expand)
Truncated icosidodecahedron
(Bevel)
Snub tetrahedron
(Snub)
Snub cube
(Snub)
Snub dodecahedron
(Snub)
Convex polyhedra
Platonic solids (regular)
• tetrahedron
• cube
• octahedron
• dodecahedron
• icosahedron
Archimedean solids
(semiregular or uniform)
• truncated tetrahedron
• cuboctahedron
• truncated cube
• truncated octahedron
• rhombicuboctahedron
• truncated cuboctahedron
• snub cube
• icosidodecahedron
• truncated dodecahedron
• truncated icosahedron
• rhombicosidodecahedron
• truncated icosidodecahedron
• snub dodecahedron
Catalan solids
(duals of Archimedean)
• triakis tetrahedron
• rhombic dodecahedron
• triakis octahedron
• tetrakis hexahedron
• deltoidal icositetrahedron
• disdyakis dodecahedron
• pentagonal icositetrahedron
• rhombic triacontahedron
• triakis icosahedron
• pentakis dodecahedron
• deltoidal hexecontahedron
• disdyakis triacontahedron
• pentagonal hexecontahedron
Dihedral regular
• dihedron
• hosohedron
Dihedral uniform
• prisms
• antiprisms
duals:
• bipyramids
• trapezohedra
Dihedral others
• pyramids
• truncated trapezohedra
• gyroelongated bipyramid
• cupola
• bicupola
• frustum
• bifrustum
• rotunda
• birotunda
• prismatoid
• scutoid
Degenerate polyhedra are in italics.
| Wikipedia |
Tetragonal trapezohedron
In geometry, a tetragonal trapezohedron, or deltohedron, is the second in an infinite series of trapezohedra, which are dual to the antiprisms. It has eight faces, which are congruent kites, and is dual to the square antiprism.
Tetragonal trapezohedron
Click on picture for large version.
Typetrapezohedra
ConwaydA4
Coxeter diagram
Faces8 kites
Edges16
Vertices10
Face configurationV4.3.3.3
Symmetry groupD4d, [2+,8], (2*4), order 16
Rotation groupD4, [2,4]+, (224), order 8
Dual polyhedronSquare antiprism
Propertiesconvex, face-transitive
In mesh generation
This shape has been used as a test case for hexahedral mesh generation,[1][2][3][4][5] simplifying an earlier test case posited by mathematician Robert Schneiders in the form of a square pyramid with its boundary subdivided into 16 quadrilaterals. In this context the tetragonal trapezohedron has also been called the cubical octahedron,[3] quadrilateral octahedron,[4] or octagonal spindle,[5] because it has eight quadrilateral faces and is uniquely defined as a combinatorial polyhedron by that property.[3] Adding four cuboids to a mesh for the cubical octahedron would also give a mesh for Schneiders' pyramid.[2] As a simply-connected polyhedron with an even number of quadrilateral faces, the cubical octahedron can be decomposed into topological cuboids with curved faces that meet face-to-face without subdividing the boundary quadrilaterals,[1][5][6] and an explicit mesh of this type has been constructed.[4] However, it is unclear whether a decomposition of this type can be obtained in which all the cuboids are convex polyhedra with flat faces.[1][5]
In art
A tetragonal trapezohedron appears in the upper left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving Stars.
Spherical tiling
The tetragonal trapezohedron also exists as a spherical tiling, with 2 vertices on the poles, and alternating vertices equally spaced above and below the equator.
Related polyhedra
Family of n-gonal trapezohedra
Trapezohedron name Digonal trapezohedron
(Tetrahedron)
Trigonal trapezohedron Tetragonal trapezohedron Pentagonal trapezohedron Hexagonal trapezohedron Heptagonal trapezohedron Octagonal trapezohedron Decagonal trapezohedron Dodecagonal trapezohedron ... Apeirogonal trapezohedron
Polyhedron image ...
Spherical tiling image Plane tiling image
Face configuration V2.3.3.3 V3.3.3.3 V4.3.3.3 V5.3.3.3 V6.3.3.3 V7.3.3.3 V8.3.3.3 V10.3.3.3 V12.3.3.3 ... V∞.3.3.3
The tetragonal trapezohedron is first in a series of dual snub polyhedra and tilings with face configuration V3.3.4.3.n.
4n2 symmetry mutations of snub tilings: 3.3.4.3.n
Symmetry
4n2
Spherical Euclidean Compact hyperbolic Paracomp.
242 342 442 542 642 742 842 ∞42
Snub
figures
Config. 3.3.4.3.2 3.3.4.3.3 3.3.4.3.4 3.3.4.3.5 3.3.4.3.6 3.3.4.3.7 3.3.4.3.8 3.3.4.3.∞
Gyro
figures
Config. V3.3.4.3.2 V3.3.4.3.3 V3.3.4.3.4 V3.3.4.3.5 V3.3.4.3.6 V3.3.4.3.7 V3.3.4.3.8 V3.3.4.3.∞
References
1. Eppstein, David (1996), "Linear complexity hexahedral mesh generation", Proceedings of the Twelfth Annual Symposium on Computational Geometry (SCG '96), New York, NY, USA: ACM, pp. 58–67, arXiv:cs/9809109, doi:10.1145/237218.237237, MR 1677595, S2CID 3266195.
2. Mitchell, S. A. (1999), "The all-hex geode-template for conforming a diced tetrahedral mesh to any diced hexahedral mesh", Engineering with Computers, 15 (3): 228–235, doi:10.1007/s003660050018, S2CID 3236051.
3. Schwartz, Alexander; Ziegler, Günter M. (2004), "Construction techniques for cubical complexes, odd cubical 4-polytopes, and prescribed dual manifolds", Experimental Mathematics, 13 (4): 385–413, doi:10.1080/10586458.2004.10504548, MR 2118264, S2CID 1741871.
4. Carbonera, Carlos D.; Shepherd, Jason F.; Shepherd, Jason F. (2006), "A constructive approach to constrained hexahedral mesh generation", Proceedings of the 15th International Meshing Roundtable, Berlin: Springer, pp. 435–452, doi:10.1007/978-3-540-34958-7_25.
5. Erickson, Jeff (2013), "Efficiently hex-meshing things with topology", Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry (SoCG '13) (PDF), New York, NY, USA: ACM, pp. 37–46, doi:10.1145/2462356.2462403, S2CID 10861924.
6. Mitchell, Scott A. (1996), "A characterization of the quadrilateral meshes of a surface which admit a compatible hexahedral mesh of the enclosed volume", STACS 96: 13th Annual Symposium on Theoretical Aspects of Computer Science Grenoble, France, February 22–24, 1996, Proceedings, Lecture Notes in Computer Science, vol. 1046, Berlin: Springer, pp. 465–476, doi:10.1007/3-540-60922-9_38, MR 1462118.
External links
• Paper model tetragonal (square) trapezohedron
• Weisstein, Eric W. "Trapezohedron". MathWorld.
Convex polyhedra
Platonic solids (regular)
• tetrahedron
• cube
• octahedron
• dodecahedron
• icosahedron
Archimedean solids
(semiregular or uniform)
• truncated tetrahedron
• cuboctahedron
• truncated cube
• truncated octahedron
• rhombicuboctahedron
• truncated cuboctahedron
• snub cube
• icosidodecahedron
• truncated dodecahedron
• truncated icosahedron
• rhombicosidodecahedron
• truncated icosidodecahedron
• snub dodecahedron
Catalan solids
(duals of Archimedean)
• triakis tetrahedron
• rhombic dodecahedron
• triakis octahedron
• tetrakis hexahedron
• deltoidal icositetrahedron
• disdyakis dodecahedron
• pentagonal icositetrahedron
• rhombic triacontahedron
• triakis icosahedron
• pentakis dodecahedron
• deltoidal hexecontahedron
• disdyakis triacontahedron
• pentagonal hexecontahedron
Dihedral regular
• dihedron
• hosohedron
Dihedral uniform
• prisms
• antiprisms
duals:
• bipyramids
• trapezohedra
Dihedral others
• pyramids
• truncated trapezohedra
• gyroelongated bipyramid
• cupola
• bicupola
• frustum
• bifrustum
• rotunda
• birotunda
• prismatoid
• scutoid
Degenerate polyhedra are in italics.
| Wikipedia |
Tetrahedral-square tiling honeycomb
In the geometry of hyperbolic 3-space, the tetrahedral-square tiling honeycomb is a paracompact uniform honeycomb, constructed from tetrahedron, cuboctahedron and square tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
Tetrahedral-square tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbol{(4,4,3,3)} or {(3,3,4,4)}
Coxeter diagrams
Cells{3,3}
{4,4}
r{4,3}
Facestriangle {3}
square {4}
Vertex figure
Rhombicuboctahedron
Coxeter group[(4,4,3,3)]
PropertiesVertex-transitive, edge-transitive
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
Cyclotruncated tetrahedral-square tiling honeycomb
Cyclotruncated tetrahedral-square tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt0,1{(4,4,3,3)}
Coxeter diagrams
Cells {4,3}
t{4,3}
{3,3}
t{4,3}
Facestriangle {3}
square {4}
octagon {8}
Vertex figure
Triangular antiprism
Coxeter group[(4,4,3,3)]
PropertiesVertex-transitive
The cyclotruncated tetrahedral-square tiling honeycomb is a paracompact uniform honeycomb, constructed from tetrahedron, cube, truncated cube and truncated square tiling cells, in a triangular antiprism vertex figure. It has a Coxeter diagram, .
See also
• Convex uniform honeycombs in hyperbolic space
• List of regular polytopes
References
• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
• Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapter 16-17: Geometries on Three-manifolds I,II)
• Norman Johnson Uniform Polytopes, Manuscript
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• N.W. Johnson: Geometries and Transformations, (2018) Chapter 13: Hyperbolic Coxeter groups
| Wikipedia |
Complete graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction).[1]
Complete graph
K7, a complete graph with 7 vertices
Verticesn
Edges$\textstyle {\frac {n(n-1)}{2}}$
Radius$\left\{{\begin{array}{ll}0&n\leq 1\\1&{\text{otherwise}}\end{array}}\right.$
Diameter$\left\{{\begin{array}{ll}0&n\leq 1\\1&{\text{otherwise}}\end{array}}\right.$
Girth$\left\{{\begin{array}{ll}\infty &n\leq 2\\3&{\text{otherwise}}\end{array}}\right.$
Automorphismsn! (Sn)
Chromatic numbern
Chromatic index
• n if n is odd
• n − 1 if n is even
Spectrum$\left\{{\begin{array}{lll}\emptyset &n=0\\\left\{0^{1}\right\}&n=1\\\left\{(n-1)^{1},-1^{n-1}\right\}&{\text{otherwise}}\end{array}}\right.$
Properties
• (n − 1)-regular
• Symmetric graph
• Vertex-transitive
• Edge-transitive
• Strongly regular
• Integral
NotationKn
Table of graphs and parameters
Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century, in the work of Ramon Llull.[2] Such a drawing is sometimes referred to as a mystic rose.[3]
Properties
The complete graph on n vertices is denoted by Kn. Some sources claim that the letter K in this notation stands for the German word komplett,[4] but the German name for a complete graph, vollständiger Graph, does not contain the letter K, and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory.[5]
Kn has n(n – 1)/2 edges (a triangular number), and is a regular graph of degree n – 1. All complete graphs are their own maximal cliques. They are maximally connected as the only vertex cut which disconnects the graph is the complete set of vertices. The complement graph of a complete graph is an empty graph.
If the edges of a complete graph are each given an orientation, the resulting directed graph is called a tournament.
Kn can be decomposed into n trees Ti such that Ti has i vertices.[6] Ringel's conjecture asks if the complete graph K2n+1 can be decomposed into copies of any tree with n edges.[7] This is known to be true for sufficiently large n.[8][9]
The number of all distinct paths between a specific pair of vertices in Kn+2 is given[10] by
$w_{n+2}=n!e_{n}=\lfloor en!\rfloor ,$
where e refers to Euler's constant, and
$e_{n}=\sum _{k=0}^{n}{\frac {1}{k!}}.$
The number of matchings of the complete graphs are given by the telephone numbers
1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152, 568504, 2390480, 10349536, 46206736, ... (sequence A000085 in the OEIS).
These numbers give the largest possible value of the Hosoya index for an n-vertex graph.[11] The number of perfect matchings of the complete graph Kn (with n even) is given by the double factorial (n – 1)!!.[12]
The crossing numbers up to K27 are known, with K28 requiring either 7233 or 7234 crossings. Further values are collected by the Rectilinear Crossing Number project.[13] Rectilinear Crossing numbers for Kn are
0, 0, 0, 0, 1, 3, 9, 19, 36, 62, 102, 153, 229, 324, 447, 603, 798, 1029, 1318, 1657, 2055, 2528, 3077, 3699, 4430, 5250, 6180, ... (sequence A014540 in the OEIS).
Geometry and topology
A complete graph with n nodes represents the edges of an (n – 1)-simplex. Geometrically K3 forms the edge set of a triangle, K4 a tetrahedron, etc. The Császár polyhedron, a nonconvex polyhedron with the topology of a torus, has the complete graph K7 as its skeleton.[15] Every neighborly polytope in four or more dimensions also has a complete skeleton.
K1 through K4 are all planar graphs. However, every planar drawing of a complete graph with five or more vertices must contain a crossing, and the nonplanar complete graph K5 plays a key role in the characterizations of planar graphs: by Kuratowski's theorem, a graph is planar if and only if it contains neither K5 nor the complete bipartite graph K3,3 as a subdivision, and by Wagner's theorem the same result holds for graph minors in place of subdivisions. As part of the Petersen family, K6 plays a similar role as one of the forbidden minors for linkless embedding.[16] In other words, and as Conway and Gordon[17] proved, every embedding of K6 into three-dimensional space is intrinsically linked, with at least one pair of linked triangles. Conway and Gordon also showed that any three-dimensional embedding of K7 contains a Hamiltonian cycle that is embedded in space as a nontrivial knot.
Examples
Complete graphs on $n$ vertices, for $n$ between 1 and 12, are shown below along with the numbers of edges:
K1: 0 K2: 1 K3: 3 K4: 6
K5: 10 K6: 15 K7: 21 K8: 28
K9: 36 K10: 45 K11: 55 K12: 66
See also
• Fully connected network, in computer networking
• Complete bipartite graph (or biclique), a special bipartite graph where every vertex on one side of the bipartition is connected to every vertex on the other side
References
1. Bang-Jensen, Jørgen; Gutin, Gregory (2018), "Basic Terminology, Notation and Results", in Bang-Jensen, Jørgen; Gutin, Gregory (eds.), Classes of Directed Graphs, Springer Monographs in Mathematics, Springer International Publishing, pp. 1–34, doi:10.1007/978-3-319-71840-8_1; see p. 17
2. Knuth, Donald E. (2013), "Two thousand years of combinatorics", in Wilson, Robin; Watkins, John J. (eds.), Combinatorics: Ancient and Modern, Oxford University Press, pp. 7–37, ISBN 978-0191630620.
3. Mystic Rose, nrich.maths.org, retrieved 23 January 2012.
4. Gries, David; Schneider, Fred B. (1993), A Logical Approach to Discrete Math, Springer-Verlag, p. 436, ISBN 0387941150.
5. Pirnot, Thomas L. (2000), Mathematics All Around, Addison Wesley, p. 154, ISBN 9780201308150.
6. Joos, Felix; Kim, Jaehoon; Kühn, Daniela; Osthus, Deryk (2019-08-05). "Optimal packings of bounded degree trees" (PDF). Journal of the European Mathematical Society. 21 (12): 3573–3647. doi:10.4171/JEMS/909. ISSN 1435-9855. S2CID 119315954. Archived (PDF) from the original on 2020-03-09. Retrieved 2020-03-09.
7. Ringel, G. (1963). Theory of Graphs and its Applications. Proceedings of the Symposium Smolenice.
8. Montgomery, Richard; Pokrovskiy, Alexey; Sudakov, Benny (2021). "A proof of Ringel's Conjecture". Geometric and Functional Analysis. 31 (3): 663–720. arXiv:2001.02665. doi:10.1007/s00039-021-00576-2.
9. Hartnett, Kevin. "Rainbow Proof Shows Graphs Have Uniform Parts". Quanta Magazine. Archived from the original on 2020-02-20. Retrieved 2020-02-20.
10. Hassani, M. "Cycles in graphs and derangements." Math. Gaz. 88, 123–126, 2004.
11. Tichy, Robert F.; Wagner, Stephan (2005), "Extremal problems for topological indices in combinatorial chemistry" (PDF), Journal of Computational Biology, 12 (7): 1004–1013, CiteSeerX 10.1.1.379.8693, doi:10.1089/cmb.2005.12.1004, PMID 16201918, archived (PDF) from the original on 2017-09-21, retrieved 2012-03-29.
12. Callan, David (2009), A combinatorial survey of identities for the double factorial, arXiv:0906.1317, Bibcode:2009arXiv0906.1317C.
13. Oswin Aichholzer. "Rectilinear Crossing Number project". Archived from the original on 2007-04-30.
14. Ákos Császár, A Polyhedron Without Diagonals. Archived 2017-09-18 at the Wayback Machine, Bolyai Institute, University of Szeged, 1949
15. Gardner, Martin (1988), Time Travel and Other Mathematical Bewilderments, W. H. Freeman and Company, p. 140, Bibcode:1988ttom.book.....G, ISBN 0-7167-1924-X
16. Robertson, Neil; Seymour, P. D.; Thomas, Robin (1993), "Linkless embeddings of graphs in 3-space", Bulletin of the American Mathematical Society, 28 (1): 84–89, arXiv:math/9301216, doi:10.1090/S0273-0979-1993-00335-5, MR 1164063, S2CID 1110662.
17. Conway, J. H.; Cameron Gordon (1983). "Knots and Links in Spatial Graphs". Journal of Graph Theory. 7 (4): 445–453. doi:10.1002/jgt.3190070410.
External links
Wikimedia Commons has media related to Complete graphs.
Look up complete graph in Wiktionary, the free dictionary.
• Weisstein, Eric W. "Complete Graph". MathWorld.
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16-cell
In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century.[1] It is also called C16, hexadecachoron,[2] or hexdecahedroid [sic?] .[3]
16-cell
(4-orthoplex)
Schlegel diagram
(vertices and edges)
TypeConvex regular 4-polytope
4-orthoplex
4-demicube
Schläfli symbol{3,3,4}
Coxeter diagram
Cells16 {3,3}
Faces32 {3}
Edges24
Vertices8
Vertex figure
Octahedron
Petrie polygonoctagon
Coxeter groupB4, [3,3,4], order 384
D4, order 192
DualTesseract
Propertiesconvex, isogonal, isotoxal, isohedral, regular, Hanner polytope
Uniform index12
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes, and is analogous to the octahedron in three dimensions. It is Coxeter's $\beta _{4}$ polytope.[4] Conway's name for a cross-polytope is orthoplex, for orthant complex. The dual polytope is the tesseract (4-cube), which it can be combined with to form a compound figure. The 16-cell has 16 cells as the tesseract has 16 vertices.
Geometry
The 16-cell is the second in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).[lower-alpha 1]
Each of its 4 successor convex regular 4-polytopes can be constructed as the convex hull of a polytope compound of multiple 16-cells: the 16-vertex tesseract as a compound of two 16-cells, the 24-vertex 24-cell as a compound of three 16-cells, the 120-vertex 600-cell as a compound of fifteen 16-cells, and the 600-vertex 120-cell as a compound of seventy-five 16-cells.[lower-alpha 2]
Regular convex 4-polytopes
Symmetry group A4 B4 F4 H4
Name 5-cell
Hyper-tetrahedron
5-point
16-cell
Hyper-octahedron
8-point
8-cell
Hyper-cube
16-point
24-cell
24-point
600-cell
Hyper-icosahedron
120-point
120-cell
Hyper-dodecahedron
600-point
Schläfli symbol {3, 3, 3} {3, 3, 4} {4, 3, 3} {3, 4, 3} {3, 3, 5} {5, 3, 3}
Coxeter mirrors
Mirror dihedrals 𝝅/3 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2 𝝅/3 𝝅/3 𝝅/4 𝝅/2 𝝅/2 𝝅/2 𝝅/4 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2 𝝅/3 𝝅/4 𝝅/3 𝝅/2 𝝅/2 𝝅/2 𝝅/3 𝝅/3 𝝅/5 𝝅/2 𝝅/2 𝝅/2 𝝅/5 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2
Graph
Vertices 5 tetrahedral 8 octahedral 16 tetrahedral 24 cubical 120 icosahedral 600 tetrahedral
Edges 10 triangular 24 square 32 triangular 96 triangular 720 pentagonal 1200 triangular
Faces 10 triangles 32 triangles 24 squares 96 triangles 1200 triangles 720 pentagons
Cells 5 tetrahedra 16 tetrahedra 8 cubes 24 octahedra 600 tetrahedra 120 dodecahedra
Tori 1 5-tetrahedron 2 8-tetrahedron 2 4-cube 4 6-octahedron 20 30-tetrahedron 12 10-dodecahedron
Inscribed 120 in 120-cell 675 in 120-cell 2 16-cells 3 8-cells 25 24-cells 10 600-cells
Great polygons 2 squares x 3 4 rectangles x 4 4 hexagons x 4 12 decagons x 6 100 irregular hexagons x 4
Petrie polygons 1 pentagon 1 octagon 2 octagons 2 dodecagons 4 30-gons 20 30-gons
Long radius $1$ $1$ $1$ $1$ $1$ $1$
Edge length ${\sqrt {\tfrac {5}{2}}}\approx 1.581$ ${\sqrt {2}}\approx 1.414$ $1$ $1$ ${\tfrac {1}{\phi }}\approx 0.618$ ${\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270$
Short radius ${\tfrac {1}{4}}$ ${\tfrac {1}{2}}$ ${\tfrac {1}{2}}$ ${\sqrt {\tfrac {1}{2}}}\approx 0.707$ ${\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926$ ${\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926$
Area $10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825$ $32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713$ $24$ $96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569$ $1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48$ $720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366$
Volume $5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329$ $16\left({\tfrac {1}{3}}\right)\approx 5.333$ $8$ $24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314$ $600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693$ $120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118$
4-Content ${\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146$ ${\tfrac {2}{3}}\approx 0.667$ $1$ $2$ ${\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863$ ${\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193$
Coordinates
Disjoint squares
xy plane
( 0, 1, 0, 0)( 0, 0,-1, 0)
( 0, 0, 1, 0)( 0,-1, 0, 0)
wz plane
( 1, 0, 0, 0)( 0, 0, 0,-1)
( 0, 0, 0, 1)(-1, 0, 0, 0)
The 16-cell is the 4-dimensional cross polytope, which means its vertices lie in opposite pairs on the 4 axes of a (w, x, y, z) Cartesian coordinate system.
The eight vertices are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by edges except opposite pairs. The edge length is √2.
The vertex coordinates form 6 orthogonal central squares lying in the 6 coordinate planes. Squares in opposite planes that do not share an axis (e.g. in the xy and wz planes) are completely disjoint (they do not intersect at any vertices).[lower-alpha 3]
The 16-cell constitutes an orthonormal basis for the choice of a 4-dimensional reference frame, because its vertices exactly define the four orthogonal axes.
Structure
The Schläfli symbol of the 16-cell is {3,3,4}, indicating that its cells are regular tetrahedra {3,3} and its vertex figure is a regular octahedron {3,4}. There are 8 tetrahedra, 12 triangles, and 6 edges meeting at every vertex. Its edge figure is a square. There are 4 tetrahedra and 4 triangles meeting at every edge.
The 16-cell is bounded by 16 cells, all of which are regular tetrahedra.[lower-alpha 5] It has 32 triangular faces, 24 edges, and 8 vertices. The 24 edges bound 6 orthogonal central squares lying on great circles in the 6 coordinate planes (3 pairs of completely orthogonal[lower-alpha 6] great squares). At each vertex, 3 great squares cross perpendicularly. The 6 edges meet at the vertex the way 6 edges meet at the apex of a canonical octahedral pyramid.[lower-alpha 4]
Rotations
A 3D projection of a 16-cell performing a simple rotation
A 3D projection of a 16-cell performing a double rotation
Rotations in 4-dimensional Euclidean space can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.[6] The 16-cell is a simple frame in which to observe 4-dimensional rotations, because each of the 16-cell's 6 great squares has another completely orthogonal great square (there are 3 pairs of completely orthogonal squares).[lower-alpha 3] Many rotations of the 16-cell can be characterized by the angle of rotation in one of its great square planes (e.g. the xy plane) and another angle of rotation in the completely orthogonal great square plane (the wz plane).[lower-alpha 10] Completely orthogonal great squares have disjoint vertices: 4 of the 16-cell's 8 vertices rotate in one plane, and the other 4 rotate independently in the completely orthogonal plane.[lower-alpha 7]
In 2 or 3 dimensions a rotation is characterized by a single plane of rotation; this kind of rotation taking place in 4-space is called a simple rotation, in which only one of the two completely orthogonal planes rotates (the angle of rotation in the other plane is 0). In the 16-cell, a simple rotation in one of the 6 orthogonal planes moves only 4 of the 8 vertices; the other 4 remain fixed. (In the simple rotation animation above, all 8 vertices move because the plane of rotation is not one of the 6 orthogonal basis planes.)
In a double rotation both sets of 4 vertices move, but independently: the angles of rotation may be different in the 2 completely orthogonal planes. If the two angles happen to be the same, a maximally symmetric isoclinic rotation takes place.[lower-alpha 17] In the 16-cell an isoclinic rotation by 90 degrees of any pair of completely orthogonal square planes takes every square plane to its completely orthogonal square plane.[lower-alpha 18]
Octahedral dipyramid
Octahedron $\beta _{3}$ 16-cell $\beta _{4}$
Orthogonal projections to skew hexagon hyperplane
The simplest construction of the 16-cell is on the 3-dimensional cross polytope, the octahedron. The octahedron has 3 perpendicular axes and 6 vertices in 3 opposite pairs (its Petrie polygon is the hexagon). Add another pair of vertices, on a fourth axis perpendicular to all 3 of the other axes. Connect each new vertex to all 6 of the original vertices, adding 12 new edges. This raises two octahedral pyramids on a shared octahedron base that lies in the 16-cell's central hyperplane.[10]
The octahedron that the construction starts with has three perpendicular intersecting squares (which appear as rectangles in the hexagonal projections). Each square intersects with each of the other squares at two opposite vertices, with two of the squares crossing at each vertex. Then two more points are added in the fourth dimension (above and below the 3-dimensional hyperplane). These new vertices are connected to all the octahedron's vertices, creating 12 new edges and three more squares (which appear edge-on as the 3 diameters of the hexagon in the projection), and three more octahedra.[lower-alpha 8]
Something unprecedented has also been created. Notice that each square no longer intersects with all of the other squares: it does intersect with four of them (with three of the squares crossing at each vertex now), but each square has one other square with which it shares no vertices: it is not directly connected to that square at all. These two separate perpendicular squares (there are three pairs of them) are like the opposite edges of a tetrahedron: perpendicular, but non-intersecting. They lie opposite each other (parallel in some sense), and they don't touch, but they also pass through each other like two perpendicular links in a chain (but unlike links in a chain they have a common center). They are an example of Clifford parallel planes, and the 16-cell is the simplest regular polytope in which they occur. Clifford parallelism[lower-alpha 12] of objects of more than one dimension (more than just curved lines) emerges here and occurs in all the subsequent 4-dimensional regular polytopes, where it can be seen as the defining relationship among disjoint regular 4-polytopes and their concentric parts. It can occur between congruent (similar) polytopes of 2 or more dimensions.[11] For example, as noted above all the subsequent convex regular 4-polytopes are compounds of multiple 16-cells; those 16-cells are Clifford parallel polytopes.
Tetrahedral constructions
The 16-cell has two Wythoff constructions from regular tetrahedra, a regular form and alternated form, shown here as nets, the second represented by tetrahedral cells of two alternating colors. The alternated form is a lower symmetry construction of the 16-cell called the demitesseract.
Wythoff's construction replicates the 16-cell's characteristic 5-cell in a kaleidoscope of mirrors. Every regular 4-polytope has its characteristic 4-orthoscheme, an irregular 5-cell.[lower-alpha 19] There are three regular 4-polytopes with tetrahedral cells: the 5-cell, the 16-cell, and the 600-cell. Although all are bounded by regular tetrahedron cells, their characteristic 5-cells (4-orthoschemes) are different tetrahedral pyramids, all based on the same characteristic irregular tetrahedron. They share the same characteristic tetrahedron (3-orthoscheme) and characteristic right triangle (2-orthoscheme) because they have the same kind of cell.[lower-alpha 20]
Characteristics of the 16-cell[13]
edge[14] arc dihedral[15]
𝒍 ${\sqrt {2}}\approx 1.414$ 90° ${\tfrac {\pi }{2}}$ 120° ${\tfrac {2\pi }{3}}$
𝟀 ${\sqrt {\tfrac {2}{3}}}\approx 0.816$ 60″ ${\tfrac {\pi }{3}}$ 60° ${\tfrac {\pi }{3}}$
𝝓 ${\sqrt {\tfrac {1}{2}}}\approx 0.707$ 45″ ${\tfrac {\pi }{4}}$ 45° ${\tfrac {\pi }{4}}$
𝟁 ${\sqrt {\tfrac {1}{6}}}\approx 0.408$ 30″ ${\tfrac {\pi }{6}}$ 60° ${\tfrac {\pi }{3}}$
$_{0}R^{3}/l$ ${\sqrt {\tfrac {3}{4}}}\approx 0.866$ 60° ${\tfrac {\pi }{3}}$ 90° ${\tfrac {\pi }{2}}$
$_{1}R^{3}/l$ ${\sqrt {\tfrac {1}{4}}}=0.5$ 45° ${\tfrac {\pi }{4}}$ 90° ${\tfrac {\pi }{2}}$
$_{2}R^{3}/l$ ${\sqrt {\tfrac {1}{12}}}\approx 0.289$ 30° ${\tfrac {\pi }{6}}$ 90° ${\tfrac {\pi }{2}}$
$_{0}R^{4}/l$ $1$
$_{1}R^{4}/l$ ${\sqrt {\tfrac {1}{2}}}\approx 0.707$
$_{2}R^{4}/l$ ${\sqrt {\tfrac {1}{3}}}\approx 0.577$
$_{3}R^{4}/l$ ${\sqrt {\tfrac {1}{4}}}=0.5$
The characteristic 5-cell of the regular 16-cell is represented by the Coxeter-Dynkin diagram , which can be read as a list of the dihedral angles between its mirror facets. It is an irregular tetrahedral pyramid based on the characteristic tetrahedron of the regular tetrahedron. The regular 16-cell is subdivided by its symmetry hyperplanes into 384 instances of its characteristic 5-cell that all meet at its center.
The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 16-cell).[lower-alpha 21] If the regular 16-cell has unit radius edge and edge length 𝒍 = ${\sqrt {2}}$, its characteristic 5-cell's ten edges have lengths ${\sqrt {\tfrac {2}{3}}}$, ${\sqrt {\tfrac {1}{2}}}$, ${\sqrt {\tfrac {1}{6}}}$ (the exterior right triangle face, the characteristic triangle 𝟀, 𝝓, 𝟁), plus ${\sqrt {\tfrac {3}{4}}}$, ${\sqrt {\tfrac {1}{4}}}$, ${\sqrt {\tfrac {1}{12}}}$ (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the characteristic radii of the regular tetrahedron), plus $1$, ${\sqrt {\tfrac {1}{2}}}$, ${\sqrt {\tfrac {1}{3}}}$, ${\sqrt {\tfrac {1}{4}}}$ (edges which are the characteristic radii of the regular 16-cell). The 4-edge path along orthogonal edges of the orthoscheme is ${\sqrt {\tfrac {1}{2}}}$, ${\sqrt {\tfrac {1}{6}}}$, ${\sqrt {\tfrac {1}{4}}}$, ${\sqrt {\tfrac {1}{4}}}$, first from a 16-cell vertex to a 16-cell edge center, then turning 90° to a 16-cell face center, then turning 90° to a 16-cell tetrahedral cell center, then turning 90° to the 16-cell center.
Helical construction
A 16-cell can be constructed (three different ways) from two Boerdijk–Coxeter helixes of eight chained tetrahedra, each bent in the fourth dimension into a ring.[16][17] The two circular helixes spiral around each other, nest into each other and pass through each other forming a Hopf link. The 16 triangle faces can be seen in a 2D net within a triangular tiling, with 6 triangles around every vertex. The purple edges represent the Petrie polygon of the 16-cell. The eight-cell ring of tetrahedra contains three octagrams of different colors, eight-edge circular paths that wind twice around the 16-cell on every third vertex of the octagram. The orange and yellow edges are two four-edge halves of one octagram, which join their ends to form a Möbius strip.
Thus the 16-cell can be decomposed into two cell-disjoint circular chains of eight tetrahedrons each, four edges long, one spiraling to the right (clockwise) and the other spiraling to the left (counterclockwise). The left-handed and right-handed cell rings fit together, nesting into each other and entirely filling the 16-cell, even though they are of opposite chirality. This decomposition can be seen in a 4-4 duoantiprism construction of the 16-cell: or , Schläfli symbol {2}⨂{2} or s{2}s{2}, symmetry [4,2+,4], order 64.
Three eight-edge paths (of different colors) spiral along each eight-cell ring, making 90° angles at each vertex. (In the Boerdijk–Coxeter helix before it is bent into a ring, the angles in different paths vary, but are not 90°.) Three paths (with three different colors and apparent angles) pass through each vertex. When the helix is bent into a ring, the segments of each eight-edge path (of various lengths) join their ends, forming a Möbius strip eight edges long along its single-sided circumference of 8𝝅, and one edge wide.[lower-alpha 16] The six four-edge halves of the three eight-edge paths each make four 90° angles, but they are not the six orthogonal great squares: they are open-ended squares, four-edge 360° helices whose open ends are antipodal vertices. The four edges come from four different great squares, and are mutually orthogonal. Combined end-to-end in pairs of the same chirality, the six four-edge paths make three eight-edge Möbius loops, helical octagrams. Each octagram is both a Petrie polygon of the 16-cell, and the helical track along which all eight vertices rotate together, in one of the 16-cell's distinct isoclinic rotations.[lower-alpha 22]
Five ways of looking at the same skew octagram[lower-alpha 23]
Edge path Petrie polygon[18] 16-cell Discrete fibration Isocline chords
Octagram{8/3}[19] Octagram{8/1} Coxeter plane B4 Octagram{8/2}=2{4} Octagram{8/4}=4{2}
The eight √2 edges of the edge-path of an isocline.[lower-alpha 24] Skew octagon of eight √2 edges. The 16-cell has 3 of these 8-vertex circuits. All 24 √2 edges and the four √4 orthogonal axes.[lower-alpha 25] Two completely orthogonal (disjoint) great squares of √2 edges.[lower-alpha 7] Eight √4 chords of an isocline (doubled).[lower-alpha 26]
Each eight-edge helix is a skew octagram{8/3} that winds twice around the 16-cell and visits every vertex before closing into a loop. Its eight edges are the circular path-near-edges of an isocline, a geodesic arc on which vertices move during an isoclinic rotation.[lower-alpha 17] The isoclines connect opposite vertices of face-bonded tetrahedral cells,[lower-alpha 15] which are also opposite vertices (antipodal vertices) of the 16-cell, so the isoclines have √4 chords.[lower-alpha 26] The isocline winds around the 16-cell twice (720°) the way the edges of the octagram{8/3} wind around twice, passing alongside each of the √2 edges once,[lower-alpha 24] and alongside each of the √4 orthogonal axes of the 16-cell twice.[lower-alpha 25] The isocline makes a circle of circumference 8𝝅.[lower-alpha 16]
The eight-cell ring is chiral: there is a right-handed form which spirals clockwise, and a left-handed form which spirals counterclockwise. The 16-cell contains one of each, so it also contains a left and a right isocline; the isocline is the circular axis around which the eight-cell ring twists. Each isocline visits all eight vertices of the 16-cell.[lower-alpha 29] Each eight-cell ring contains half of the 16 cells, but all 8 vertices; the two rings share the vertices, as they nest into each other and fit together. They also share the 24 edges, though left and right octagram helices are different eight-edge paths.[lower-alpha 30]
Because there are three pairs of completely orthogonal great squares,[lower-alpha 3] there are three congruent ways to compose a 16-cell from two eight-cell rings. The 16-cell contains three left-right pairs of eight-cell rings in different orientations, with each cell ring containing its axial isocline.[lower-alpha 22] Each left-right pair of isoclines is the track of a left-right pair of distinct isoclinic rotations: the rotations in one pair of completely orthogonal invariant planes of rotation.[lower-alpha 7] At each vertex, there are three great squares and six octagram isoclines that cross at the vertex and share a 16-cell axis chord.[lower-alpha 31]
As a configuration
This configuration matrix represents the 16-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 16-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
${\begin{bmatrix}{\begin{matrix}8&6&12&8\\2&24&4&4\\3&3&32&2\\4&6&4&16\end{matrix}}\end{bmatrix}}$
Tessellations
One can tessellate 4-dimensional Euclidean space by regular 16-cells. This is called the 16-cell honeycomb and has Schläfli symbol {3,3,4,3}. Hence, the 16-cell has a dihedral angle of 120°.[21] Each 16-cell has 16 neighbors with which it shares a tetrahedron, 24 neighbors with which it shares only an edge, and 72 neighbors with which it shares only a single point. Twenty-four 16-cells meet at any given vertex in this tessellation.
The dual tessellation, the 24-cell honeycomb, {3,4,3,3}, is made of regular 24-cells. Together with the tesseractic honeycomb {4,3,3,4} these are the only three regular tessellations of R4.
Projections
orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
Graph
Dihedral symmetry [12/3] [4]
The cell-first parallel projection of the 16-cell into 3-space has a cubical envelope. The closest and farthest cells are projected to inscribed tetrahedra within the cube, corresponding with the two possible ways to inscribe a regular tetrahedron in a cube. Surrounding each of these tetrahedra are 4 other (non-regular) tetrahedral volumes that are the images of the 4 surrounding tetrahedral cells, filling up the space between the inscribed tetrahedron and the cube. The remaining 6 cells are projected onto the square faces of the cube. In this projection of the 16-cell, all its edges lie on the faces of the cubical envelope.
The cell-first perspective projection of the 16-cell into 3-space has a triakis tetrahedral envelope. The layout of the cells within this envelope are analogous to that of the cell-first parallel projection.
The vertex-first parallel projection of the 16-cell into 3-space has an octahedral envelope. This octahedron can be divided into 8 tetrahedral volumes, by cutting along the coordinate planes. Each of these volumes is the image of a pair of cells in the 16-cell. The closest vertex of the 16-cell to the viewer projects onto the center of the octahedron.
Finally the edge-first parallel projection has a shortened octahedral envelope, and the face-first parallel projection has a hexagonal bipyramidal envelope.
4 sphere Venn diagram
A 3-dimensional projection of the 16-cell and 4 intersecting spheres (a Venn diagram of 4 sets) are topologically equivalent.
The 16 cells ordered by number of intersecting spheres (from 0 to 4) (see all cells and k-faces)
4 sphere Venn diagram and 16-cell projection in the same orientation
Symmetry constructions
The 16-cell's symmetry group is denoted B4.
There is a lower symmetry form of the 16-cell, called a demitesseract or 4-demicube, a member of the demihypercube family, and represented by h{4,3,3}, and Coxeter diagrams or . It can be drawn bicolored with alternating tetrahedral cells.
It can also be seen in lower symmetry form as a tetrahedral antiprism, constructed by 2 parallel tetrahedra in dual configurations, connected by 8 (possibly elongated) tetrahedra. It is represented by s{2,4,3}, and Coxeter diagram: .
It can also be seen as a snub 4-orthotope, represented by s{21,1,1}, and Coxeter diagram: or .
With the tesseract constructed as a 4-4 duoprism, the 16-cell can be seen as its dual, a 4-4 duopyramid.
Name Coxeter diagram Schläfli symbol Coxeter notation Order Vertex figure
Regular 16-cell {3,3,4} [3,3,4]384
Demitesseract
Quasiregular 16-cell
=
=
h{4,3,3}
{3,31,1}
[31,1,1] = [1+,4,3,3]192
Alternated 4-4 duoprism 2s{4,2,4} [[4,2+,4]]64
Tetrahedral antiprism s{2,4,3} [2+,4,3]48
Alternated square prism prism sr{2,2,4} [(2,2)+,4]16
Snub 4-orthotope = s{21,1,1} [2,2,2]+ = [21,1,1]+8
4-fusil
{3,3,4} [3,3,4]384
{4}+{4} or 2{4} [[4,2,4]] = [8,2+,8]128
{3,4}+{ } [4,3,2]96
{4}+2{ } [4,2,2]32
{ }+{ }+{ }+{ } or 4{ } [2,2,2]16
Related complex polygons
The Möbius–Kantor polygon is a regular complex polygon 3{3}3, , in $\mathbb {C} ^{2}$ shares the same vertices as the 16-cell. It has 8 vertices, and 8 3-edges.[22][23]
The regular complex polygon, 2{4}4, , in $\mathbb {C} ^{2}$ has a real representation as a 16-cell in 4-dimensional space with 8 vertices, 16 2-edges, only half of the edges of the 16-cell. Its symmetry is 4[4]2, order 32.[24]
Orthographic projections of 2{4}4 polygon
In B4 Coxeter plane, 2{4}4 has 8 vertices and 16 2-edges, shown here with 4 sets of colors.
The 8 vertices are grouped in 2 sets (shown red and blue), each only connected with edges to vertices in the other set, making this polygon a complete bipartite graph, K4,4.[25]
Related uniform polytopes and honeycombs
The regular 16-cell and tesseract are the regular members of a set of 15 uniform 4-polytopes with the same B4 symmetry. The 16-cell is also one of the uniform polytopes of D4 symmetry.
The 16-cell is also related to the cubic honeycomb, order-4 dodecahedral honeycomb, and order-4 hexagonal tiling honeycomb which all have octahedral vertex figures.
It belongs to the sequence of {3,3,p} 4-polytopes which have tetrahedral cells. The sequence includes three regular 4-polytopes of Euclidean 4-space, the 5-cell {3,3,3}, 16-cell {3,3,4}, and 600-cell {3,3,5}), and the order-6 tetrahedral honeycomb {3,3,6} of hyperbolic space.
It is first in a sequence of quasiregular polytopes and honeycombs h{4,p,q}, and a half symmetry sequence, for regular forms {p,3,4}.
See also
• 24-cell
• 4-polytope
• D4 polytope
Regular 4-polytopes
Convex
5-cell8-cell16-cell24-cell120-cell600-cell
• {3,3,3}
• pentachoron
• 4-simplex
• {4,3,3}
• tesseract
• 4-cube
• {3,3,4}
• hexadecachoron
• 4-orthoplex
• {3,4,3}
• icositetrachoron
• octaplex
• {5,3,3}
• hecatonicosachoron
• dodecaplex
• {3,3,5}
• hexacosichoron
• tetraplex
Star
icosahedral
120-cell
small
stellated
120-cell
great
120-cell
grand
120-cell
great
stellated
120-cell
grand
stellated
120-cell
great grand
120-cell
great
icosahedral
120-cell
grand
600-cell
great grand
stellated 120-cell
• {3,5,5/2}
• icosaplex
• {5/2,5,3}
• stellated dodecaplex
• {5,5/2,5}
• great dodecaplex
• {5,3,5/2}
• grand dodecaplex
• {5/2,3,5}
• great stellated dodecaplex
• {5/2,5,5/2}
• grand stellated dodecaplex
• {5,5/2,3}
• great grand dodecaplex
• {3,5/2,5}
• great icosaplex
• {3,3,5/2}
• grand tetraplex
• {5/2,3,3}
• great grand stellated dodecaplex
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
Notes
1. The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more content[5] within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 16-cell is the 8-point 4-polytope: second in the ascending sequence that runs from 5-point 4-polytope to 600-point 4-polytope.
2. There are 2 and only 2 16-cells inscribed in the 8-cell (tesseract), 3 and only 3 16-cells inscribed in the 24-cell, 75 distinct 16-cells (but only 15 disjoint 16-cells) inscribed in the 600-cell, and 675 distinct 16-cells (but only 75 disjoint 16-cells) inscribed in the 120-cell.
3. In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or completely orthogonal to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.
4. Each vertex in the 16-cell is the apex of an octahedral pyramid, the base of which is the octahedron formed by the 6 other vertices to which the apex is connected by edges. The 16-cell can be deconstructed (four different ways) into two octahedral pyramids by cutting it in half through one of its four octahedral central hyperplanes. Looked at from inside the curved 3 dimensional volume of its boundary surface of 16 face-bonded tetrahedra, the 16-cell's vertex figure is an octahedron. In 4 dimensions, the vertex octahedron is actually an octahedral pyramid. The apex of the octahedral pyramid (the vertex where the 6 edges meet) is not actually at the center of the octahedron: it is displaced radially outwards in the fourth dimension, out of the hyperplane defined by the octahedron's 6 vertices. The 6 edges around the vertex make an orthogonal 3-axis cross in 3 dimensions (and in the 3-dimensional projection of the 4-pyramid), but the 3 lines are actually bent 90 degrees in the fourth dimension where they meet in an apex.
5. The boundary surface of a 16-cell is a finite 3-dimensional space consisting of 16 tetrahedra arranged face-to-face (four around one). It is a closed, tightly curved (non-Euclidean) 3-space, within which we can move straight through 4 tetrahedra in any direction and arrive back in the tetrahedron where we started. We can visualize moving around inside this tetrahedral jungle gym, climbing from one tetrahedron into another on its 24 struts (its edges), and never being able to get out (or see out) of the 16 tetrahedra no matter what direction we go (or look). We are always on (or in) the surface of the 16-cell, never inside the 16-cell itself (nor outside it). We can see that the 6 edges around each vertex radiate symmetrically in 3 dimensions and form an orthogonal 3-axis cross, just as the radii of an octahedron do (so we say the vertex figure of the 16-cell is the octahedron).[lower-alpha 4]
6. Two flat planes A and B of a Euclidean space of four dimensions are called completely orthogonal if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.[lower-alpha 3]
7. Completely orthogonal great squares are non-intersecting and rotate independently because the great circles on which their vertices lie are Clifford parallel.[lower-alpha 12] They are √2 apart at each pair of nearest vertices (and in the 16-cell all the pairs except antipodal pairs are nearest). The two squares cannot intersect at all because they lie in planes which intersect at only one point: the center of the 16-cell.[lower-alpha 3] Because they are perpendicular and share a common center, the two squares are obviously not parallel and separate in the usual way of parallel squares in 3 dimensions; rather they are connected like adjacent square links in a chain, each passing through the other without intersecting at any points, forming a Hopf link.
8. Three great squares meet at each vertex (and at its opposite vertex) in the 16-cell. Each of them has a different completely orthogonal square.[lower-alpha 6] Thus there are three great squares completely orthogonal to each vertex and its opposite vertex (each axis). They form an octahedron (a central hyperplane). Every axis line in the 16-cell is completely orthogonal to a central octahedron hyperplane, as every great square plane is completely orthogonal to another great square plane.[lower-alpha 3] The axis and the octahedron intersect only at one point (the center of the 16-cell), as each pair of completely orthogonal great squares intersects only at one point (the center of the 16-cell). Each central octahedron is also the octahedral vertex figure of two of the eight vertices: the two on its completely orthogonal axis.
9. The three incompletely orthogonal great squares which intersect at each vertex of the 16-cell form the vertex's octahedral vertex figure.[lower-alpha 4] Any two of them, together with the completely orthogonal square of the third, also form an octahedron: a central octahedral hyperplane.[lower-alpha 8] In the 16-cell, each octahedral vertex figure is also a central octahedral hyperplane.
10. Each great square vertex is √2 distant from two of the square's other vertices, and √4 distant from its opposite vertex. The other four vertices of the 16-cell (also √2 distant) are the vertices of the square's completely orthogonal square.[lower-alpha 7] Each 16-cell vertex is a vertex of three orthogonal great squares which intersect there. Each of them has a different completely orthogonal square. Thus there are three great squares completely orthogonal to each vertex: squares that the vertex is not part of.[lower-alpha 9]
11. Each great square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal[lower-alpha 6] to only one of them. Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal. There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not chiral. A pair of isoclinic (Clifford parallel) planes is either a left pair or a right pair unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).[20] Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. Because planes separated by a 90° isoclinic rotation are 180° apart, the plane to the left and the plane to the right are the same plane.[lower-alpha 18]
12. Clifford parallels are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.[7] A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the 3-sphere.[8] In the 16-cell the corresponding vertices of completely orthogonal great circle squares are all √2 apart, so these squares are Clifford parallel polygons.[lower-alpha 11] Note that only the vertices of the great squares (the points on the great circle) are √2 apart; points on the edges of the squares (on chords of the circle) are closer together.
13. Opposite vertices in a unit-radius 4-polytope correspond to the opposite vertices of an 8-cell hypercube (tesseract). The long diagonal of this radially equilateral 4-cube is √4. In a 90° isoclinic rotation each vertex of the 16-cell is displaced to its antipodal vertex, traveling along a helical geodesic arc of length 𝝅 (180°), to a vertex √4 away along the long diameter of the unit-radius 4-polytope (16-cell or tesseract), the same displacement as if it had been displaced √1 four times, traveling along a path of four successive orthogonal edges of a tesseract.
14. There are six different two-edge paths connecting a pair of antipodal vertices along the edges of a great square. The left isoclinic rotation runs diagonally between three of them, and the right isoclinic rotation runs diagonally between the other three. These diagonals are the straight lines (geodesics) connecting opposite vertices of face-bonded tetrahedral cells in the left-handed eight-cell ring and the right-handed eight-cell ring, respectively.
15. Successive vertices visited by an isocline are two edges apart along a great circle, in adjacent cells. In the 16-cell they are the opposite vertices of two face-bonded tetrahedral cells. The two vertices are connected by three two-edge great circle paths along edges of the tetrahedra, but the isocline is the shortest path between them: the geodesic path. On the arc of the isocline (a straight line through 3-space) the two vertices are √16/3 ≈ 2.309 apart, compared to the path along the two √2 edges ≈ 2.828. The vertices are only √4 = 2.0 apart in 4-space, because they are antipodal vertices.[lower-alpha 26]
16. An isocline is a circle of special kind corresponding to a pair of Villarceau circles linked in a Möbius loop. It curves through four dimensions instead of just two. All ordinary circles have a 2𝝅 circumference, but the 16-cell's isocline is a circle with an 8𝝅 circumference (over eight 180° chords). An isocline is a geodesic circle like an ordinary great circle in the plane, but to avoid confusion we always refer to it as an isocline and reserve the term great circle for a geodesic circle in the plane.
17. In an isoclinic rotation, all 6 orthogonal planes are displaced in two orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted sideways by that same angle. An isoclinic displacement (also known as a Clifford displacement) is 4-dimensionally diagonal. Points are displaced an equal distance in four orthogonal directions at once, and displaced a total Pythagorean distance equal to the square root of four times the square of that distance. All vertices of a regular 4-polytope are displaced to a vertex at least two edge lengths away. For example, when the unit-radius 16-cell rotates isoclinically 90° in a great square invariant plane, it also rotates 90° in the completely orthogonal great square invariant plane.[lower-alpha 3] The great square plane also tilts sideways 90° to occupy its completely orthogonal plane. (By isoclinic symmetry, every great square rotates 90° and tilts sideways 90° into its completely orthogonal plane.) Each vertex (in every great square) is displaced to its antipodal vertex, at a distance of √1 in each of four orthogonal directions, a total distance of √4.[lower-alpha 13] The original and displaced vertex are two edge lengths apart by three[lower-alpha 14] different paths along two edges of a great square. But the isocline (the geodesic arc the vertex follows during the isoclinic rotation) does not run along edges: it runs between these different edge-paths diagonally, on a geodesic (shortest arc) between the original and displaced vertices.[lower-alpha 15] This isoclinic geodesic arc is not a segment of an ordinary great circle; it does not lie in the plane of any great square. It is a helical 180° arc that bends in a circle in two completely orthogonal planes at once. This Möbius circle does not lie in any plane or intersect any vertices between the original and the displaced vertex.[lower-alpha 16]
18. The 90 degree isoclinic rotation of two completely orthogonal planes takes them to each other. In such a rotation of a rigid 16-cell, all 6 orthogonal planes rotate by 90 degrees, and also tilt sideways by 90 degrees to their completely orthogonal (Clifford parallel)[lower-alpha 12] plane.[9] The corresponding vertices of the two completely orthogonal great squares are √4 (180°) apart; the great squares (Clifford parallel polytopes) are √4 (180°) apart; but the two completely orthogonal planes are 90° apart, in the two orthogonal angles that separate them. If the isoclinic rotation is continued through another 90°, each vertex completes a 360° rotation and each great square returns to its original plane, but in a different orientation (axes swapped): it has been turned "upside down" on the surface of the 16-cell (which is now "inside out"). Continuing through a second 360° isoclinic rotation (through four 90° by 90° isoclinic steps, a 720° rotation) returns everything to its original place and orientation.
19. An orthoscheme is a chiral irregular simplex with right triangle faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own facets (its mirror walls). Every regular polytope can be dissected radially into instances of its characteristic orthoscheme surrounding its center. The characteristic orthoscheme has the shape described by the same Coxeter-Dynkin diagram as the regular polytope without the generating point ring.
20. A regular polytope of dimension k has a characteristic k-orthoscheme, and also a characteristic (k-1)-orthoscheme. A regular 4-polytope has a characteristic 5-cell (4-orthoscheme) into which it is subdivided by its (3-dimensional) hyperplanes of symmetry, and also a characteristic tetrahedron (3-orthoscheme) into which its surface is subdivided by its cells' (2-dimensional) planes of symmetry. After subdividing its (3-dimensional) surface into characteristic tetrahedra surrounding each cell center, its (4-dimensional) interior can be subdivided into characteristic 5-cells by adding radii joining the vertices of the surface characteristic tetrahedra to the 4-polytope's center.[12] The interior tetrahedra and triangles thus formed will also be orthoschemes.
21. The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.
22. The 16-cell can be constructed from two cell-disjoint eight-cell rings in three different ways; it has three orientations of its pair of rings. Each orientation "contains" a distinct left-right pair of isoclinic rotations, and also a pair of completely orthogonal great squares (Clifford parallel fibers), so each orientation is a discrete fibration of the 16-cell. Each eight-cell ring contains three axial octagrams which have different orientations (they exchange roles) in the three discrete fibrations and six distinct isoclinic rotations (three left and three right) through the cell rings. Three octagrams (of different colors) can be seen in the illustration of a single cell ring, one in the role of Petrie polygon, one as the right isocline, and one as the left isocline. Because each octagram plays three roles, there are exactly six distinct isoclines in the 16-cell, not 18.
23. All five views are the same orthogonal projection of the 16-cell into the same plane (a circular cross-section of the eight-cell ring cylinder), looking along the central axis of the cut ring cylinder pictured above, from one end of the cylinder. The only difference is which √2 edges and √4 isocline chords are omitted for focus. The different colors of √2 edges appear to be of different lengths because they are oblique to the viewer at different angles. Vertices are numbered 1 (top) through 8 in counterclockwise order.
24. There is a path between any two adjacent isocline vertices along four mutually orthogonal √2 edges, making three left 90° turns or three right 90° turns, and thus forming an open square (a square helix). This roundabout path is characteristic of the isocline because the helical arc over the √4 chord curves along it, missing the three vertices on it. These four-edge paths can be seen in a regular octagram{8/3} in which eight √2 edges wind twice around the 16-cell under their invisible 90° isocline arc segments. Notice that the endpoints of four-edge path segments are antipodal vertices (connected by a √4 chord).
25. Each isocline has the eight √2 edges of its edge-path, and eight √4 chords that connect every 3rd vertex on the hexagram{8/3}, vertices that have a twisted path of four mutually orthogonal √2 edges connecting them. The isocline curves smoothly around in a helix, over the √4 chord, and alongside the four orthogonal √2 edges of the edge-path, but it does not actually touch the three vertices of that four-edge path where it makes sharp right-angled turns.[lower-alpha 24] Each √2 edge is an edge of a great square, that is completely orthogonal to another great square, in which the √4 chord is a diagonal.
26. Successive vertices visited by an isocline are √4 apart in 4-space because they are antipodal vertices.[lower-alpha 13] The isocline's √4 chords can be seen in an octagram4{2} running straight through the center of the 16-cell under their invisible 180° isocline arcs. Each orthogonal axis is used twice as a chord in each 8-chord isocline. The two uses of each axis have different (but congruent) 180° isocline arcs: each 180° arc is a helical half-circle path that winds around alongside a unique four-edge path of hexagram{8/3} edges.[lower-alpha 24] The two half-circles have the same chirality (they both wind either clockwise or counterclockwise), so the isocline can be made by nesting the two half-circle helices together to form a circular double helix, and joining the open ends of the two half-circles together to make a Möbius strip whose "single edge" runs through all eight vertices. The two half-circle arcs are completely orthogonal (all their corresponding points are 180° apart), but their chords are coincident (the same √4 axis).
27. For another example of the left and right isoclines of a rotation visiting the same set of vertices, see the characteristic isoclinic rotation of the 5-cell. Although in these two special cases left and right isoclines of the same rotation visit the same set of vertices, they still take very different rotational paths because they visit the same vertices in different sequences.
28. Except in the 5-cell and 16-cell,[lower-alpha 27] a pair of left and right isocline circles have disjoint vertices: the left and right isocline helices are non-intersecting parallels but counter-rotating, forming a special kind of double helix which cannot occur in three dimensions (where counter-rotating helices of the same radius must intersect).
29. In the 16-cell each single isocline winds through all 8 vertices: an entire fibration of two completely orthogonal great squares.[lower-alpha 11] The 5-cell and the 16-cell are the only regular 4-polytopes where each discrete fibration has just one isocline fiber.[lower-alpha 28]
30. The left and right isoclines intersect each other at every vertex; but with respect to a different set of vertex pairs which are √2 apart, they can be considered to be Clifford parallel. There is also a sense in which they are completely orthogonal.[lower-alpha 11]
31. This is atypical for isoclinic rotations generally; normally both the left and right isoclines do not occur at the same vertex: there are two disjoint sets of vertices reachable only by the left or right rotation respectively.[lower-alpha 28] The left and right isoclines of the 16-cell form a very special double helix: unusual not just because it is circular, but because its different left and right helices twist around each other through the same set of antipodal vertices,[lower-alpha 29] not through the two disjoint subsets of antipodal vertices, as the isocline pairs do in most isoclinic rotations found in nature.[lower-alpha 27] Isoclinic rotations in completely orthogonal invariant planes are special.[lower-alpha 11] To see how and why they are special, visualize two completely orthogonal invariant planes of rotation, each rotating by some rotation angle and tilting sideways by the same rotation angle into a different plane entirely.[lower-alpha 17] Only when the rotation angle is 90°, that different plane in which the tilting invariant plane lands will be the completely orthogonal invariant plane itself. The destination plane of the rotation is the completely orthogonal invariant plane. The 90° isoclinic rotation is the only rotation which takes the completely orthogonal invariant planes to each other.[lower-alpha 18] This reciprocity is the reason both left and right rotations go to the same place.
Citations
1. Coxeter 1973, p. 141, § 7-x. Historical remarks.
2. N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249
3. Matila Ghyka, The Geometry of Art and Life (1977), p.68
4. Coxeter 1973, pp. 120=121, § 7.2. See illustration Fig 7.2B.
5. Coxeter 1973, pp. 292–293, Table I(ii): The sixteen regular polytopes {p,q,r} in four dimensions; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.
6. Kim & Rote 2016, p. 6, § 5. Four-Dimensional Rotations.
7. Tyrrell & Semple 1971, pp. 5–6, § 3. Clifford's original definition of parallelism.
8. Kim & Rote 2016, pp. 7–10, § 6. Angles between two Planes in 4-Space.
9. Kim & Rote 2016, pp. 8–10, Relations to Clifford Parallelism.
10. Coxeter 1973, p. 121, § 7.21. See illustration Fig 7.2B: "$\beta _{4}$ is a four-dimensional dipyramid based on $\beta _{3}$ (with its two apices in opposite directions along the fourth dimension)."
11. Tyrrell & Semple 1971.
12. Coxeter 1973, p. 130, § 7.6; "simplicial subdivision".
13. Coxeter 1973, pp. 292–293, Table I(ii); "16-cell, 𝛽4".
14. Coxeter 1973, p. 139, § 7.9 The characteristic simplex.
15. Coxeter 1973, p. 290, Table I(ii); "dihedral angles".
16. Coxeter 1970, p. 45, Table 2: Reflexible honeycombs and their groups; Honeycomb [3,3,4]4 is a tiling of the 3-sphere by 2 rings of 8 tetrahedral cells.
17. Banchoff 2013.
18. Coxeter 1973, pp. 292–293, Table I(ii); 24-cell h1.
19. Coxeter 1973, pp. 292–293, Table I(ii); 24-cell h2.
20. Kim & Rote 2016, pp. 7–8, § 6 Angles between two Planes in 4-Space; Left and Right Pairs of Isoclinic Planes.
21. Coxeter 1973, p. 293.
22. Coxeter 1991, pp. 30, 47.
23. Coxeter & Shephard 1992.
24. Coxeter 1991, p. 108.
25. Coxeter 1991, p. 114.
References
• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• H.S.M. Coxeter:
• Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover.
• Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge: Cambridge University Press.
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 Kaleidoscopes: Selected Writings of H.S.M. Coxeter | Wiley
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Coxeter, H.S.M.; Shephard, G.C. (1992). "Portraits of a family of complex polytopes". Leonardo. 25 (3/4): 239–244. doi:10.2307/1575843. JSTOR 1575843. S2CID 124245340.
• Coxeter, H.S.M. (1970), "Twisted Honeycombs", Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, Providence, Rhode Island: American Mathematical Society, 4
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Kim, Heuna; Rote, Günter (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].
• Tyrrell, J. A.; Semple, J.G. (1971). Generalized Clifford parallelism. Cambridge University Press. ISBN 0-521-08042-8.
• Banchoff, Thomas F. (2013). "Torus Decompostions of Regular Polytopes in 4-space". In Senechal, Marjorie (ed.). Shaping Space. Springer New York. pp. 257–266. doi:10.1007/978-0-387-92714-5_20. ISBN 978-0-387-92713-8.
External links
• Weisstein, Eric W. "16-Cell". MathWorld.
• Der 16-Zeller (16-cell) Marco Möller's Regular polytopes in R4 (German)
• Description and diagrams of 16-cell projections
• Klitzing, Richard. "4D uniform polytopes (polychora) x3o3o4o – hex".
| Wikipedia |
Tetrahedral bipyramid
In 4-dimensional geometry, the tetrahedral bipyramid is the direct sum of a tetrahedron and a segment, {3,3} + { }. Each face of a central tetrahedron is attached with two tetrahedra, creating 8 tetrahedral cells, 16 triangular faces, 14 edges, and 6 vertices,.[1] A tetrahedral bipyramid can be seen as two tetrahedral pyramids augmented together at their base.
Tetrahedral bipyramid
Orthogonal projection.
4 red vertices and 6 blue edges make central tetrahedron. 2 yellow vertices are bipyramid apexes.
Type Polyhedral bipyramid
Schläfli symbol {3,3} + { }
dt{2,3,3}
Coxeter diagram
Cells 8 {3,3} (4+4)
Faces 16 {3} (4+6+6)
Edges 14 (6+4+4)
Vertices 6 (4+2)
Dual Tetrahedral prism
Symmetry group [2,3,3], order 48
Properties convex, regular-faced, Blind polytope
It is the dual of a tetrahedral prism, , so it can also be given a Coxeter-Dynkin diagram, , and both have Coxeter notation symmetry [2,3,3], order 48.
Being convex with all regular cells (tetrahedra) means that it is a Blind polytope.
This bipyramid exists as the cells of the dual of the uniform rectified 5-simplex, and rectified 5-cube or the dual of any uniform 5-polytope with a tetrahedral prism vertex figure. And, as well, it exists as the cells of the dual to the rectified 24-cell honeycomb.
See also
• Triangular bipyramid - A lower dimensional analogy of the tetrahedral bipyramid.
• Octahedral bipyramid - A lower symmetry form of the as 16-cell.
• Cubic bipyramid
• Dodecahedral bipyramid
• Icosahedral bipyramid
References
1. https://www.bendwavy.org/klitzing/incmats/tedpy.htm
• Klitzing, Richard, "Johnson solids, Blind polytopes, and CRFs", Polytopes, retrieved 2022-11-14
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Tetrahedral cupola
In 4-dimensional geometry, the tetrahedral cupola is a polychoron bounded by one tetrahedron, a parallel cuboctahedron, connected by 10 triangular prisms, and 4 triangular pyramids.[1]
Tetrahedral cupola
Schlegel diagram
Type Polyhedral cupola
Schläfli symbol {3,3} v rr{3,3}
Cells 16 1 rr{3,3}
1+4 {3,3}
4+6 {}×{3}
Faces 42 24 triangles
18 squares
Edges 42
Vertices 16
Dual
Symmetry group [3,3,1], order 24
Properties convex, regular-faced
Related polytopes
The tetrahedral cupola can be sliced off from a runcinated 5-cell, on a hyperplane parallel to a tetrahedral cell. The cuboctahedron base passes through the center of the runcinated 5-cell, so the Tetrahedral cupola contains half of the tetrahedron and triangular prism cells of the runcinated 5-cell. The cupola can be seen in A2 and A3 Coxeter plane orthogonal projection of the runcinated 5-cell:
A3 Coxeter plane
Runcinated 5-cell Tetrahedron
(Cupola top)
Cuboctahedron
(Cupola base)
A2 Coxeter plane
See also
• Tetrahedral pyramid (5-cell)
References
1. Convex Segmentochora Dr. Richard Klitzing, Symmetry: Culture and Science, Vol. 11, Nos. 1-4, 139-181, 2000 (4.23 tetrahedron || cuboctahedron)
External links
• Segmentochora: tetaco, tet || co, K-4.23
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Tetrahedral prism
In geometry, a tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedral cells: 2 tetrahedra connected by 4 triangular prisms. It has 14 faces: 8 triangular and 6 square. It has 16 edges and 8 vertices.
Tetrahedral prism
Schlegel diagram
TypePrismatic uniform 4-polytope
Uniform index48
Schläfli symbolt{2,3,3} = {}×{3,3} = h{4,3}×{}
s{2,4}×{}
sr{2,2}×{}
Coxeter diagram =
Cells2 (3.3.3)
4 (3.4.4)
Faces8 {3}
6 {4}
Edges16
Vertices8
Vertex configuration
Equilateral-triangular pyramid
DualTetrahedral bipyramid
Symmetry group[3,3,2], order 48
[4,2+,2], order 16
[(2,2)+,2], order 8
Propertiesconvex
Net
It is one of 18 uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids and Archimedean solids.
Images
An orthographic projection showing the pair of parallel tetrahedra projected as a quadrilateral divided into yellow and blue triangular faces. Each tetrahedra also have two other uncolored triangles across the opposite diagonal.
Transparent Schlegel diagram seen as one tetrahedron nested inside another, with 4 triangular prisms between pairs of triangular faces.
Rotating on 2 different planes
Alternative names
1. Tetrahedral dyadic prism (Norman W. Johnson)
2. Tepe (Jonathan Bowers: for tetrahedral prism)
3. Tetrahedral hyperprism
4. Digonal antiprismatic prism
5. Digonal antiprismatic hyperprism
Structure
The tetrahedral prism is bounded by two tetrahedra and four triangular prisms. The triangular prisms are joined to each other via their square faces, and are joined to the two tetrahedra via their triangular faces.
Projections
The tetrahedron-first orthographic projection of the tetrahedral prism into 3D space has a tetrahedral projection envelope. Both tetrahedral cells project onto this tetrahedron, while the triangular prisms project to its faces.
The triangular-prism-first orthographic projection of the tetrahedral prism into 3D space has a projection envelope in the shape of a triangular prism. The two tetrahedral cells are projected onto the triangular ends of the prism, each with a vertex that projects to the center of the respective triangular face. An edge connects these two vertices through the center of the projection. The prism may be divided into three non-uniform triangular prisms that meet at this edge; these 3 volumes correspond with the images of three of the four triangular prismic cells. The last triangular prismic cell projects onto the entire projection envelope.
The edge-first orthographic projection of the tetrahedral prism into 3D space is identical to its triangular-prism-first parallel projection.
The square-face-first orthographic projection of the tetrahedral prism into 3D space has a cuboidal envelope (see diagram). Each triangular prismic cell projects onto half of the cuboidal volume, forming two pairs of overlapping images. The tetrahedral cells project onto the top and bottom square faces of the cuboid.
Related polytopes
It is the first in an infinite series of uniform antiprismatic prisms.
Convex p-gonal antiprismatic prisms
Name s{2,2}×{} s{2,3}×{} s{2,4}×{} s{2,5}×{} s{2,6}×{} s{2,7}×{} s{2,8}×{} s{2,p}×{}
Coxeter
diagram
Image
Vertex
figure
Cells 2 s{2,2}
(2) {2}×{}={4}
4 {3}×{}
2 s{2,3}
2 {3}×{}
6 {3}×{}
2 s{2,4}
2 {4}×{}
8 {3}×{}
2 s{2,5}
2 {5}×{}
10 {3}×{}
2 s{2,6}
2 {6}×{}
12 {3}×{}
2 s{2,7}
2 {7}×{}
14 {3}×{}
2 s{2,8}
2 {8}×{}
16 {3}×{}
2 s{2,p}
2 {p}×{}
2p {3}×{}
Net
The tetrahedral prism, -131, is first in a dimensional series of uniform polytopes, expressed by Coxeter as k31 series. The tetrahedral prism is the vertex figure for the second, the rectified 5-simplex. The fifth figure is a Euclidean honeycomb, 331, and the final is a noncompact hyperbolic honeycomb, 431. Each uniform polytope in the sequence is the vertex figure of the next.
k31 dimensional figures
n 4 5 6 7 8 9
Coxeter
group
A3A1 A5 D6 E7 ${\tilde {E}}_{7}$ = E7+ ${\bar {T}}_{8}$=E7++
Coxeter
diagram
Symmetry [3−1,3,1] [30,3,1] [31,3,1] [32,3,1] [33,3,1] [34,3,1]
Order 48 720 23,040 2,903,040 ∞
Graph - -
Name −131 031 131 231 331 431
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
External links
• 6. Convex uniform prismatic polychora - Model 48, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora) x x3o3o - tepe".
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Tetrahedrally diminished dodecahedron
In geometry, a tetrahedrally diminished[lower-alpha 1] dodecahedron (also tetrahedrally stellated icosahedron or propello tetrahedron[1]) is a topologically self-dual polyhedron made of 16 vertices, 30 edges, and 16 faces (4 equilateral triangles and 12 identical quadrilaterals).[2]
Dorman Luke self-dual form
Tetrahedrally stellated icosahedron
Tetrahedrally diminished dodecahedron
Conway polyhedron notationpT
Faces16: 4 {3} + 12 quadrilaterals
Edges30
Vertices16
Vertex configuration3.4.4.4
4.4.4
Symmetry groupT, [3,3]+, (332), order 12
Dual polyhedronSelf-dual
Propertiesconvex
Nets
A canonical form exists with two edge lengths at 0.849 : 1.057, assuming that the radius of the midsphere is 1. The kites remain isosceles.
It has chiral tetrahedral symmetry, and so its geometry can be constructed from pyritohedral symmetry of the pseudoicosahedron with 4 faces stellated, or from the pyritohedron, with 4 vertices diminished. Within its tetrahedral symmetry, it has geometric varied proportions. By Dorman Luke dual construction, a unique geometric proportion can be defined. The kite faces have edges of length ratio ~ 1:0.633.
Topologically, the triangles are always equilateral, while the quadrilaterals are irregular, although the two adjacent edges that meet at the vertices of a tetrahedron are equal.
As a self-dual hexadecahedron, it is one of 302404 forms, 1476 with at least order 2 symmetry, and the only one with tetrahedral symmetry.[3]
As a diminished regular dodecahedron, with 4 vertices removed, the quadrilaterals faces are trapezoids.
As a stellation of the regular icosahedron it is one of 32 stellations defined with tetrahedral symmetry. It has kite faces.[4]
In Conway polyhedron notation, it can represented as pT, applying George W. Hart's propeller operator to a regular tetrahedron.[5]
Related polytopes and honeycombs
This polyhedron represents the vertex figure of a hyperbolic uniform honeycomb, the partially diminished icosahedral honeycomb, pd{3,5,3}, with 12 pentagonal antiprisms and 4 dodecahedron cells meeting at every vertex.
Vertex figure projected as Schlegel diagram
Notes
1. It is also less accurately called a tetrahedrally truncated dodecahedron
References
1. Sculpture Based on Propellorized Polyhedra
2. Tetrahedrally Stellated Icosahedron
3. Self-Dual Hexadecahedra
4. Tetrahedral Stellations of the Icosahedron
5. Conway Notation for Polyhedra
External links
• tetrahedrally truncated dodecahedron and stellated icosahedron
• Generation of an icosahedron by the intersection of five tetrahedra: geometrical and crystallographic features of the intermediate polyhedra
• VRML model as truncated regular dodecahedron
• VRML model as tetrahedrally stellated icosahedron
Polyhedra
Listed by number of faces and type
1–10 faces
• Monohedron
• Dihedron
• Trihedron
• Tetrahedron
• Pentahedron
• Hexahedron
• Heptahedron
• Octahedron
• Enneahedron
• Decahedron
11–20 faces
• Hendecahedron
• Dodecahedron
• Tridecahedron
• Tetradecahedron
• Pentadecahedron
• Hexadecahedron
• Heptadecahedron
• Octadecahedron
• Enneadecahedron
• Icosahedron
>20 faces
• Icositetrahedron (24)
• Triacontahedron (30)
• Hexecontahedron (60)
• Enneacontahedron (90)
• Hectotriadiohedron (132)
• Apeirohedron (∞)
elemental things
• face
• edge
• vertex
• uniform polyhedron (two infinite groups and 75)
• regular polyhedron (9)
• quasiregular polyhedron (7)
• semiregular polyhedron (two infinite groups and 59)
convex polyhedron
• Platonic solid (5)
• Archimedean solid (13)
• Catalan solid (13)
• Johnson solid (92)
non-convex polyhedron
• Kepler–Poinsot polyhedron (4)
• Star polyhedron (infinite)
• Uniform star polyhedron (57)
prismatoids
• prism
• antiprism
• frustum
• cupola
• wedge
• pyramid
• parallelepiped
Convex polyhedra
Platonic solids (regular)
• tetrahedron
• cube
• octahedron
• dodecahedron
• icosahedron
Archimedean solids
(semiregular or uniform)
• truncated tetrahedron
• cuboctahedron
• truncated cube
• truncated octahedron
• rhombicuboctahedron
• truncated cuboctahedron
• snub cube
• icosidodecahedron
• truncated dodecahedron
• truncated icosahedron
• rhombicosidodecahedron
• truncated icosidodecahedron
• snub dodecahedron
Catalan solids
(duals of Archimedean)
• triakis tetrahedron
• rhombic dodecahedron
• triakis octahedron
• tetrakis hexahedron
• deltoidal icositetrahedron
• disdyakis dodecahedron
• pentagonal icositetrahedron
• rhombic triacontahedron
• triakis icosahedron
• pentakis dodecahedron
• deltoidal hexecontahedron
• disdyakis triacontahedron
• pentagonal hexecontahedron
Dihedral regular
• dihedron
• hosohedron
Dihedral uniform
• prisms
• antiprisms
duals:
• bipyramids
• trapezohedra
Dihedral others
• pyramids
• truncated trapezohedra
• gyroelongated bipyramid
• cupola
• bicupola
• frustum
• bifrustum
• rotunda
• birotunda
• prismatoid
• scutoid
Degenerate polyhedra are in italics.
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Tetrahedroid
In algebraic geometry, a tetrahedroid (or tétraédroïde) is a special kind of Kummer surface studied by Cayley (1846), with the property that the intersections with the faces of a fixed tetrahedron are given by two conics intersecting in four nodes. Tetrahedroids generalize Fresnel's wave surface.
Not to be confused with tetrahedron.
References
• Cayley, Arthur (1846), "Sur la surface des ondes", Journal de Mathématiques Pures et Appliquées, 11: 291–296, Collected papers vol 1 pages 302–305
• Hudson, R. W. H. T. (1990) [First published 1905], Kummer's quartic surface, Cambridge Mathematical Library, Cambridge University Press, ISBN 978-0-521-39790-2, MR 1097176
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Tetrahedral-cubic honeycomb
In the geometry of hyperbolic 3-space, the tetrahedron-cube honeycomb is a compact uniform honeycomb, constructed from cube, tetrahedron, and cuboctahedron cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
Tetrahedron-cube honeycomb
TypeCompact uniform honeycomb
Schläfli symbol{(4,3,3,3)} or {(3,3,3,4)}
Coxeter diagram or or
Cells{3,3}
{4,3}
r{4,3}
Facestriangular {3}
square {4}
Vertex figure
rhombicuboctahedron
Coxeter group[(4,3,3,3)]
PropertiesVertex-transitive, edge-transitive
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
Images
Wide-angle perspective view
Centered on cube
See also
• Convex uniform honeycombs in hyperbolic space
• List of regular polytopes
• Hyperbolic tetrahedral-octahedral honeycomb
References
• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
• Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapter 16-17: Geometries on Three-manifolds I,II)
• Norman Johnson Uniform Polytopes, Manuscript
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• N.W. Johnson: Geometries and Transformations, (2018) Chapter 13: Hyperbolic Coxeter groups
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Tetrahedral-dodecahedral honeycomb
In the geometry of hyperbolic 3-space, the tetrahedral-dodecahedral honeycomb is a compact uniform honeycomb, constructed from dodecahedron, tetrahedron, and icosidodecahedron cells, in a rhombitetratetrahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
Tetrahedral-dodecahedral honeycomb
TypeCompact uniform honeycomb
Schläfli symbol{(5,3,3,3)} or {(3,3,3,5)}
Coxeter diagram or or
Cells{3,3}
{5,3}
r{5,3}
Facestriangular {3}
pentagon {5}
Vertex figure
rhombitetratetrahedron
Coxeter group[(5,3,3,3)]
PropertiesVertex-transitive, edge-transitive
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
Images
Wide-angle perspective views
Centered on dodecahedron
Centered on icosidodecahedron
See also
• Convex uniform honeycombs in hyperbolic space
• List of regular polytopes
References
• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
• Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapter 16-17: Geometries on Three-manifolds I,II)
• Norman Johnson Uniform Polytopes, Manuscript
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• N.W. Johnson: Geometries and Transformations, (2018) Chapter 13: Hyperbolic Coxeter groups
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Tetrahedral-icosahedral honeycomb
In the geometry of hyperbolic 3-space, the tetrahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from icosahedron, tetrahedron, and octahedron cells, in an icosidodecahedron vertex figure. It has a single-ring Coxeter diagram , and is named by its two regular cells.
Tetrahedral-icosahedral honeycomb
TypeCompact uniform honeycomb
Semiregular honeycomb
Schläfli symbol{(3,3,5,3)}
Coxeter diagram or or
Cells{3,3}
{3,5}
r{3,3}
Facestriangle {3}
Vertex figure
rhombicosidodecahedron
Coxeter group[(5,3,3,3)]
PropertiesVertex-transitive, edge-transitive
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
It represents a semiregular honeycomb as defined by all regular cells, although from the Wythoff construction, the octahedron comes from the rectified tetrahedron .
Images
Wide-angle perspective views
Centered on octahedron
See also
• Convex uniform honeycombs in hyperbolic space
• List of regular polytopes
References
• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
• Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapter 16-17: Geometries on Three-manifolds I,II)
• Norman Johnson Uniform Polytopes, Manuscript
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• N.W. Johnson: Geometries and Transformations, (2018) Chapter 13: Hyperbolic Coxeter groups
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Hyperbolic tetrahedral-octahedral honeycomb
In the geometry of hyperbolic 3-space, the tetrahedron-octahedron honeycomb is a compact uniform honeycomb, constructed from octahedron and tetrahedron cells, in a rhombicuboctahedron vertex figure.
Tetrahedron-octahedron honeycomb
TypeCompact uniform honeycomb
Semiregular honeycomb
Schläfli symbol{(3,4,3,3)} or {(3,3,4,3)}
Coxeter diagram or or
Cells{3,3}
{3,4}
Facestriangular {3}
Vertex figure
rhombicuboctahedron
Coxeter group[(4,3,3,3)]
PropertiesVertex-transitive, edge-transitive
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
It represents a semiregular honeycomb as defined by all regular cells, although from the Wythoff construction, rectified tetrahedral r{3,3}, becomes the regular octahedron {3,4}.
Images
Wide-angle perspective view
Centered on octahedron
See also
• Convex uniform honeycombs in hyperbolic space
• List of regular polytopes
• Tetrahedral-octahedral honeycomb - similar Euclidean honeycomb,
• Tetrahedral-cubic honeycomb
References
• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
• Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapter 16-17: Geometries on Three-manifolds I,II)
• Norman Johnson Uniform Polytopes, Manuscript
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• N.W. Johnson: Geometries and Transformations, (2015) Chapter 13: Hyperbolic Coxeter groups
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Compound of ten tetrahedra
The compound of ten tetrahedra is one of the five regular polyhedral compounds. This polyhedron can be seen as either a stellation of the icosahedron or a compound. This compound was first described by Edmund Hess in 1876.
Compound of ten tetrahedra
Typeregular compound
Coxeter symbol2{5,3}[10{3,3}]2{3,5}[1]
IndexUC6, W25
Elements
(As a compound)
10 tetrahedra:
F = 40, E = 60, V = 20
Dual compoundSelf-dual
Symmetry groupicosahedral (Ih)
Subgroup restricting to one constituentchiral tetrahedral (T)
It can be seen as a faceting of a regular dodecahedron.
As a compound
It can also be seen as the compound of ten tetrahedra with full icosahedral symmetry (Ih). It is one of five regular compounds constructed from identical Platonic solids.
It shares the same vertex arrangement as a dodecahedron.
The compound of five tetrahedra represents two chiral halves of this compound (it can therefore be seen as a "compound of two compounds of five tetrahedra").
It can be made from the compound of five cubes by replacing each cube with a stella octangula on the cube's vertices (which results in a "compound of five compounds of two tetrahedra").
As a stellation
This polyhedron is a stellation of the icosahedron, and given as Wenninger model index 25.
Stellation diagramStellation coreConvex hull
Icosahedron
Dodecahedron
As a facetting
It is also a facetting of the dodecahedron, as shown at left. Concave pentagrams can be seen on the compound where the pentagonal faces of the dodecahedron are positioned.
As a simple polyhedron
If it is treated as a simple non-convex polyhedron without self-intersecting surfaces, it has 180 faces (120 triangles and 60 concave quadrilaterals), 122 vertices (60 with degree 3, 30 with degree 4, 12 with degree 5, and 20 with degree 12), and 300 edges, giving an Euler characteristic of 122-300+180 = +2.
See also
• Compound of five tetrahedra
References
1. Regular polytopes, p.98
• Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.
• Coxeter, Harold Scott MacDonald; Du Val, P.; Flather, H. T.; Petrie, J. F. (1999). The fifty-nine icosahedra (3rd ed.). Tarquin. ISBN 978-1-899618-32-3. MR 0676126. (1st Edn University of Toronto (1938))
• H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, 3.6 The five regular compounds, pp.47-50, 6.2 Stellating the Platonic solids, pp.96-104
External links
• Weisstein, Eric W. "Tetrahedron 10-Compound". MathWorld.
• VRML model:
• Compounds of 5 and 10 Tetrahedra by Sándor Kabai, The Wolfram Demonstrations Project.
• Klitzing, Richard. "3D compound".
Notable stellations of the icosahedron
Regular Uniform duals Regular compounds Regular star Others
(Convex) icosahedron Small triambic icosahedron Medial triambic icosahedron Great triambic icosahedron Compound of five octahedra Compound of five tetrahedra Compound of ten tetrahedra Great icosahedron Excavated dodecahedron Final stellation
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.
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Compound of six tetrahedra
The compound of six tetrahedra is a uniform polyhedron compound. It's composed of a symmetric arrangement of 6 tetrahedra. It can be constructed by inscribing a stella octangula within each cube in the compound of three cubes, or by stellating each octahedron in the compound of three octahedra.
Compound of six tetrahedra
TypeUniform compound
Convex hullNonuniform truncated octahedron
IndexUC3
Polyhedra6 tetrahedra
Faces24 triangles
Edges36
Vertices24
Symmetry groupoctahedral (Oh)
Subgroup restricting to one constituent2-fold antiprismatic (D2d)
It is one of only five polyhedral compounds (along with the compound of two great dodecahedra, the compound of five great dodecahedra, the compound of two small stellated dodecahedra, and the compound of five small stellated dodecahedra) which is vertex-transitive and face-transitive but not edge-transitive.
References
• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79: 447–457, doi:10.1017/S0305004100052440, MR 0397554.
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Tetrahedron packing
In geometry, tetrahedron packing is the problem of arranging identical regular tetrahedra throughout three-dimensional space so as to fill the maximum possible fraction of space.
Currently, the best lower bound achieved on the optimal packing fraction of regular tetrahedra is 85.63%.[1] Tetrahedra do not tile space,[2] and an upper bound below 100% (namely, 1 − (2.6...)·10−25) has been reported.[3]
Historical results
Aristotle claimed that tetrahedra could fill space completely.[4] [5]
In 2006, Conway and Torquato showed that a packing fraction about 72% can be obtained by constructing a non-Bravais lattice packing of tetrahedra (with multiple particles with generally different orientations per repeating unit), and thus they showed that the best tetrahedron packing cannot be a lattice packing (with one particle per repeating unit such that each particle has a common orientation).[6] These packing constructions almost doubled the optimal Bravais-lattice-packing fraction 36.73% obtained by Hoylman.[7] In 2007 and 2010, Chaikin and coworkers experimentally showed that tetrahedron-like dice can randomly pack in a finite container up to a packing fraction between 75% and 76%.[8] In 2008, Chen was the first to propose a packing of hard, regular tetrahedra that packed more densely than spheres, demonstrating numerically a packing fraction of 77.86%.[9][10] A further improvement was made in 2009 by Torquato and Jiao, who compressed Chen's structure using a computer algorithm to a packing fraction of 78.2021%.[11]
In mid-2009 Haji-Akbari et al. showed, using MC simulations of initially random systems that at packing densities >50% an equilibrium fluid of hard tetrahedra spontaneously transforms to a dodecagonal quasicrystal, which can be compressed to 83.24%. They also reported a glassy, disordered packing at densities exceeding 78%. For a periodic approximant to a quasicrystal with an 82-tetrahedron unit cell, they obtained a packing density as high as 85.03%.[12]
In late 2009, a new, much simpler family of packings with a packing fraction of 85.47% was discovered by Kallus, Elser, and Gravel.[13] These packings were also the basis of a slightly improved packing obtained by Torquato and Jiao at the end of 2009 with a packing fraction of 85.55%,[14] and by Chen, Engel, and Glotzer in early 2010 with a packing fraction of 85.63%.[1] The Chen, Engel and Glotzer result currently stands as the densest known packing of hard, regular tetrahedra. Surprisingly, the square-triangle tiling[12] packs denser than this double lattice of triangular bipyramids when tetrahedra are slightly rounded (the Minkowski sum of a tetrahedron and a sphere), making the 82-tetrahedron crystal the largest unit cell for a densest packing of identical particles to date.[15]
Relationship to other packing problems
Because the earliest lower bound known for packings of tetrahedra was less than that of spheres, it was suggested that the regular tetrahedra might be a counterexample to Ulam's conjecture that the optimal density for packing congruent spheres is smaller than that for any other convex body. However, the more recent results have shown that this is not the case.
See also
• Packing problem
• Disphenoid tetrahedral honeycomb - an isohedral packing of irregular tetrahedra in 3-space.
• The triakis truncated tetrahedral honeycomb is cell-transitive and based on a regular tetrahedron.
References
1. Chen, Elizabeth R.; Engel, Michael; Glotzer, Sharon C. (2010). "Dense crystalline dimer packings of regular tetrahedra". Discrete & Computational Geometry. 44 (2): 253–280. arXiv:1001.0586. doi:10.1007/s00454-010-9273-0. S2CID 18523116.
2. Struik, D. J. (1925). "Het probleem 'De Impletione Loci'". Nieuw Archief voor Wiskunde. 2nd ser. 15: 121–134. JFM 52.0002.04.
3. Simon Gravel; Veit Elser; Yoav Kallus (2010). "Upper bound on the packing density of regular tetrahedra and octahedra". Discrete & Computational Geometry. 46 (4): 799–818. arXiv:1008.2830. doi:10.1007/s00454-010-9304-x. S2CID 18908213.
4. Jeffrey Lagarias and Chuanming Zong (2012-12-04). "Mysteries in Packing Regular Tetrahedra" (PDF).
5. News Release (2014-12-03). "Jeffrey Lagarias and Chuanming Zong to receive 2015 Conant Prize".
6. Conway, J. H. (2006). "Packing, tiling, and covering with tetrahedra". Proceedings of the National Academy of Sciences. 103 (28): 10612–10617. Bibcode:2006PNAS..10310612C. doi:10.1073/pnas.0601389103. PMC 1502280. PMID 16818891.
7. Hoylman, Douglas J. (1970). "The densest lattice packing of tetrahedra". Bulletin of the American Mathematical Society. 76: 135–138. doi:10.1090/S0002-9904-1970-12400-4. hdl:10150/288016.
8. Jaoshvili, Alexander; Esakia, Andria; Porrati, Massimo; Chaikin, Paul M. (2010). "Experiments on the Random Packing of Tetrahedral Dice". Physical Review Letters. 104 (18): 185501. Bibcode:2010PhRvL.104r5501J. doi:10.1103/PhysRevLett.104.185501. hdl:10919/24495. PMID 20482187.
9. Chen, Elizabeth R. (2008). "A Dense Packing of Regular Tetrahedra". Discrete & Computational Geometry. 40 (2): 214–240. arXiv:0908.1884. doi:10.1007/s00454-008-9101-y. S2CID 32166668.
10. Cohn, Henry (2009). "Mathematical physics: A tight squeeze". Nature. 460 (7257): 801–802. Bibcode:2009Natur.460..801C. doi:10.1038/460801a. PMID 19675632. S2CID 5157975.
11. Torquato, S.; Jiao, Y. (2009). "Dense packings of the Platonic and Archimedean solids". Nature. 460 (7257): 876–879. arXiv:0908.4107. Bibcode:2009Natur.460..876T. doi:10.1038/nature08239. PMID 19675649. S2CID 52819935.
12. Haji-Akbari, Amir; Engel, Michael; Keys, Aaron S.; Zheng, Xiaoyu; Petschek, Rolfe G.; Palffy-Muhoray, Peter; Glotzer, Sharon C. (2009). "Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra". Nature. 462 (7274): 773–777. arXiv:1012.5138. Bibcode:2009Natur.462..773H. doi:10.1038/nature08641. PMID 20010683. S2CID 4412674.
13. Kallus, Yoav; Elser, Veit; Gravel, Simon (2010). "Dense Periodic Packings of Tetrahedra with Small Repeating Units". Discrete & Computational Geometry. 44 (2): 245–252. arXiv:0910.5226. doi:10.1007/s00454-010-9254-3. S2CID 13385357.
14. Torquato, S.; Jiao, Y. (2009). "Analytical Constructions of a Family of Dense Tetrahedron Packings and the Role of Symmetry". arXiv:0912.4210 [cond-mat.stat-mech].
15. Jin, Weiwei; Lu, Peng; Li, Shuixiang (December 2015). "Evolution of the dense packings of spherotetrahedral particles: from ideal tetrahedra to spheres". Scientific Reports. 5 (1): 15640. Bibcode:2015NatSR...515640J. doi:10.1038/srep15640. PMC 4614866. PMID 26490670.
External links
• Packing Tetrahedrons, and Closing in on a Perfect Fit, NYTimes
• Efficient shapes, The Economist
• Pyramids are the best shape for packing, New Scientist
Packing problems
Abstract packing
• Bin
• Set
Circle packing
• In a circle / equilateral triangle / isosceles right triangle / square
• Apollonian gasket
• Circle packing theorem
• Tammes problem (on sphere)
Sphere packing
• Apollonian
• Finite
• In a sphere
• In a cube
• In a cylinder
• Close-packing
• Kissing number
• Sphere-packing (Hamming) bound
Other 2-D packing
• Square packing
Other 3-D packing
• Tetrahedron
• Ellipsoid
Puzzles
• Conway
• Slothouber–Graatsma
| Wikipedia |
Tetrahemihexahedron
In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.[1] Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin diagram is (although this is a double covering of the tetrahemihexahedron).
Tetrahemihexahedron
TypeUniform star polyhedron
ElementsF = 7, E = 12
V = 6 (χ = 1)
Faces by sides4{3}+3{4}
Coxeter diagram (double-covering)
Wythoff symbol3/2 3 | 2 (double-covering)
Symmetry groupTd, [3,3], *332
Index referencesU04, C36, W67
Dual polyhedronTetrahemihexacron
Vertex figure
3.4.3/2.4
Bowers acronymThah
It is the only non-prismatic uniform polyhedron with an odd number of faces. Its Wythoff symbol is 3/2 3 | 2, but that represents a double covering of the tetrahemihexahedron with eight triangles and six squares, paired and coinciding in space. (It can more intuitively be seen as two coinciding tetrahemihexahedra.)
It is a hemipolyhedron. The "hemi" part of the name means some of the faces form a group with half as many members as some regular polyhedron—here, three square faces form a group with half as many faces as the regular hexahedron, better known as the cube—hence hemihexahedron. Hemi faces are also oriented in the same direction as the regular polyhedron's faces. The three square faces of the tetrahemihexahedron are, like the three facial orientations of the cube, mutually perpendicular.
The "half-as-many" characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. Visually, each square is divided into four right triangles, with two visible from each side.
Related surfaces
It is a non-orientable surface. It is unique as the only uniform polyhedron with an Euler characteristic of 1 and is hence a projective polyhedron, yielding a representation of the real projective plane[2] very similar to the Roman surface.
Roman surface
Related polyhedra
It has the same vertices and edges as the regular octahedron. It also shares 4 of the 8 triangular faces of the octahedron, but has three additional square faces passing through the centre of the polyhedron.
Octahedron
Tetrahemihexahedron
The dual figure is the tetrahemihexacron.
It is 2-covered by the cuboctahedron,[2] which accordingly has the same abstract vertex figure (2 triangles and two squares: 3.4.3.4) and twice the vertices, edges, and faces. It has the same topology as the abstract polyhedron hemi-cuboctahedron.
Cuboctahedron
Tetrahemihexahedron
It may also be constructed as a crossed triangular cuploid. All cuploids and their duals are topologically projective planes.[3]
Family of star-cuploids
n⁄d 3 5 7
2
Crossed triangular cuploid
Pentagrammic cuploid
Heptagrammic cuploid
4 —
Crossed pentagonal cuploid
Crossed heptagrammic cuploid
Tetrahemihexacron
Tetrahemihexacron
TypeStar polyhedron
Face—
ElementsF = 6, E = 12
V = 7 (χ = 1)
Symmetry groupTd, [3,3], *332
Index referencesDU04
dual polyhedronTetrahemihexahedron
The tetrahemihexacron is the dual of the tetrahemihexahedron, and is one of nine dual hemipolyhedra.
Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at infinity.[4] In Magnus Wenninger's Dual Models, they are represented with intersecting prisms, each extending in both directions to the same vertex at infinity, in order to maintain symmetry. In practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, called stellation to infinity. However, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions.
Topologically it is considered to contain seven vertices. The three vertices considered at infinity (the real projective plane at infinity) correspond directionally to the three vertices of the hemi-octahedron, an abstract polyhedron. The other four vertices exist at alternate corners of a central cube (a demicube, in this case a tetrahedron).
References
1. Maeder, Roman. "04: tetrahemihexahedron". MathConsult.
2. (Richter)
3. Polyhedral Models of the Projective Plane, Paul Gailiunas, Bridges 2018 Conference Proceedings
4. (Wenninger 2003, p. 101)
• Richter, David A., Two Models of the Real Projective Plane
• Wenninger, Magnus (2003) [1983], Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208 (Page 101, Duals of the (nine) hemipolyhedra)
External links
• Eric W. Weisstein, Tetrahemihexahedron (Uniform polyhedron) at MathWorld.
• Uniform polyhedra and duals
• Paper model
• Great Stella: software used to create main image on this page
Star-polyhedra navigator
Kepler-Poinsot
polyhedra
(nonconvex
regular polyhedra)
• small stellated dodecahedron
• great dodecahedron
• great stellated dodecahedron
• great icosahedron
Uniform truncations
of Kepler-Poinsot
polyhedra
• dodecadodecahedron
• truncated great dodecahedron
• rhombidodecadodecahedron
• truncated dodecadodecahedron
• snub dodecadodecahedron
• great icosidodecahedron
• truncated great icosahedron
• nonconvex great rhombicosidodecahedron
• great truncated icosidodecahedron
Nonconvex uniform
hemipolyhedra
• tetrahemihexahedron
• cubohemioctahedron
• octahemioctahedron
• small dodecahemidodecahedron
• small icosihemidodecahedron
• great dodecahemidodecahedron
• great icosihemidodecahedron
• great dodecahemicosahedron
• small dodecahemicosahedron
Duals of nonconvex
uniform polyhedra
• medial rhombic triacontahedron
• small stellapentakis dodecahedron
• medial deltoidal hexecontahedron
• small rhombidodecacron
• medial pentagonal hexecontahedron
• medial disdyakis triacontahedron
• great rhombic triacontahedron
• great stellapentakis dodecahedron
• great deltoidal hexecontahedron
• great disdyakis triacontahedron
• great pentagonal hexecontahedron
Duals of nonconvex
uniform polyhedra with
infinite stellations
• tetrahemihexacron
• hexahemioctacron
• octahemioctacron
• small dodecahemidodecacron
• small icosihemidodecacron
• great dodecahemidodecacron
• great icosihemidodecacron
• great dodecahemicosacron
• small dodecahemicosacron
| Wikipedia |
Tetraheptagonal tiling
In geometry, the tetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of r{4,7}.
Tetraheptagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration(4.7)2
Schläfli symbolr{7,4} or ${\begin{Bmatrix}7\\4\end{Bmatrix}}$
rr{7,7}
Wythoff symbol2 | 7 4
7 7 | 2
Coxeter diagram
Symmetry group[7,4], (*742)
[7,7], (*772)
DualOrder-7-4 rhombille tiling
PropertiesVertex-transitive edge-transitive
Symmetry
A half symmetry [1+,4,7] = [7,7] construction exists, which can be seen as two colors of heptagons. This coloring can be called a rhombiheptaheptagonal tiling.
The dual tiling is made of rhombic faces and has a face configuration V4.7.4.7.
Related polyhedra and tiling
*n42 symmetry mutations of quasiregular tilings: (4.n)2
Symmetry
*4n2
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
[ni,4]
Figures
Config. (4.3)2 (4.4)2 (4.5)2 (4.6)2 (4.7)2 (4.8)2 (4.∞)2 (4.ni)2
Uniform heptagonal/square tilings
Symmetry: [7,4], (*742) [7,4]+, (742) [7+,4], (7*2) [7,4,1+], (*772)
{7,4} t{7,4} r{7,4} 2t{7,4}=t{4,7} 2r{7,4}={4,7} rr{7,4} tr{7,4} sr{7,4} s{7,4} h{4,7}
Uniform duals
V74 V4.14.14 V4.7.4.7 V7.8.8 V47 V4.4.7.4 V4.8.14 V3.3.4.3.7 V3.3.7.3.7 V77
Uniform heptaheptagonal tilings
Symmetry: [7,7], (*772) [7,7]+, (772)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
{7,7} t{7,7}
r{7,7} 2t{7,7}=t{7,7} 2r{7,7}={7,7} rr{7,7} tr{7,7} sr{7,7}
Uniform duals
V77 V7.14.14 V7.7.7.7 V7.14.14 V77 V4.7.4.7 V4.14.14 V3.3.7.3.7
Dimensional family of quasiregular polyhedra and tilings: 7.n.7.n
Symmetry
*7n2
[n,7]
Hyperbolic... Paracompact Noncompact
*732
[3,7]
*742
[4,7]
*752
[5,7]
*762
[6,7]
*772
[7,7]
*872
[8,7]...
*∞72
[∞,7]
[iπ/λ,7]
Coxeter
Quasiregular
figures
configuration
3.7.3.7
4.7.4.7
7.5.7.5
7.6.7.6
7.7.7.7
7.8.7.8
7.∞.7.∞
7.∞.7.∞
See also
Wikimedia Commons has media related to Uniform tiling 4-7-4-7.
• Uniform tilings in hyperbolic plane
• List of regular polytopes
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Tetrakis hexahedron
In geometry, a tetrakis hexahedron (also known as a tetrahexahedron, hextetrahedron, tetrakis cube, and kiscube[2]) is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid.
Tetrakis hexahedron
(Click here for rotating model)
TypeCatalan solid
Coxeter diagram
Conway notationkC
Face typeV4.6.6
isosceles triangle
Faces24
Edges36
Vertices14
Vertices by type6{4}+8{6}
Symmetry groupOh, B3, [4,3], (*432)
Rotation groupO, [4,3]+, (432)
Dihedral angle143°07′48″
arccos(−4/5)
Propertiesconvex, face-transitive
Truncated octahedron
(dual polyhedron)
Net
Dual compound of truncated octahedron and tetrakis hexahedron. The woodcut on the left is from Perspectiva Corporum Regularium (1568) by Wenzel Jamnitzer.
Die and crystal model
Drawing and crystal model of variant with tetrahedral symmetry called hexakis tetrahedron [1]
It can be called a disdyakis hexahedron or hexakis tetrahedron as the dual of an omnitruncated tetrahedron, and as the barycentric subdivision of a tetrahedron.[3]
Cartesian coordinates
Cartesian coordinates for the 14 vertices of a tetrakis hexahedron centered at the origin, are the points (±3/2, 0, 0), (0, ±3/2, 0), (0, 0, ±3/2) and (±1, ±1, ±1).
The length of the shorter edges of this tetrakis hexahedron equals 3/2 and that of the longer edges equals 2. The faces are acute isosceles triangles. The larger angle of these equals $\arccos(1/9)\approx 83.620\,629\,791\,56^{\circ }$ and the two smaller ones equal $\arccos(2/3)\approx 48.189\,685\,104\,22^{\circ }$.
Orthogonal projections
The tetrakis hexahedron, dual of the truncated octahedron has 3 symmetry positions, two located on vertices and one mid-edge.
Orthogonal projections
Projective
symmetry
[2] [4] [6]
Tetrakis
hexahedron
Truncated
octahedron
Uses
Naturally occurring (crystal) formations of tetrahexahedra are observed in copper and fluorite systems.
Polyhedral dice shaped like the tetrakis hexahedron are occasionally used by gamers.
A 24-cell viewed under a vertex-first perspective projection has a surface topology of a tetrakis hexahedron and the geometric proportions of the rhombic dodecahedron, with the rhombic faces divided into two triangles.
The tetrakis hexahedron appears as one of the simplest examples in building theory. Consider the Riemannian symmetric space associated to the group SL4(R). Its Tits boundary has the structure of a spherical building whose apartments are 2-dimensional spheres. The partition of this sphere into spherical simplices (chambers) can be obtained by taking the radial projection of a tetrakis hexahedron.
Symmetry
With Td, [3,3] (*332) tetrahedral symmetry, the triangular faces represent the 24 fundamental domains of tetrahedral symmetry. This polyhedron can be constructed from 6 great circles on a sphere. It can also be seen by a cube with its square faces triangulated by their vertices and face centers and a tetrahedron with its faces divided by vertices, mid-edges, and a central point.
Truncated
octahedron
Disdyakis
hexahedron
Deltoidal
dodecahedron
Rhombic
hexahedron
Tetrahedron
Spherical polyhedron
(see rotating model) Orthographic projections from 2-, 3- and 4-fold axes
The edges of the spherical tetrakis hexahedron belong to six great circles, which correspond to mirror planes in tetrahedral symmetry. They can be grouped into three pairs of orthogonal circles (which typically intersect on one coordinate axis each). In the images below these square hosohedra are colored red, green and blue.
Stereographic projections
2-fold 3-fold 4-fold
Dimensions
If we denote the edge length of the base cube by a, the height of each pyramid summit above the cube is a/4. The inclination of each triangular face of the pyramid versus the cube face is arctan(1/2), approximately 26.565° (sequence A073000 in the OEIS). One edge of the isosceles triangles has length a, the other two have length 3a/4, which follows by applying the Pythagorean theorem to height and base length. This yields an altitude of √5a/4 in the triangle (OEIS: A204188). Its area is √5a2/8, and the internal angles are arccos(2/3) (approximately 48.1897°) and the complementary 180° − 2 arccos(2/3) (approximately 83.6206°).
The volume of the pyramid is a3/12; so the total volume of the six pyramids and the cube in the hexahedron is 3a3/2.
Kleetope
It can be seen as a cube with square pyramids covering each square face; that is, it is the Kleetope of the cube.
Cubic pyramid
It is very similar to the 3D net for a 4D cubic pyramid, as the net for a square based is a square with triangles attached to each edge, the net for a cubic pyramid is a cube with square pyramids attached to each face.
Related polyhedra and tilings
Uniform octahedral polyhedra
Symmetry: [4,3], (*432) [4,3]+
(432)
[1+,4,3] = [3,3]
(*332)
[3+,4]
(3*2)
{4,3} t{4,3} r{4,3}
r{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3} sr{4,3} h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{3,4}
s{31,1}
=
=
=
=
or
=
or
=
Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35
*n32 symmetry mutation of truncated tilings: n.6.6
Sym.
*n42
[n,3]
Spherical Euclid. Compact Parac. Noncompact hyperbolic
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
[12i,3] [9i,3] [6i,3]
Truncated
figures
Config. 2.6.6 3.6.6 4.6.6 5.6.6 6.6.6 7.6.6 8.6.6 ∞.6.6 12i.6.6 9i.6.6 6i.6.6
n-kis
figures
Config. V2.6.6 V3.6.6 V4.6.6 V5.6.6 V6.6.6 V7.6.6 V8.6.6 V∞.6.6 V12i.6.6 V9i.6.6 V6i.6.6
It is a polyhedra in a sequence defined by the face configuration V4.6.2n. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any n ≥ 7.
With an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors.
Each face on these domains also corresponds to the fundamental domain of a symmetry group with order 2,3,n mirrors at each triangle face vertex.
*n32 symmetry mutation of omnitruncated tilings: 4.6.2n
Sym.
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]
*∞32
[∞,3]
[12i,3]
[9i,3]
[6i,3]
[3i,3]
Figures
Config. 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞ 4.6.24i 4.6.18i 4.6.12i 4.6.6i
Duals
Config. V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞ V4.6.24i V4.6.18i V4.6.12i V4.6.6i
See also
• Disdyakis triacontahedron
• Disdyakis dodecahedron
• Kisrhombille tiling
• Compound of three octahedra
• Deltoidal icositetrahedron, another 24-face Catalan solid.
References
1. Hexakistetraeder in German, see e.g. Meyers page and Brockhaus page. The same drawing appears in Brockhaus and Efron as преломленный пирамидальный тетраэдр (refracted pyramidal tetrahedron).
2. Conway, Symmetries of Things, p.284
3. Langer, Joel C.; Singer, David A. (2010), "Reflections on the lemniscate of Bernoulli: the forty-eight faces of a mathematical gem", Milan Journal of Mathematics, 78 (2): 643–682, doi:10.1007/s00032-010-0124-5, MR 2781856
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208 (The thirteen semiregular convex polyhedra and their duals, Page 14, Tetrakishexahedron)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Tetrakis hexahedron)
External links
• Eric W. Weisstein, Tetrakis hexahedron (Catalan solid) at MathWorld.
• Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
• VRML model
• Conway Notation for Polyhedra Try: "dtO" or "kC"
• Tetrakis Hexahedron – Interactive Polyhedron model
• The Uniform Polyhedra
Catalan solids
Tetrahedron
(Dual)
Tetrahedron
(Seed)
Octahedron
(Dual)
Cube
(Seed)
Icosahedron
(Dual)
Dodecahedron
(Seed)
Triakis tetrahedron
(Needle)
Triakis tetrahedron
(Kis)
Triakis octahedron
(Needle)
Tetrakis hexahedron
(Kis)
Triakis icosahedron
(Needle)
Pentakis dodecahedron
(Kis)
Rhombic hexahedron
(Join)
Rhombic dodecahedron
(Join)
Rhombic triacontahedron
(Join)
Deltoidal dodecahedron
(Ortho)
Disdyakis hexahedron
(Meta)
Deltoidal icositetrahedron
(Ortho)
Disdyakis dodecahedron
(Meta)
Deltoidal hexecontahedron
(Ortho)
Disdyakis triacontahedron
(Meta)
Pentagonal dodecahedron
(Gyro)
Pentagonal icositetrahedron
(Gyro)
Pentagonal hexecontahedron
(Gyro)
Archimedean duals
Tetrahedron
(Seed)
Tetrahedron
(Dual)
Cube
(Seed)
Octahedron
(Dual)
Dodecahedron
(Seed)
Icosahedron
(Dual)
Truncated tetrahedron
(Truncate)
Truncated tetrahedron
(Zip)
Truncated cube
(Truncate)
Truncated octahedron
(Zip)
Truncated dodecahedron
(Truncate)
Truncated icosahedron
(Zip)
Tetratetrahedron
(Ambo)
Cuboctahedron
(Ambo)
Icosidodecahedron
(Ambo)
Rhombitetratetrahedron
(Expand)
Truncated tetratetrahedron
(Bevel)
Rhombicuboctahedron
(Expand)
Truncated cuboctahedron
(Bevel)
Rhombicosidodecahedron
(Expand)
Truncated icosidodecahedron
(Bevel)
Snub tetrahedron
(Snub)
Snub cube
(Snub)
Snub dodecahedron
(Snub)
Convex polyhedra
Platonic solids (regular)
• tetrahedron
• cube
• octahedron
• dodecahedron
• icosahedron
Archimedean solids
(semiregular or uniform)
• truncated tetrahedron
• cuboctahedron
• truncated cube
• truncated octahedron
• rhombicuboctahedron
• truncated cuboctahedron
• snub cube
• icosidodecahedron
• truncated dodecahedron
• truncated icosahedron
• rhombicosidodecahedron
• truncated icosidodecahedron
• snub dodecahedron
Catalan solids
(duals of Archimedean)
• triakis tetrahedron
• rhombic dodecahedron
• triakis octahedron
• tetrakis hexahedron
• deltoidal icositetrahedron
• disdyakis dodecahedron
• pentagonal icositetrahedron
• rhombic triacontahedron
• triakis icosahedron
• pentakis dodecahedron
• deltoidal hexecontahedron
• disdyakis triacontahedron
• pentagonal hexecontahedron
Dihedral regular
• dihedron
• hosohedron
Dihedral uniform
• prisms
• antiprisms
duals:
• bipyramids
• trapezohedra
Dihedral others
• pyramids
• truncated trapezohedra
• gyroelongated bipyramid
• cupola
• bicupola
• frustum
• bifrustum
• rotunda
• birotunda
• prismatoid
• scutoid
Degenerate polyhedra are in italics.
| Wikipedia |
Tetrakis cuboctahedron
In geometry, the tetrakis cuboctahedron is a convex polyhedron with 32 triangular faces, 48 edges, and 18 vertices. It is a dual of the truncated rhombic dodecahedron.
Tetrakis cuboctahedron
Faces32 triangles (2 types)
Edges48 (2 types)
Vertices18 (2 types)
Vertex configuration(6) 35
(12) 36
Conway notationk4aC
Symmetry groupOctahedral (Oh)
Dual polyhedronchamfered cube
Propertiesconvex
Net
Its name comes from a topological construction from the cuboctahedron with the kis operator applied to the square faces. In this construction, all the vertices are assumed to be the same distance from the center, while in general octahedral symmetry can be maintain even with the 6 order-4 vertices at a different distance from the center as the other 12.
Related polyhedra
It can also be topologically constructed from the octahedron, dividing each triangular face into 4 triangles by adding mid-edge vertices (an ortho operation). From this construction, all 32 triangles will be equilateral.
This polyhedron can be confused with a slightly smaller Catalan solid, the tetrakis hexahedron, which has only 24 triangles, 32 edges, and 14 vertices.
• Octahedron with edges bisected and faces divided into subtriangles of the tetrakis cuboctahedron
• Cuboctahedron
• Tetrakis hexahedron
• The nonconvex octahemioctahedron looks like a concave tetrakis cuboctahedron with inverted square pyramids meeting at the polyhedron center.
See also
• Pentakis icosidodecahedron
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
• Chapter 21: Naming the Archimedean and Catalan polyhedra and Tilings (p284)
External links
• VTML polyhedral generator Try "k4aC" (Conway polyhedron notation)
| Wikipedia |
Tetrakis square tiling
In geometry, the tetrakis square tiling is a tiling of the Euclidean plane. It is a square tiling with each square divided into four isosceles right triangles from the center point, forming an infinite arrangement of lines. It can also be formed by subdividing each square of a grid into two triangles by a diagonal, with the diagonals alternating in direction, or by overlaying two square grids, one rotated by 45 degrees from the other and scaled by a factor of √2.
Tetrakis square tiling
TypeDual semiregular tiling
Faces45-45-90 triangle
Coxeter diagram
Symmetry groupp4m, [4,4], *442
Rotation groupp4, [4,4]+, (442)
Dual polyhedronTruncated square tiling
Face configurationV4.8.8
Propertiesface-transitive
Conway, Burgiel, and Goodman-Strauss call it a kisquadrille,[1] represented by a kis operation that adds a center point and triangles to replace the faces of a square tiling (quadrille). It is also called the Union Jack lattice because of the resemblance to the UK flag of the triangles surrounding its degree-8 vertices.[2]
It is labeled V4.8.8 because each isosceles triangle face has two types of vertices: one with 4 triangles, and two with 8 triangles.
As a dual uniform tiling
It is the dual tessellation of the truncated square tiling which has one square and two octagons at each vertex.[3]
Applications
A 5 × 9 portion of the tetrakis square tiling is used to form the board for the Malagasy board game Fanorona. In this game, pieces are placed on the vertices of the tiling, and move along the edges, capturing pieces of the other color until one side has captured all of the other side's pieces. In this game, the degree-4 and degree-8 vertices of the tiling are called respectively weak intersections and strong intersections, a distinction that plays an important role in the strategy of the game.[4] A similar board is also used for the Brazilian game Adugo, and for the game of Hare and Hounds.
The tetrakis square tiling was used for a set of commemorative postage stamps issued by the United States Postal Service in 1997, with an alternating pattern of two different stamps. Compared to the simpler pattern for triangular stamps in which all diagonal perforations are parallel to each other, the tetrakis pattern has the advantage that, when folded along any of its perforations, the other perforations line up with each other, making repeated folding possible.[5]
This tiling also forms the basis for a commonly used "pinwheel", "windmill", and "broken dishes" patterns in quilting.[6][7][8]
Symmetry
The symmetry type is:
• with the coloring: cmm; a primitive cell is 8 triangles, a fundamental domain 2 triangles (1/2 for each color)
• with the dark triangles in black and the light ones in white: p4g; a primitive cell is 8 triangles, a fundamental domain 1 triangle (1/2 each for black and white)
• with the edges in black and the interiors in white: p4m; a primitive cell is 2 triangles, a fundamental domain 1/2
The edges of the tetrakis square tiling form a simplicial arrangement of lines, a property it shares with the triangular tiling and the kisrhombille tiling.
These lines form the axes of symmetry of a reflection group (the wallpaper group [4,4], (*442) or p4m), which has the triangles of the tiling as its fundamental domains. This group is isomorphic to, but not the same as, the group of automorphisms of the tiling, which has additional axes of symmetry bisecting the triangles and which has half-triangles as its fundamental domains.
There are many small index subgroups of p4m, [4,4] symmetry (*442 orbifold notation), that can be seen in relation to the Coxeter diagram, with nodes colored to correspond to reflection lines, and gyration points labeled numerically. Rotational symmetry is shown by alternately white and blue colored areas with a single fundamental domain for each subgroup is filled in yellow. Glide reflections are given with dashed lines.
Subgroups can be expressed as Coxeter diagrams, along with fundamental domain diagrams.
Small index subgroups of p4m, [4,4], (*442)
index 1 2 4
Fundamental
domain
diagram
Coxeter notation
Coxeter diagram
[1,4,1,4,1] = [4,4]
[1+,4,4]
=
[4,4,1+]
=
[4,1+,4]
=
[1+,4,4,1+]
=
[4+,4+] = [(4,4+,2+)]
Orbifold *442 *2222 22×
Semidirect subgroups
index 2 4
Diagram
Coxeter [4,4+]
[4+,4]
[(4,4,2+)]
[1+,4,1+,4]=[(2+,4,4)]
= =
[4,1+,4,1+]=[(4,4,2+)]
= =
Orbifold 4*2 2*22
Direct subgroups
Index 2 4 8
Diagram
Coxeter [4,4]+
[1+,4,4+] = [4,4+]+
=
[4+,4,1+] = [4+,4]+
=
[(4,1+,4,2+)] = [(4,4,2+)]+
=
[1+,4,1+,4,1+] = [(4+,4+,2+)] = [4+,4+]+
=
Orbifold 442 2222
See also
• Tilings of regular polygons
• List of uniform tilings
• Percolation threshold
Notes
1. Conway, John; Burgiel, Heidi; Goodman-Strauss, Chaim (2008), "Chapter 21: Naming Archimedean and Catalan polyhedra and tilings", The Symmetries of Things, AK Peters, p. 288, ISBN 978-1-56881-220-5
2. Stephenson, John, "Ising Model with Antiferromagnetic Next-Nearest-Neighbor Coupling: Spin Correlations and Disorder Points", Phys. Rev. B, 1 (11): 4405–4409, doi:10.1103/PhysRevB.1.4405.
3. Weisstein, Eric W. "Dual tessellation". MathWorld.
4. Bell, R. C. (1983), "Fanorona", The Boardgame Book, Exeter Books, pp. 150–151, ISBN 0-671-06030-9
5. Frederickson, Greg N. (2006), Piano-Hinged Dissections, A K Peters, p. 144.
6. The Quilting Bible, Creative Publishing International, 1997, p. 55, ISBN 9780865732001.
7. Zieman, Nancy (2011), Quilt With Confidence, Krause Publications, p. 66, ISBN 9781440223556.
8. Fassett, Kaffe (2007), Kaffe Fassett's Kaleidoscope of Quilts: Twenty Designs from Rowan for Patchwork and Quilting, Taunton Press, p. 96, ISBN 9781561589388.
References
Wikimedia Commons has media related to Tetrakis square tiling.
• Grünbaum, Branko & Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (Chapter 2.1: Regular and uniform tilings, p. 58-65)
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 40. ISBN 0-486-23729-X.
• Keith Critchlow, Order in Space: A design source book, 1970, p. 77-76, pattern 8
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Tetractys
The tetractys (Greek: τετρακτύς), or tetrad,[1] or the tetractys of the decad[2] is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row, which is the geometrical representation of the fourth triangular number. As a mystical symbol, it was very important to the secret worship of Pythagoreanism. There were four seasons, and the number was also associated with planetary motions and music.[3]
Pythagorean symbol
1. The first four numbers symbolize the musica universalis and the Cosmos as:
1. Monad – Unity
2. Dyad – Power – Limit/Unlimited (peras/apeiron)
3. Triad – Harmony
4. Tetrad – Kosmos[4]
2. The four rows add up to ten, which was unity of a higher order (The Dekad).
3. The Tetractys symbolizes the four classical elements—air, fire, water, and earth.
4. The Tetractys represented the organization of space:
1. the first row represented zero dimensions (a point)
2. the second row represented one dimension (a line of two points)
3. the third row represented two dimensions (a plane defined by a triangle of three points)
4. the fourth row represented three dimensions (a tetrahedron defined by four points)
A prayer of the Pythagoreans shows the importance of the Tetractys (sometimes called the "Mystic Tetrad"), as the prayer was addressed to it.
Bless us, divine number, thou who generated gods and men! O holy, holy Tetractys, thou that containest the root and source of the eternally flowing creation! For the divine number begins with the profound, pure unity until it comes to the holy four; then it begets the mother of all, the all-comprising, all-bounding, the first-born, the never-swerving, the never-tiring holy ten, the keyholder of all.[5]
As a portion of the secret religion, initiates were required to swear a secret oath by the Tetractys. They then served as novices, which required them to observe silence for a period of five years.
The Pythagorean oath also mentioned the Tetractys:
By that pure, holy, four lettered name on high,
nature's eternal fountain and supply,
the parent of all souls that living be,
by him, with faith find oath, I swear to thee.
It is said[6][7][8] that the Pythagorean musical system was based on the Tetractys as the rows can be read as the ratios of 4:3 (perfect fourth), 3:2 (perfect fifth), 2:1 (octave), forming the basic intervals of the Pythagorean scales. That is, Pythagorean scales are generated from combining pure fourths (in a 4:3 relation), pure fifths (in a 3:2 relation), and the simple ratios of the unison 1:1 and the octave 2:1. Note that the diapason, 2:1 (octave), and the diapason plus diapente, 3:1 (compound fifth or perfect twelfth), are consonant intervals according to the tetractys of the decad, but that the diapason plus diatessaron, 8:3 (compound fourth or perfect eleventh), is not.[9][10]
The Tetractys [also known as the decad] is an equilateral triangle formed from the sequence of the first ten numbers aligned in four rows. It is both a mathematical idea and a metaphysical symbol that embraces within itself—in seedlike form—the principles of the natural world, the harmony of the cosmos, the ascent to the divine, and the mysteries of the divine realm. So revered was this ancient symbol that it inspired ancient philosophers to swear by the name of the one who brought this gift to humanity.
Kabbalist symbol
In the work by anthropologist Raphael Patai entitled The Hebrew Goddess, the author argues that the tetractys and its mysteries influenced the early Kabbalah.[11] A Hebrew tetractys has the letters of the Tetragrammaton inscribed on the ten positions of the tetractys, from right to left. It has been argued that the Kabbalistic Tree of Life, with its ten spheres of emanation, is in some way connected to the tetractys, but its form is not that of a triangle. The occultist Dion Fortune writes:
The point is assigned to Kether;
the line to Chokmah;
the two-dimensional plane to Binah;
consequently the three-dimensional solid naturally falls to Chesed.[12]
The relationship between geometrical shapes and the first four Sephirot is analogous to the geometrical correlations in Tetraktys, shown above under #Pythagorean symbol, and unveils the relevance of the Tree of Life with the Tetraktys.
Tarot card reading arrangement
In a Tarot reading, the various positions of the tetractys provide a representation for forecasting future events by signifying according to various occult disciplines, such as Alchemy.[13] Below is only a single variation for interpretation.
The first row of a single position represents the Premise of the reading, forming a foundation for understanding all the other cards.
The second row of two positions represents the cosmos and the individual and their relationship.
• The Light Card to the right represents the influence of the cosmos leading the individual to an action.
• The Dark Card to the left represents the reaction of the cosmos to the actions of the individual.
The third row of three positions represents three kinds of decisions an individual must make.
• The Creator Card is rightmost, representing new decisions and directions that may be made.
• The Sustainer Card is in the middle, representing decisions to keep balance, and things that should not change.
• The Destroyer Card is leftmost, representing old decisions and directions that should not be continued.
The fourth row of four positions represents the four Greek elements.
• The Fire card is rightmost, representing dynamic creative force, ambitions, and personal will.
• The Air card is to the right middle, representing the mind, thoughts, and strategies toward goals.
• The Water card is to the left middle, representing the emotions, feelings, and whims.
• The Earth card is leftmost, representing physical realities of day to day living.
Occurrence
The tetractys occurs (generally coincidentally) in the following:
• the baryon decuplet
• an archbishop's coat of arms
• the arrangement of bowling pins in ten-pin bowling
• the arrangement of billiard balls in ten-ball pool
• a Chinese checkers board
• the "Christmas Tree" formation in association football
In poetry
In English-language poetry, a tetractys is a syllable-counting form with five lines. The first line has one syllable, the second has two syllables, the third line has three syllables, the fourth line has four syllables, and the fifth line has ten syllables.[14] A sample tetractys would look like this:
Mantrum
Your /
fury /
confuses /
us all greatly. /
Volatile, big-bodied tots are selfish. //
The tetractys was created by Ray Stebbing, who said the following about his newly created form:
"The tetractys could be Britain's answer to the haiku. Its challenge is to express a complete thought, profound or comic, witty or wise, within the narrow compass of twenty syllables.[15]
See also
• Pascal's triangle
References
1. The Theosophical Glossary, Forgotten Books, p. 302, ISBN 9781440073915
2. Eduard Zeller. Outlines of the History of Greek Philosophy (13 ed.). p. 36.
3. Dimitra Karamanides (2005), Pythagoras: pioneering mathematician and musical theorist of Ancient Greece, The Rosen Publishing Group, p. 65, ISBN 9781404205000
4. The Pythagorean Sourcebook and Library by Kenneth Sylvan Guthrie
5. Dantzig, Tobias ([1930], 2005) Number. The Language of Science. p. 42
6. Introduction to Arithmetic – Nicomachus
7. Bruhn, Siglind (2005), The Musical Order of the World: Kepler, Hesse, Hindemith-Siglind Bruhn, Pendragon Press, ISBN 9781576471173
8. A Dictionary of Greek and Roman Antiquities(1890) – William Smith, LLD, William Wayte, G. E. Marindin, Ed.
9. Plutarch, De animae procreatione in Timaeo – Goodwin, Ed.(lang.:English)
10. Pennick, Nigel (January 2012), Sacred Architecture of London – Nigel Pennick, Aeon Books, ISBN 9781904658627
11. Patai, Raphael (1967). The Hebrew Goddess. Wayne State University Press. ISBN 0-8143-2271-9. - Chapter V - The Kabbalistic Tetrad
12. The Mystical Qabalah, Dion Fortune, Chapter XVIII, 25
13. "Tetractys Spread".
14. English Syllable Counters
15. Search result for Tetractys
Further reading
• von Franz, Marie-Louise. Number and Time: Reflections Leading Towards a Unification of Psychology and Physics. Rider & Company, London, 1974. ISBN 0-09-121020-8
• Fideler, D. ed. The Pythagorean Sourcebook and Library Archived 2015-05-09 at the Wayback Machine. Phanes Press, 1987.
• The Theoretic Arithmetic of the Pythagoreans – Thomas Taylor
External links
• Examples of Tetractys poems
Ancient Greek philosophical concepts
• Adiaphora (indifferent)
• Apeiron (infinite)
• Aporia (problem)
• Arche (first principle)
• Arete (excellence)
• Ataraxia (tranquility)
• Cosmos (order)
• Diairesis (division)
• Doxa (opinion)
• Episteme (knowledge)
• Ethos (character)
• Eudaimonia (flourishing)
• Logos (reason)
• Mimesis (imitation)
• Monad (unit)
• Nous (intellect)
• Ousia (substance)
• Pathos (passion)
• Phronesis (prudence)
• Physis (nature)
• Sophia (wisdom)
• Sophrosyne (temperance)
• Techne (craft)
• Telos (goal)
• Thumos (temper)
| Wikipedia |
Tetraoctagonal tiling
In geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.
Tetraoctagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration(4.8)2
Schläfli symbolr{8,4} or ${\begin{Bmatrix}8\\4\end{Bmatrix}}$
rr{8,8}
rr(4,4,4)
t0,1,2,3(∞,4,∞,4)
Wythoff symbol2 | 8 4
Coxeter diagram or
or
Symmetry group[8,4], (*842)
[8,8], (*882)
[(4,4,4)], (*444)
[(∞,4,∞,4)], (*4242)
DualOrder-8-4 quasiregular rhombic tiling
PropertiesVertex-transitive edge-transitive
Constructions
There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,4] or (*842) orbifold symmetry. Removing the mirror between the order 2 and 4 points, [8,4,1+], gives [8,8], (*882). Removing the mirror between the order 2 and 8 points, [1+,8,4], gives [(4,4,4)], (*444). Removing both mirrors, [1+,8,4,1+], leaves a rectangular fundamental domain, [(∞,4,∞,4)], (*4242).
Four uniform constructions of 4.8.4.8
Name Tetra-octagonal tiling Rhombi-octaoctagonal tiling
Image
Symmetry [8,4]
(*842)
[8,8] = [8,4,1+]
(*882)
=
[(4,4,4)] = [1+,8,4]
(*444)
=
[(∞,4,∞,4)] = [1+,8,4,1+]
(*4242)
= or
Schläfli r{8,4} rr{8,8}
=r{8,4}1/2
r(4,4,4)
=r{4,8}1/2
t0,1,2,3(∞,4,∞,4)
=r{8,4}1/4
Coxeter = = = or
Symmetry
The dual tiling has face configuration V4.8.4.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4242), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*42) orbifold.
Related polyhedra and tiling
*n42 symmetry mutations of quasiregular tilings: (4.n)2
Symmetry
*4n2
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
[ni,4]
Figures
Config. (4.3)2 (4.4)2 (4.5)2 (4.6)2 (4.7)2 (4.8)2 (4.∞)2 (4.ni)2
Dimensional family of quasiregular polyhedra and tilings: (8.n)2
Symmetry
*8n2
[n,8]
Hyperbolic... Paracompact Noncompact
*832
[3,8]
*842
[4,8]
*852
[5,8]
*862
[6,8]
*872
[7,8]
*882
[8,8]...
*∞82
[∞,8]
[iπ/λ,8]
Coxeter
Quasiregular
figures
configuration
3.8.3.8
4.8.4.8
8.5.8.5
8.6.8.6
8.7.8.7
8.8.8.8
8.∞.8.∞
8.∞.8.∞
Uniform octagonal/square tilings
[8,4], (*842)
(with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries)
(And [(∞,4,∞,4)] (*4242) index 4 subsymmetry)
=
=
=
=
=
=
=
=
=
=
=
{8,4} t{8,4}
r{8,4} 2t{8,4}=t{4,8} 2r{8,4}={4,8} rr{8,4} tr{8,4}
Uniform duals
V84 V4.16.16 V(4.8)2 V8.8.8 V48 V4.4.4.8 V4.8.16
Alternations
[1+,8,4]
(*444)
[8+,4]
(8*2)
[8,1+,4]
(*4222)
[8,4+]
(4*4)
[8,4,1+]
(*882)
[(8,4,2+)]
(2*42)
[8,4]+
(842)
=
=
=
=
=
=
h{8,4} s{8,4} hr{8,4} s{4,8} h{4,8} hrr{8,4} sr{8,4}
Alternation duals
V(4.4)4 V3.(3.8)2 V(4.4.4)2 V(3.4)3 V88 V4.44 V3.3.4.3.8
Uniform octaoctagonal tilings
Symmetry: [8,8], (*882)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
{8,8} t{8,8}
r{8,8} 2t{8,8}=t{8,8} 2r{8,8}={8,8} rr{8,8} tr{8,8}
Uniform duals
V88 V8.16.16 V8.8.8.8 V8.16.16 V88 V4.8.4.8 V4.16.16
Alternations
[1+,8,8]
(*884)
[8+,8]
(8*4)
[8,1+,8]
(*4242)
[8,8+]
(8*4)
[8,8,1+]
(*884)
[(8,8,2+)]
(2*44)
[8,8]+
(882)
= = = =
=
=
=
h{8,8} s{8,8} hr{8,8} s{8,8} h{8,8} hrr{8,8} sr{8,8}
Alternation duals
V(4.8)8 V3.4.3.8.3.8 V(4.4)4 V3.4.3.8.3.8 V(4.8)8 V46 V3.3.8.3.8
Uniform (4,4,4) tilings
Symmetry: [(4,4,4)], (*444) [(4,4,4)]+
(444)
[(1+,4,4,4)]
(*4242)
[(4+,4,4)]
(4*22)
t0(4,4,4)
h{8,4}
t0,1(4,4,4)
h2{8,4}
t1(4,4,4)
{4,8}1/2
t1,2(4,4,4)
h2{8,4}
t2(4,4,4)
h{8,4}
t0,2(4,4,4)
r{4,8}1/2
t0,1,2(4,4,4)
t{4,8}1/2
s(4,4,4)
s{4,8}1/2
h(4,4,4)
h{4,8}1/2
hr(4,4,4)
hr{4,8}1/2
Uniform duals
V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V8.8.8 V3.4.3.4.3.4 V88 V(4,4)3
See also
Wikimedia Commons has media related to Uniform tiling 4-8-4-8.
• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch
| Wikipedia |
Tetraapeirogonal tiling
In geometry, the tetraapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of r{∞,4}.
tetraapeirogonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration(4.∞)2
Schläfli symbolr{∞,4} or ${\begin{Bmatrix}\infty \\4\end{Bmatrix}}$
rr{∞,∞} or $r{\begin{Bmatrix}\infty \\\infty \end{Bmatrix}}$
Wythoff symbol2 | ∞ 4
∞ | ∞ 2
Coxeter diagram
or
Symmetry group[∞,4], (*∞42)
[∞,∞], (*∞∞2)
DualOrder-4-infinite rhombille tiling
PropertiesVertex-transitive edge-transitive
Uniform constructions
There are 3 lower symmetry uniform construction, one with two colors of apeirogons, one with two colors of squares, and one with two colors of each:
Symmetry (*∞42)
[∞,4]
(*∞33)
[1+,∞,4] = [(∞,4,4)]
(*∞∞2)
[∞,4,1+] = [∞,∞]
(*∞2∞2)
[1+,∞,4,1+]
Coxeter = = =
Schläfli r{∞,4} r{4,∞}1⁄2 r{∞,4}1⁄2=rr{∞,∞} r{∞,4}1⁄4
Coloring
Dual
Symmetry
The dual to this tiling represents the fundamental domains of *∞2∞2 symmetry group. The symmetry can be doubled by adding mirrors on either diagonal of the rhombic domains, creating *∞∞2 and *∞44 symmetry.
Related polyhedra and tiling
*n42 symmetry mutations of quasiregular tilings: (4.n)2
Symmetry
*4n2
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
[ni,4]
Figures
Config. (4.3)2 (4.4)2 (4.5)2 (4.6)2 (4.7)2 (4.8)2 (4.∞)2 (4.ni)2
Paracompact uniform tilings in [∞,4] family
{∞,4} t{∞,4} r{∞,4} 2t{∞,4}=t{4,∞} 2r{∞,4}={4,∞} rr{∞,4} tr{∞,4}
Dual figures
V∞4 V4.∞.∞ V(4.∞)2 V8.8.∞ V4∞ V43.∞ V4.8.∞
Alternations
[1+,∞,4]
(*44∞)
[∞+,4]
(∞*2)
[∞,1+,4]
(*2∞2∞)
[∞,4+]
(4*∞)
[∞,4,1+]
(*∞∞2)
[(∞,4,2+)]
(2*2∞)
[∞,4]+
(∞42)
=
=
h{∞,4} s{∞,4} hr{∞,4} s{4,∞} h{4,∞} hrr{∞,4} s{∞,4}
Alternation duals
V(∞.4)4 V3.(3.∞)2 V(4.∞.4)2 V3.∞.(3.4)2 V∞∞ V∞.44 V3.3.4.3.∞
Paracompact uniform tilings in [∞,∞] family
=
=
=
=
=
=
=
=
=
=
=
=
{∞,∞} t{∞,∞} r{∞,∞} 2t{∞,∞}=t{∞,∞} 2r{∞,∞}={∞,∞} rr{∞,∞} tr{∞,∞}
Dual tilings
V∞∞ V∞.∞.∞ V(∞.∞)2 V∞.∞.∞ V∞∞ V4.∞.4.∞ V4.4.∞
Alternations
[1+,∞,∞]
(*∞∞2)
[∞+,∞]
(∞*∞)
[∞,1+,∞]
(*∞∞∞∞)
[∞,∞+]
(∞*∞)
[∞,∞,1+]
(*∞∞2)
[(∞,∞,2+)]
(2*∞∞)
[∞,∞]+
(2∞∞)
h{∞,∞} s{∞,∞} hr{∞,∞} s{∞,∞} h2{∞,∞} hrr{∞,∞} sr{∞,∞}
Alternation duals
V(∞.∞)∞ V(3.∞)3 V(∞.4)4 V(3.∞)3 V∞∞ V(4.∞.4)2 V3.3.∞.3.∞
See also
Wikimedia Commons has media related to Uniform tiling 4-i-4-i.
• List of uniform planar tilings
• Tilings of regular polygons
• Uniform tilings in hyperbolic plane
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, "The Hyperbolic Archimedean Tessellations")
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
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• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Polystick
In recreational mathematics, a polystick (or polyedge) is a polyform with a line segment (a 'stick') as the basic shape. A polystick is a connected set of segments in a regular grid. A square polystick is a connected subset of a regular square grid. A triangular polystick is a connected subset of a regular triangular grid. Polysticks are classified according to how many line segments they contain.[1]
The name "polystick" seems to have been first coined by Brian R. Barwell.[2]
The names "polytrig"[3] and "polytwigs"[4] has been proposed by David Goodger to simplify the phrases "triangular-grid polysticks" and "hexagonal-grid polysticks," respectively. Colin F. Brown has used an earlier term "polycules" for the hexagonal-grid polysticks due to their appearance resembling the spicules of sea sponges.[5]
There is no standard term for line segments built on other regular tilings, an unstructured grid, or a simple connected graph, but both "polynema" and "polyedge" have been proposed.[6]
When reflections are considered distinct we have the one-sided polysticks. When rotations and reflections are not considered to be distinct shapes, we have the free polysticks. Thus, for example, there are 7 one-sided square tristicks because two of the five shapes have left and right versions.[7][8]
Square Polysticks
SticksNameFree OEIS: A019988One-Sided OEIS: A151537
1monostick11
2distick22
3tristick57
4tetrastick1625
5pentastick5599
6hexastick222416
7heptastick9501854
Hexagonal Polysticks
SticksNameFree OEIS: A197459One-Sided OEIS: A197460
1monotwig11
2ditwig11
3tritwigs34
4tetratwigs46
5pentatwigs1219
6hexatwigs2749
7heptatwigs78143
Triangular Polysticks
SticksNameFree OEIS: A159867One-Sided OEIS: A151539
1monostick11
2distick33
3tristick1219
4tetrastick60104
5pentastick375719
6hexastick26135123
7heptastick1907437936
The set of n-sticks that contain no closed loops is equivalent, with some duplications, to the set of (n+1)-ominos, as each vertex at the end of every line segment can be replaced with a single square of a polyomino. For example, the set of tristicks is equivalent to the set of Tetrominos. In general, an n-stick with m loops is equivalent to a (n−m+1)-omino (as each loop means that one line segment does not add a vertex to the figure).
Diagram
References
1. Weisstein, Eric W. "Polystick." From MathWorld
2. Brian R. Barwell, "Polysticks," Journal of Recreational Mathematics volume 22 issue 3 (1990), p.165-175
3. David Goodger, "An Introduction to Polytrigs (Triangular-Grid Polysticks)," (2015), http://puzzler.sourceforge.net/docs/polytrigs-intro.html
4. David Goodger, "An Introduction to Polytwigs (Hexagonal-Grid Polysticks)," (2015), http://puzzler.sourceforge.net/docs/polytwigs-intro.html
5. David Goodger, "An Introduction to Polytwigs (Hexagonal-Grid Polysticks)," (2015), http://puzzler.sourceforge.net/docs/polytwigs-intro.html
6. "Polynema -- from Wolfram MathWorld".
7. Weisstein, Eric W. "Polystick." From MathWorld
8. Counting polyforms, at the Solitaire Laboratory
External links
• Polysticks Puzzles & Solutions, at Polyforms Puzzler
• Covering the Aztec Diamond with One-sided Tetrasticks, Alfred Wassermann, University of Bayreuth, Germany
• Polypolylines, at Math Magic
Polyforms
Polyominoes
• Domino
• Tromino
• Tetromino
• Pentomino
• Hexomino
• Heptomino
• Octomino
• Nonomino
• Decomino
Higher dimensions
• Polyominoid
• Polycube
Others
• Polyabolo
• Polydrafter
• Polyhex
• Polyiamond
• Pseudo-polyomino
• Polystick
Games and puzzles
• Blokus
• Soma cube
• Snake cube
• Tangram
• Hexastix
• Tantrix
• Tetris
WikiProject Portal
| Wikipedia |
Complete quadrangle
In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six lines connecting the six pairs of points. Dually, a complete quadrilateral is a system of four lines, no three of which pass through the same point, and the six points of intersection of these lines. The complete quadrangle was called a tetrastigm by Lachlan (1893), and the complete quadrilateral was called a tetragram; those terms are occasionally still used.
Diagonals
The six lines of a complete quadrangle meet in pairs to form three additional points called the diagonal points of the quadrangle. Similarly, among the six points of a complete quadrilateral there are three pairs of points that are not already connected by lines; the line segments connecting these pairs are called diagonals. For points and lines in the Euclidean plane, the diagonal points cannot lie on a single line, and the diagonals cannot have a single point of triple crossing. Due to the discovery of the Fano plane, a finite geometry in which the diagonal points of a complete quadrangle are collinear, some authors have augmented the axioms of projective geometry with Fano's axiom that the diagonal points are not collinear,[1] while others have been less restrictive.
A set of contracted expressions for the parts of a complete quadrangle were introduced by G. B. Halsted: He calls the vertices of the quadrangle dots, and the diagonal points he calls codots. The lines of the projective space are called straights, and in the quadrangle they are called connectors. The "diagonal lines" of Coxeter are called opposite connectors by Halsted. Opposite connectors cross at a codot. The configuration of the complete quadrangle is a tetrastim.[2]
Projective properties
As systems of points and lines in which all points belong to the same number of lines and all lines contain the same number of points, the complete quadrangle and the complete quadrilateral both form projective configurations; in the notation of projective configurations, the complete quadrangle is written as (4362) and the complete quadrilateral is written (6243), where the numbers in this notation refer to the numbers of points, lines per point, lines, and points per line of the configuration. The projective dual of a complete quadrangle is a complete quadrilateral, and vice versa. For any two complete quadrangles, or any two complete quadrilaterals, there is a unique projective transformation taking one of the two configurations into the other.[3]
Karl von Staudt reformed mathematical foundations in 1847 with the complete quadrangle when he noted that a "harmonic property" could be based on concomitants of the quadrangle: When each pair of opposite sides of the quadrangle intersect on a line, then the diagonals intersect the line at projective harmonic conjugate positions. The four points on the line deriving from the sides and diagonals of the quadrangle are called a harmonic range. Through perspectivity and projectivity, the harmonic property is stable. Developments of modern geometry and algebra note the influence of von Staudt on Mario Pieri and Felix Klein .
Euclidean properties
In the Euclidean plane, the four lines of a complete quadrilateral must not include any pairs of parallel lines, so that every pair of lines has a crossing point.
Wells (1991) describes several additional properties of complete quadrilaterals that involve metric properties of the Euclidean plane, rather than being purely projective. The midpoints of the diagonals are collinear, and (as proved by Isaac Newton) also collinear with the center of a conic that is tangent to all four lines of the quadrilateral. Any three of the lines of the quadrilateral form the sides of a triangle; the orthocenters of the four triangles formed in this way lie on a second line, perpendicular to the one through the midpoints. The circumcircles of these same four triangles meet in a point. In addition, the three circles having the diagonals as diameters belong to a common pencil of circles[4] the axis of which is the line through the orthocenters.
The polar circles of the triangles of a complete quadrilateral form a coaxal system.[5]: p. 179
See also
• Newton line
• Nine-point conic
• Quadrilateral
Notes
1. Hartshorne 1967; Coxeter 1987, p. 15.
2. G. B. Halsted (1906) Synthetic Projective Geometry, page 14 via Internet Archive
3. Coxeter 1987, p. 51
4. Wells writes incorrectly that the three circles meet in a pair of points, but, as can be seen in Alexander Bogomolny's animation of the same results, the pencil can be hyperbolic instead of elliptic, in which case the circles do not intersect.
5. Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960).
References
• Coxeter, H. S. M. (1987). Projective Geometry, 2nd ed. Springer-Verlag. ISBN 0-387-96532-7.
• Hartshorne, Robin (1967). Foundations of Projective Geometry. W. A. Benjamin. pp. 53–6.
• Lachlan, Robert (1893). An Elementary Treatise on Modern Pure Geometry. London, New York: Macmillan and Co. Link from Cornell University Historical Math Monographs. See in particular tetrastigm, page 85, and tetragram, page 90.
• Wells, David (1991). The Penguin Dictionary of Curious and Interesting Geometry. Penguin. pp. 35–36. ISBN 0-14-011813-6.
External links
• "Quadrangle, complete", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Bogomolny, Alexander. "The Complete Quadrilateral". Cut-the-Knot.
• Weisstein, Eric W. "Complete Quadrangle". MathWorld.
Incidence structures
Representation
• Incidence matrix
• Incidence graph
Fields
• Combinatorics
• Block design
• Steiner system
• Geometry
• Incidence
• Projective plane
• Graph theory
• Hypergraph
• Statistics
• Blocking
Configurations
• Complete quadrangle
• Fano plane
• Möbius–Kantor configuration
• Pappus configuration
• Hesse configuration
• Desargues configuration
• Reye configuration
• Schläfli double six
• Cremona–Richmond configuration
• Kummer configuration
• Grünbaum–Rigby configuration
• Klein configuration
• Dual
Theorems
• Sylvester–Gallai theorem
• De Bruijn–Erdős theorem
• Szemerédi–Trotter theorem
• Beck's theorem
• Bruck–Ryser–Chowla theorem
Applications
• Design of experiments
• Kirkman's schoolgirl problem
| Wikipedia |
Tetrated dodecahedron
In geometry, the tetrated dodecahedron is a near-miss Johnson solid. It was first discovered in 2002 by Alex Doskey. It was then independently rediscovered in 2003, and named, by Robert Austin.[1]
Tetrated dodecahedron
TypeNear-miss Johnson solid
Faces4 equilateral triangles
12 isosceles triangles
12 pentagons
Edges54
Vertices28
Vertex configuration4 (5.5.5)
12 (3.5.3.5)
12 (3.3.5.5)
Symmetry groupTd
Propertiesconvex
Net
It has 28 faces: twelve regular pentagons arranged in four panels of three pentagons each, four equilateral triangles (shown in blue), and six pairs of isosceles triangles (shown in yellow). All edges of the tetrated dodecahedron have the same length, except for the shared bases of these isosceles triangles, which are approximately 1.07 times as long as the other edges. This polyhedron has tetrahedral symmetry.
Topologically, as a near-miss Johnson solid, the four triangles corresponding to the face planes of a tetrahedron are always equilateral, while the pentagons and the other triangles only have reflection symmetry.
Related polyhedra
Dodecahedron
(Platonic solid)
Icosidodecahedron
(Archimedean solid)
Pentagonal
orthobirotunda
(Johnson solid)
See also
• Tetrahedrally diminished dodecahedron
Notes
1. Tetrated dodecahedra
Near-miss Johnson solids
Truncated forms
• Truncated triakis tetrahedron
• Chamfered cube (Truncated rhombic dodecahedron)
• Chamfered dodecahedron (Truncated rhombic triacontahedron)
Other forms
• Tetrated dodecahedron
• Rectified truncated icosahedron
• Pentahexagonal pyritoheptacontatetrahedron
| Wikipedia |
Tetraview
A tetraview is an attempt to graph a complex function of a complex variable, by a method invented by Davide P. Cervone.
A graph of a real function of a real variable is the set of ordered pairs (x,y) such that y = f(x). This is the ordinary two-dimensional Cartesian graph studied in school algebra.
Every complex number has both a real part and an imaginary part, so one complex variable is two-dimensional and a pair of complex variables is four-dimensional. A tetraview is an attempt to give a picture of a four-dimensional object using a two-dimensional representation—either on a piece of paper or on a computer screen, showing a still picture consisting of five views, one in the center and one at each corner. This is roughly analogous to a picture of a three-dimensional object by giving a front view, a side view, and a view from above.
A picture of a three-dimensional object is a projection of that object from three dimensions into two dimensions. A tetraview is set of five projections, first from four dimensions into three dimensions, and then from three dimensions into two dimensions.
A complex function w = f(z), where z = a + bi and w = c + di are complex numbers, has a graph in four-space (four dimensional space) R4 consisting of all points (a, b, c, d) such that c + di = f(a + bi).
To construct a tetraview, we begin with the four points (1,0,0,0), (0, 1, 0, 0), (0, 0, 1, 0), and (0, 0, 0, 1), which are vertices of a spherical tetrahedron on the unit three-sphere S3 in R4.
We project the four-dimensional graph onto the three-dimensional sphere along one of the four coordinate axes, and then give a two-dimensional picture of the resulting three-dimensional graph. This provides the four corner graph. The graph in the center is a similar picture "taken" from the point of view of the origin.
External links
• http://www.math.union.edu/~dpvc/professional/art/tetra-exp.html
• http://www.maa.org/cvm/1998/01/sbtd/article/tour/tetra-Z3/tetra-Z3.html
| Wikipedia |
Tetromino
A tetromino is a geometric shape composed of four squares, connected orthogonally (i.e. at the edges and not the corners).[1][2] Tetrominoes, like dominoes and pentominoes, are a particular type of polyomino. The corresponding polycube, called a tetracube, is a geometric shape composed of four cubes connected orthogonally.
A popular use of tetrominoes is in the video game Tetris created by the Soviet game designer Alexey Pajitnov, which refers to them as tetriminos.[3] The tetrominoes used in the game are specifically the one-sided tetrominoes.
Types of tetrominoes
Free tetrominoes
Polyominos are formed by joining unit squares along their edges. A free polyomino is a polyomino considered up to congruence. That is, two free polyominos are the same if there is a combination of translations, rotations, and reflections that turns one into the other. A free tetromino is a free polyomino made from four squares. There are five free tetrominoes.
The free tetrominoes have the following symmetry:
• Straight: vertical and horizontal reflection symmetry, and two points of rotational symmetry
• Square: vertical and horizontal reflection symmetry, and four points of rotational symmetry
• T: vertical reflection symmetry only
• L: no symmetry
• S and Z: two points of rotational symmetry only
"straight tetromino"
"square tetromino"
"T-tetromino"
"L-tetromino"
"skew tetromino"
One-sided tetrominoes
One-sided tetrominoes are tetrominoes that may be translated and rotated but not reflected. They are used by, and are overwhelmingly associated with, Tetris. There are seven distinct one-sided tetrominoes. These tetrominoes are named by the letter of the alphabet they most closely resemble. The "I", "O", and "T" tetrominoes have reflectional symmetry, so it does not matter whether they are considered as free tetrominoes or one-sided tetrominoes. The remaining four tetrominoes, "J", "L", "S", and "Z", exhibit a phenomenon called chirality. J and L are reflections of each other, and S and Z are reflections of each other.
As free tetrominoes, J is equivalent to L, and S is equivalent to Z. But in two dimensions and without reflections, it is not possible to transform J into L or S into Z.
I
O
T
J
L
S
Z
Fixed tetrominoes
The fixed tetrominoes allow only translation, not rotation or reflection. There are two distinct fixed I-tetrominoes, four J, four L, one O, two S, four T, and two Z, for a total of 19 fixed tetrominoes:
Tiling a rectangle
Filling a rectangle with one set of tetrominoes
A single set of free tetrominoes or one-sided tetrominoes cannot fit in a rectangle. This can be shown with a proof similar to the mutilated chessboard argument. A 5×4 rectangle with a checkerboard pattern has 20 squares, containing 10 light squares and 10 dark squares, but a complete set of free tetrominoes has either 11 dark squares and 9 light squares, or 11 light squares and 9 dark squares. This is due to the T tetromino having either 3 dark squares and one light square, or 3 light squares and one dark square, while all other tetrominoes each have 2 dark squares and 2 light squares. Similarly, a 7×4 rectangle has 28 squares, containing 14 squares of each shade, but the set of one-sided tetrominoes has either 15 dark squares and 13 light squares, or 15 light squares and 13 dark squares. By extension, any odd number of sets for either type cannot fit in a rectangle. Additionally, the 19 fixed tetrominoes cannot fit in a 4×19 rectangle. This was discovered by exhausting all possibilities in a computer search.
The free tetrominoes (left side of line) have 11 dark squares and 9 light squares.
The one-sided tetrominoes (all 7 shown above) have 15 dark squares and 13 light squares.
A 5×4 board has 10 squares each color.
A 7×4 board has 14 squares each color.
Filling a modified rectangle with one set of tetrominoes
However, all three sets of tetrominoes can fit rectangles with holes:
• All 5 free tetrominoes fit a 7×3 rectangle with a hole.
• All 7 one-sided tetrominoes fit a 6×5 rectangle with two holes of the same "checkerboard color".
• All 19 fixed tetrominoes fit a 11×7 rectangle with a hole.
Free tetrominoes in a rectangle with one hole
One-sided tetrominoes in a rectangle with two holes
Fixed tetrominoes in rectangle with one hole
Filling a rectangle with two sets of tetrominoes
Two sets of free or one-sided tetrominoes can fit into a rectangle in different ways, as shown below:
Two sets of free tetrominoes in a 5×8 rectangle
Two sets of free tetrominoes in a 4×10 rectangle
Two sets of one-sided tetrominoes in a 8×7 rectangle
Two sets of one-sided tetrominoes in a 14×4 rectangle
Etymology
The name "tetromino" is a combination of the prefix tetra- 'four' (from Ancient Greek τετρα-), and "domino". The name was introduced by Solomon W. Golomb in 1953 along with other nomenclature related to polyominos.[4][1]
Filling a box with tetracubes
Each of the five free tetrominoes has a corresponding tetracube, which is the tetromino extruded by one unit. J and L are the same tetracube, as are S and Z, because one may be rotated around an axis parallel to the tetromino's plane to form the other. Three more tetracubes are possible, all created by placing a unit cube on the bent tricube:
I
"straight tetracube"
O
"square tetracube"
T
"T-tetracube"
L
"L-tetracube"
J is the same as L in 3D
S
"skew tetracube"
Z is the same as S in 3D
B
"Branch"
D
"Right Screw"
F
"Left Screw"
The tetracubes can be packed into two-layer 3D boxes in several different ways, based on the dimensions of the box and criteria for inclusion. They are shown in both a pictorial diagram and a text diagram. For boxes using two sets of the same pieces, the pictorial diagram depicts each set as a lighter or darker shade of the same color. The text diagram depicts each set as having a capital or lower-case letter. In the text diagram, the top layer is on the left, and the bottom layer is on the right.
1.) 2×4×5 box filled with two sets of free tetrominoes:
Z Z T t I l T T T i
L Z Z t I l l l t i
L z z t I o o z z i
L L O O I o o O O i
2.) 2×2×10 box filled with two sets of free tetrominoes:
L L L z z Z Z T O O o o z z Z Z T T T l
L I I I I t t t O O o o i i i i t l l l
3.) 2×4×4 box filled with one set of all tetrominoes:
F T T T F Z Z B
F F T B Z Z B B
O O L D L L L D
O O D D I I I I
4.) 2×2×8 box filled with one set of all tetrominoes:
D Z Z L O T T T D L L L O B F F
D D Z Z O B T F I I I I O B B F
5.) 2×2×7 box filled with tetrominoes, with mirror-image pieces removed:
L L L Z Z B B L C O O Z Z B
C I I I I T B C C O O T T T
See also
• Soma cube
Previous and next orders
• Tromino
• Pentomino
References
1. Golomb, Solomon W. (1994). Polyominoes (2nd ed.). Princeton, New Jersey: Princeton University Press. ISBN 0-691-02444-8.
2. Redelmeier, D. Hugh (1981). "Counting polyominoes: yet another attack". Discrete Mathematics. 36 (2): 191–203. doi:10.1016/0012-365X(81)90237-5.
3. "About Tetris", Tetris.com. Retrieved 2014-04-19.
4. Darling, David. "Polyomino". daviddarling.info. Retrieved May 23, 2020.
External links
• Vadim Gerasimov, "Tetris: the story."; The story of Tetris
• The Father of Tetris (Web Archive copy of the page here)
Tetris
Main games
Console
• Tetris (Atari Games)
• Tetris (NES)
• Tetris Classic
• Tetris Battle Gaiden
• Tetris & Dr. Mario
• Tetris Plus
• The Grand Master
• Magical Tetris Challenge
• Tetris 64
• The New Tetris
• The Next Tetris
• Tetris Worlds
• Tetris Evolution
• Tetris Zone
• Tetris Splash
• Tetris Friends
• Tetris Party
• Tetris Ultimate
• Puyo Puyo Tetris
• Tetris Effect
• Tetris 99
• Puyo Puyo Tetris 2
Handheld
• Tetris (Game Boy)
• V-Tetris
• 3D Tetris
• Tetris DS
• Tetris (EA)
• Tetris: Axis
Variants
• Blockout
• Welltris
• Hatris
• Faces...tris III
• Wordtris
• Tetris 2 (Nintendo)
• Tetris Attack
• TetriNET
• Tetrisphere
• BreakThru!
• Pokémon Tetris
• Tetris Giant
People
• Thor Aackerlund
• Minoru Arakawa
• Vadim Gerasimov
• Jonas Neubauer
• Alexey Pajitnov
• Vladimir Pokhilko
• Henk Rogers
Related
• Classic Tetris World Championship
• Ecstasy of Order: The Tetris Masters
• ELORG
• "Korobeiniki"
• Apple TV+ film
• The Tetris Company
• Blue Planet Software
• Tetris effect
• Tetris Holding, LLC v. Xio Interactive, Inc.
• Tetris Online, Inc.
• Tetris: The Games People Play
• Tetromino
• Category
Polyforms
Polyominoes
• Domino
• Tromino
• Tetromino
• Pentomino
• Hexomino
• Heptomino
• Octomino
• Nonomino
• Decomino
Higher dimensions
• Polyominoid
• Polycube
Others
• Polyabolo
• Polydrafter
• Polyhex
• Polyiamond
• Pseudo-polyomino
• Polystick
Games and puzzles
• Blokus
• Soma cube
• Snake cube
• Tangram
• Hexastix
• Tantrix
• Tetris
WikiProject Portal
| Wikipedia |
Tetsuji Shioda
Tetsuji Shioda (塩田 徹治) is a Japanese mathematician who introduced Shioda modular surfaces and who used Mordell–Weil lattices to give examples of dense sphere packings. He was an invited speaker at the ICM in 1990.
References
• Home page of Tetsuji Shioda
• Tetsuji Shioda at the Mathematics Genealogy Project
• Tetsuji Shioda in the Oberwolfach photo collection
Authority control
International
• ISNI
• VIAF
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• France
• BnF data
• Germany
• Israel
• Belgium
• United States
• Japan
• Netherlands
Academics
• MathSciNet
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• zbMATH
Other
• IdRef
| Wikipedia |
Tevian Dray
Tevian Dray (born March 17, 1956) is an American mathematician who has worked in general relativity, mathematical physics, geometry, and both science and mathematics education. He was elected a Fellow of the American Physical Society in 2010.
Tevian Dray
Born (1956-03-17) March 17, 1956
Washington, DC, United States
Nationality United States
Alma materMassachusetts Institute of Technology BS 1976; University of California, Berkeley Ph.D 1981
SpouseCorinne A. Manogue
AwardsHaimo Distinguished Teaching Award, Mathematical Association of America, 2017
Scientific career
InstitutionsOregon State University
Doctoral advisorRainer K. Sachs
He has primarily worked in the area of classical general relativity. His research results include confirmation of the existence of solutions of Einstein's equation containing gravitational radiation, the use of computer algebra to classify exact solutions of Einstein's equation, an analysis of a class of gravitational shock waves (including one of the few known exact 2-body solutions in general relativity), and the study of signature change, a possible model for the Big Bang. More recently, his work has focused on applications of the octonions to the theory of fundamental particles.
He was a graduate student under Rainer K. Sachs at Berkeley, where he received his Ph.D. in 1981, although much of his dissertation research was done in collaboration with Abhay Ashtekar. The context of his dissertation, titled The Asymptotic Structure of a Family of Einstein-Maxwell Solutions focused on families of spacetimes which describe accelerating black holes, and which contain gravitational radiation. This demonstrated the existence of exact radiating solutions to the Einstein field equations.[1]
He is currently a professor of mathematics at Oregon State University. In addition to his ongoing work in mathematical physics, he has made significant contributions in science education, where he directs the Vector Calculus Bridge Project, [2] an attempt to teach vector calculus the way it is used by scientists and engineers, and is part of the development team of the Paradigms Project, [3] a complete restructuring of the undergraduate physics major around several core "paradigms". He has written a book [4] on special relativity and a sequel on general relativity using differential forms. ,[5] and is coauthor of The Geometry of the Octonions released in 2015.[6]
Bibliography
• Abhay Ashtekar & Tevian Dray (1981). "On the Existence of Solutions to Einstein's Equation with Non-Zero Bondi News". Commun. Math. Phys. 79 (4): 581–589. Bibcode:1981CMaPh..79..581A. doi:10.1007/BF01209313. S2CID 121427482.
• Tevian Dray & Gerard 't Hooft (1985). "The Effect of Spherical Shells of Matter on the Schwarzschild Black Hole". Commun. Math. Phys. 99 (4): 613–625. Bibcode:1985CMaPh..99..613D. doi:10.1007/BF01215912. hdl:1874/4753. S2CID 122717417.
• Paul C. W. Davies; Tevian Dray & Corinne A. Manogue (1996). "Detecting the Rotating Quantum Vacuum". Phys. Rev. D. 53 (8): 4382–4387. arXiv:gr-qc/9601034. Bibcode:1996PhRvD..53.4382D. doi:10.1103/PhysRevD.53.4382. PMID 10020436. S2CID 2114187.
• Tevian Dray; George Ellis; Charles Hellaby & Corinne A. Manogue (1997). "Gravity and Signature Change". Gen. Rel. Grav. 29 (5): 591–597. arXiv:gr-qc/9610063. Bibcode:1997GReGr..29..591D. doi:10.1023/A:1018895302693. S2CID 7617543.
• (2012) Tevian Dray, The Geometry of Special Relativity (A K Peters/CRC Press) ISBN 978-1466510470[7]
• (2014) Tevian Dray, Differential Forms and The Geometry of General Relativity (A K Peters/CRC Press) ISBN 978-1466510005[8]
• (2015) Tevian Dray and Corinne A. Manogue, The Geometry of the Octonions (World Scientific) ISBN 978-9814401814[6]
References
1. "Tevian Dray's Dissertation".
2. "Bridging the Vector Calculus Gap".
3. "Start - Portfolios Wiki".
4. "Bookinfo - Geometry of Special Relativity".
5. "Differential Forms and the Geometry of General Relativity".
6. Reviews of The Geometry of the Octonions:
• Elduque, Alberto (2015), Mathematical Reviews, doi:10.1142/8456, ISBN 978-981-4401-81-4, MR 3361898{{citation}}: CS1 maint: untitled periodical (link)
• Brezov, Danail (2015), "Review" (PDF), J. Geom. Symmetry Phys., 39: 99–101
• Hunacek, Mark (June 2015), "Review", MAA Reviews
7. "The Geometry of Special Relativity". A K Peters/CRC Press. Retrieved 17 April 2014.
8. "Differential Forms and The Geometry of General Relativity". A K Peters/CRC Press. Retrieved 4 January 2015.
External links
• Tevian Dray's home page
• Vector Calculus Bridge Project
• Paradigms Project
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| Wikipedia |
Texas Math and Science Coaches Association
The Texas Math and Science Coaches Association or TMSCA is an organization for coaches of academic University Interscholastic League teams in Texas middle schools and high schools, specifically those that compete in mathematics and science-related tests.
Events
There are four events in the TMSCA at both the middle and high school level: Number Sense, General Mathematics, Calculator Applications, and General Science.
Number Sense is an 80-question exam that students are given only 10 minutes to solve. Additionally, no scratch work or paper calculations are allowed. These questions range from simple calculations such as 99+98 to more complicated operations such as 1001×1938. Each calculation is able to be done with a certain trick or shortcut that makes the calculations easier. The high school exam includes calculus and other difficult topics in the questions also with the same rules applied as to the middle school version. It is well known that the grading for this event is particularly stringent as errors such as writing over a line or crossing out potential answers are considered as incorrect answers.
General Mathematics is a 50-question exam that students are given only 40 minutes to solve. These problems are usually more challenging than questions on the Number Sense test, and the General Mathematics word problems take more thinking to figure out. Every problem correct is worth 5 points, and for every problem incorrect, 2 points are deducted. Tiebreakers are determined by the person that misses the first problem and by percent accuracy.
Calculator Applications is an 80-question exam that students are given only 30 minutes to solve. This test requires practice on the calculator, knowledge of a few crucial formulas, and much speed and intensity. Memorizing formulas, tips, and tricks will not be enough. In this event, plenty of practice is necessary in order to master the locations of the keys and develop the speed necessary. All correct questions are worth 5 points and all incorrect questions or skipped questions that are before the last answered questions will lose 4 points; answers are to be given with three significant figures.
Science is a 50-question exam that is solved in 40 minutes at the middle school level or a 60-question exam that is solved in a 2-hour time limit at the high school level. Tiebreakers are determined by the person that misses the first problem and by percent accuracy. As the name suggests, the test focuses on the science subjects learned in the middle school or high school level depending on the student's grade and the version of the test being taken.
Competitions
Individual schools that are members of TMSCA can host invitational competitions using TMSCA-released tests. Many schools use this as a fund-raising opportunity for their competitive math program.
TMSCA also hosts two statewide competitions for member schools each year, one at the middle school level and one at the high school level, as well as a qualification competition at the middle school level prior to the state competition, also known as the Regional Qualifier. These statewide competitions are held at the University of Texas at San Antonio campus each spring. These competitions can often serve as practice for statewide UIL tournaments, which occur shortly after, and for middle school students are their only opportunity to compete at the state level (UIL competitions at the middle school level do not go beyond district). At the statewide competition, students have the opportunity to win scholarships based on their performance at the meet.
Grading
For the General Mathematics and General Science contests in middle school, 5 points are awarded for each correct answer and 2 points are deducted for each incorrect answer. In the high school contest, 6 points are awarded for each correct answer and 2 points are deducted for each incorrect answer. The real way to calculate the score is to multiply the number of questions you attempted by 5 and subtract 7 for each incorrect question. Unanswered questions do not affect the score. Thus, competitors are penalized for guessing incorrectly. (For both General Mathematics and General Science a perfect score is a 250.)
On the Number Sense test, scoring is 5 times the last question answered (a student answering 32 questions would be awarded 160 points) and after that 9 points are deducted for incorrect answers, problems skipped up to the last attempted question and markovers/erasures, (so if the student above missed one and skipped three questions the student would end up with 124 points). Number sense tests are also checked for possible scratch work, overwrites, and erasures (bluntly called "markovers"), which if found could result in questions being counted as incorrect or tests being disqualified. (For both Number Sense and Calculator, a perfect score is a 400.)
The Calculator Applications test multiplies 5 times the last question answered and deducts 9 points for incorrect or skipped questions, similar to Number Sense, but scratch work, markovers/erasures, and the use of a calculator is allowed.
Results
At almost all TMSCA competitions, students are ranked against each other in their specific grade level. For example, all eighth graders compete against each other, all seventh graders compete against each other, and so on and so forth. This ensures parity of competition since students in higher grades generally tend to score higher than students in the lower grades. Particularly at the high school level, there is a stark contrast between freshmen with little real math and science experience and seniors, who presumably have taken or are taking advanced placement science courses and calculus.
References
External links
• Official website
| Wikipedia |
Thabit number
In number theory, a Thabit number, Thâbit ibn Qurra number, or 321 number is an integer of the form $3\cdot 2^{n}-1$ for a non-negative integer n.
Thabit prime
Named afterThābit ibn Qurra
Conjectured no. of termsInfinite
Subsequence ofThabit numbers
First terms2, 5, 11, 23, 47, 191, 383, 6143, 786431
OEIS indexA007505
The first few Thabit numbers are:
2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, ... (sequence A055010 in the OEIS)
The 9th century mathematician, physician, astronomer and translator Thābit ibn Qurra is credited as the first to study these numbers and their relation to amicable numbers.[1]
Properties
The binary representation of the Thabit number 3·2n−1 is n+2 digits long, consisting of "10" followed by n 1s.
The first few Thabit numbers that are prime (Thabit primes or 321 primes):
2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, ... (sequence A007505 in the OEIS)
As of July 2023, there are 67 known prime Thabit numbers. Their n values are:[2][3][4][5]
0, 1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 55, 64, 76, 94, 103, 143, 206, 216, 306, 324, 391, 458, 470, 827, 1274, 3276, 4204, 5134, 7559, 12676, 14898, 18123, 18819, 25690, 26459, 41628, 51387, 71783, 80330, 85687, 88171, 97063, 123630, 155930, 164987, 234760, 414840, 584995, 702038, 727699, 992700, 1201046, 1232255, 2312734, 3136255, 4235414, 6090515, 11484018, 11731850, 11895718, 16819291, 17748034, 18196595, 18924988, 20928756, ... (sequence A002235 in the OEIS)
The primes for 234760 ≤ n ≤ 3136255 were found by the distributed computing project 321 search.[6]
In 2008, PrimeGrid took over the search for Thabit primes.[7] It is still searching and has already found all currently known Thabit primes with n ≥ 4235414.[8] It is also searching for primes of the form 3·2n+1, such primes are called Thabit primes of the second kind or 321 primes of the second kind.
The first few Thabit numbers of the second kind are:
4, 7, 13, 25, 49, 97, 193, 385, 769, 1537, 3073, 6145, 12289, 24577, 49153, 98305, 196609, 393217, 786433, 1572865, ... (sequence A181565 in the OEIS)
The first few Thabit primes of the second kind are:
7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657, 221360928884514619393, ... (sequence A039687 in the OEIS)
Their n values are:
1, 2, 5, 6, 8, 12, 18, 30, 36, 41, 66, 189, 201, 209, 276, 353, 408, 438, 534, 2208, 2816, 3168, 3189, 3912, 20909, 34350, 42294, 42665, 44685, 48150, 54792, 55182, 59973, 80190, 157169, 213321, 303093, 362765, 382449, 709968, 801978, 916773, 1832496, 2145353, 2291610, 2478785, 5082306, 7033641, 10829346, 16408818, ... (sequence A002253 in the OEIS)
Connection with amicable numbers
When both n and n−1 yield Thabit primes (of the first kind), and $9\cdot 2^{2n-1}-1$ is also prime, a pair of amicable numbers can be calculated as follows:
$2^{n}(3\cdot 2^{n-1}-1)(3\cdot 2^{n}-1)$ and $2^{n}(9\cdot 2^{2n-1}-1).$
For example, n = 2 gives the Thabit prime 11, and n−1 = 1 gives the Thabit prime 5, and our third term is 71. Then, 22=4, multiplied by 5 and 11 results in 220, whose divisors add up to 284, and 4 times 71 is 284, whose divisors add up to 220.
The only known n satisfying these conditions are 2, 4 and 7, corresponding to the Thabit primes 11, 47 and 383 given by n, the Thabit primes 5, 23 and 191 given by n−1, and our third terms are 71, 1151 and 73727. (The corresponding amicable pairs are (220, 284), (17296, 18416) and (9363584, 9437056))
Generalization
For integer b ≥ 2, a Thabit number base b is a number of the form (b+1)·bn − 1 for a non-negative integer n. Also, for integer b ≥ 2, a Thabit number of the second kind base b is a number of the form (b+1)·bn + 1 for a non-negative integer n.
The Williams numbers are also a generalization of Thabit numbers. For integer b ≥ 2, a Williams number base b is a number of the form (b−1)·bn − 1 for a non-negative integer n.[9] Also, for integer b ≥ 2, a Williams number of the second kind base b is a number of the form (b−1)·bn + 1 for a non-negative integer n.
For integer b ≥ 2, a Thabit prime base b is a Thabit number base b that is also prime. Similarly, for integer b ≥ 2, a Williams prime base b is a Williams number base b that is also prime.
Every prime p is a Thabit prime of the first kind base p, a Williams prime of the first kind base p+2, and a Williams prime of the second kind base p; if p ≥ 5, then p is also a Thabit prime of the second kind base p−2.
It is a conjecture that for every integer b ≥ 2, there are infinitely many Thabit primes of the first kind base b, infinitely many Williams primes of the first kind base b, and infinitely many Williams primes of the second kind base b; also, for every integer b ≥ 2 that is not congruent to 1 modulo 3, there are infinitely many Thabit primes of the second kind base b. (If the base b is congruent to 1 modulo 3, then all Thabit numbers of the second kind base b are divisible by 3 (and greater than 3, since b ≥ 2), so there are no Thabit primes of the second kind base b.)
The exponent of Thabit primes of the second kind cannot congruent to 1 mod 3 (except 1 itself), the exponent of Williams primes of the first kind cannot congruent to 4 mod 6, and the exponent of Williams primes of the second kind cannot congruent to 1 mod 6 (except 1 itself), since the corresponding polynomial to b is a reducible polynomial. (If n ≡ 1 mod 3, then (b+1)·bn + 1 is divisible by b2 + b + 1; if n ≡ 4 mod 6, then (b−1)·bn − 1 is divisible by b2 − b + 1; and if n ≡ 1 mod 6, then (b−1)·bn + 1 is divisible by b2 − b + 1) Otherwise, the corresponding polynomial to b is an irreducible polynomial, so if Bunyakovsky conjecture is true, then there are infinitely many bases b such that the corresponding number (for fixed exponent n satisfying the condition) is prime. ((b+1)·bn − 1 is irreducible for all nonnegative integer n, so if Bunyakovsky conjecture is true, then there are infinitely many bases b such that the corresponding number (for fixed exponent n) is prime)
Pierpont numbers $3^{m}\cdot 2^{n}+1$ are a generalization of Thabit numbers of the second kind $3\cdot 2^{n}+1$.
References
1. Rashed, Roshdi (1994). The development of Arabic mathematics: between arithmetic and algebra. Vol. 156. Dordrecht, Boston, London: Kluwer Academic Publishers. p. 277. ISBN 0-7923-2565-6.
2. http://www.mersenneforum.org/321search/How%20many%20digits%20these%20primes%20have.html
3. "PrimePage Primes: 3 · 2^4235414 - 1".
4. http://primes.utm.edu/primes/lists/short.txt
5. PrimeGrid Primes search for 3*2^n - 1.
6. http://www.mersenneforum.org/321search/The%20status%20of%20the%20search.html
7. "PrimePage Bios: 321search".
8. http://primes.utm.edu/primes/lists/short.txt
9. List of Williams primes (of the first kind) base 3 to 2049 (for exponent ≥ 1)
External links
• Weisstein, Eric W. "Thâbit ibn Kurrah Number". MathWorld.
• Weisstein, Eric W. "Thâbit ibn Kurrah Prime". MathWorld.
• Chris Caldwell, The Largest Known Primes Database at The Prime Pages
• A Thabit prime of the first kind base 2: (2+1)·211895718 − 1
• A Thabit prime of the second kind base 2: (2+1)·210829346 + 1
• A Williams prime of the first kind base 2: (2−1)·274207281 − 1
• A Williams prime of the first kind base 3: (3−1)·31360104 − 1
• A Williams prime of the second kind base 3: (3−1)·31175232 + 1
• A Williams prime of the first kind base 10: (10−1)·10383643 − 1
• A Williams prime of the first kind base 113: (113−1)·113286643 − 1
• List of Williams primes
• PrimeGrid’s 321 Prime Search, about the discovery of the Thabit prime of the first kind base 2: (2+1)·26090515 − 1
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| Wikipedia |
Ted Hurley
Ted Hurley (born Thaddeus C. Hurley in 1944) is an Irish mathematician specialising in algebra, specifically in group theory, group rings, cryptography, coding theory, and computer algebra. Most of his academic career was spent at University College Galway (later renamed National University of Ireland Galway, or simply NUI Galway). He was Head of Discipline of Mathematics there from 1996 to 2010.[1]
Education
Ted Hurley was born in September 1945 in Tuam, Co. Galway, Ireland, to James Hurley and Bridget Walsh. He earned his BSc (1965) and MSc (1966) from University College Galway (UCG), also winning the Peel Prize in Geometry and the Sir Joseph Larmor Prize. He was awarded a National University of Ireland Travelling Studentship Prize (1966), and was then appointed a Tutorial Research Fellow at Royal Holloway College, University of London, while conducting his doctoral research at nearby Queen Mary College. His 1970 thesis on "Representations of Some Relatively Free Groups in Power Series Rings" was done under Karl W. Gruenberg.[2]
Career
Hurley taught at Imperial College in London (1970-1971) and then at the University of Sheffield (1971-1974), before returning to Ireland. Hurley was a founding member of the Irish Mathematical Society in 1976, and served as its inaugural secretary (1977-1979).[3] After six years on the staff at University College Dublin (UCD), in 1980 he secured a position at his alma mater, University College Galway (later known as the National University of Ireland Galway), from which he officially retired in 2010. He was Lecturer in Mathematics there from 1980 to 1988, Associate Professor from 1988 to 1996, and Professor of Mathematics from 1996 on. He was also Head of Discipline of Mathematics from 1996 to 2010. He has been active in the years since formal retirement, publishing frequently.
He has also been a vocal public commentator on mathematics' education, including the importance of numeracy and mathematics to our lives, in the Irish print media[4] [5] and has also discussed these issues on popular national radio shows.[6]
Mathematics
Hurley's work was originally mostly in group theory, specifically on structural features of infinite groups (relatively free groups, commutators and powers in groups), and also group rings. Later, his interests expanded to include algebraic coding theory and cryptography.[7]
He has supervised three PhD students[2] and has co-edited several conference proceedings.[8]
Selected papers
• 2021 "Unique builders for classes of matrices Special Matrices", Hurley, Ted. Special Matrices, vol. 9, no. 1, 2021, pp. 52-65.
• 2018 "Coding theory: the unit-derived methodology". Hurley T., Hurley D., International Journal of Information and Coding Theory, 5 (1):55-80
• 2018 "Quantum error-correcting codes: the unit design strategy". Hurley T., Hurley D., Hurley B. International Journal of Information and Coding Theory, 5 (2):169-182
• 2017 "Solving underdetermined systems with error-correcting codes". Hurley, T. International Journal of Information and Coding Theory, 4 (4)
• 2014 "Cryptographic schemes, key exchange, public key". Hurley, Ted. International Journal Of Pure And Applied Mathematics, 6 (93):897-927
• 2014 "Algebraic Structures for Communications". Hurley, Ted (2014). Contemporary Mathematics, (611):59-78
• 2014 "Systems of MDS codes from units and idempotents". Hurley, Barry and Hurley, Ted (2014). Discrete Mathematics, (335):81-91
• 2011 "Group ring cryptography". Hurley, B.,Hurley, T. (2011). Int. J. Pure Appl. Math, 69 (1):67-86
• 2009 "Convolutional codes from units in matrix and group rings". Hurley, T. (2009). Int. J. Pure Appl. Math, 50 (3):431-463]
• 2006 "Group Rings And Rings Of Matrices". Hurley T., Hurley D., International Journal of Information and Coding Theory, Vol 31 No. 3 2006, 319-335
• 2000 "Groups related to Fox subgroups". Hurley, T,Sehgal, S (2000). Communications In Algebra, 28 :1051-1059
• 1996 "The modular Fox subgroups". Hurley, T.C.,Sehgal, S.K. (1996). Commun. Algebra, 24 (14):4563-4580
• 1991 "The Lie Dimension Subgroup Conjecture*", Thaddeus C. Hurley and Sudarshan K. Sehgal, Journal Of Algebra 143, 46-56
• 1990 "On the Class of the Stability Group of a Series of Subgroups", Ted Hurley. Journal of the London Mathematical Society s2-41(1)
• 1986 "On commutators and powers in groups II". Hurley, T.C. (1986). Arch. Math, 46 (5):385-386
Conference proceedings edited
• Groups '93 Galway/St Andrews (1995, in 2 volumes), selected papers from the international conference "Groups 1993 Galway/St Andrews", University College Galway, August 1993. Edited by T. C. Hurley, S. J. Tobin, J. J. Ward, C. M. Campbell & E. F. Robertson, Cambridge (LMS Lecture Note Series 211).
• The First Irish Conference on the Mathematical Foundations of Computer Science and Information Technology (MFCSIT2000) (2002), papers from conference held in Cork, Ireland, 20th and 21st July, 2000. Edited by Ted Hurley, Micheal Mac an Airchinnigh, Michel Schellekens and Anthony Seda. Elsevier Science (Electronic Notes in Theoretical Computer Science, Vol 40)
• The Second Irish Conference on the Mathematical Foundations of Computer Science and Information Technology (MFCSIT2002) (2003), papers from conference held in Galway, Ireland, 18th and 19th of July, 2002. Edited by Ted Hurley, Sharon Flynn, Micheal Mac an Airchinnigh, Niall Madden, Michael McGettrick, Michel Schellekens and Anthony Seda. Elsevier Science (Electronic Notes in Theoretical Computer Science, Vol 74)
• The Third Irish Conference on the Mathematical Foundations of Computer Science and Information Technology (MFCSIT2004) (2006), papers from conference held in Dublin, Ireland, 22nd and 23rd July, 2004. Edited by Ted Hurley, Micheal Mac an Airchinnigh, Michel Schellekens, Anthony Seda & Glenn Strong. Elsevier Science (Electronic Notes in Theoretical Computer Science, Vol 161)
• The Fourth Irish Conference on the Mathematical Foundations of Computer Science and Information Technology (MFCSIT2006) (2009), papers from conference held at University College Cork, Ireland, 1st to 5th August July 2006. Edited by Ted Hurley, Micheal Mac an Airchinnigh, Michel Schellekens, Anthony Seda, Glenn Strong and Menouer Boubekeur. Elsevier Science (Electronic Notes in Theoretical Computer Science, Vol 225)
References
1. Mathematics, Statistics & Applied Mathematics: Professor Ted Hurley National University of Ireland Galway
2. Thaddeus (Ted) Christopher Hurley at the Mathematics Genealogy Project
3. The Origins of the Irish Mathematical Society by T. T. West, Irish Math. Soc. Bulletin 51 (2003), 73–75 73
4. ‘I’m useless at maths’ should never be a boast, by Ted Hurley, The Irish Times, 16 Feb 2015
5. The war on rote learning just doesn’t add up, by Ted Hurley, The Irish Times, 11 May 2015
6. Maths graduates ideally placed for dynamism by Anne Byrne, The Irish Times, 05 March, 1996
7. T C Hurley papers MacTutor
8. British Mathematical Colloquium held at Galway: 6-9 April 2009: Organized by T Hurley MacTutor
External links
• Ted Hurley at the Mathematics Genealogy Project
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
| Wikipedia |
Book (graph theory)
In graph theory, a book graph (often written $B_{p}$ ) may be any of several kinds of graph formed by multiple cycles sharing an edge.
Not to be confused with book embedding.
Variations
One kind, which may be called a quadrilateral book, consists of p quadrilaterals sharing a common edge (known as the "spine" or "base" of the book). That is, it is a Cartesian product of a star and a single edge.[1][2] The 7-page book graph of this type provides an example of a graph with no harmonious labeling.[2]
A second type, which might be called a triangular book, is the complete tripartite graph K1,1,p. It is a graph consisting of $p$ triangles sharing a common edge.[3] A book of this type is a split graph. This graph has also been called a $K_{e}(2,p)$[4] or a thagomizer graph (after thagomizers, the spiked tails of stegosaurian dinosaurs, because of their pointy appearance in certain drawings) and their graphic matroids have been called thagomizer matroids.[5] Triangular books form one of the key building blocks of line perfect graphs.[6]
The term "book-graph" has been employed for other uses. Barioli[7] used it to mean a graph composed of a number of arbitrary subgraphs having two vertices in common. (Barioli did not write $B_{p}$ for his book-graph.)
Within larger graphs
Given a graph $G$, one may write $bk(G)$ for the largest book (of the kind being considered) contained within $G$.
Theorems on books
Denote the Ramsey number of two triangular books by $r(B_{p},\ B_{q}).$ This is the smallest number $r$ such that for every $r$-vertex graph, either the graph itself contains $B_{p}$ as a subgraph, or its complement graph contains $B_{q}$ as a subgraph.
• If $1\leq p\leq q$, then $r(B_{p},\ B_{q})=2q+3$.[8]
• There exists a constant $c=o(1)$ such that $r(B_{p},\ B_{q})=2q+3$ whenever $q\geq cp$.
• If $p\leq q/6+o(q)$, and $q$ is large, the Ramsey number is given by $2q+3$.
• Let $C$ be a constant, and $k=Cn$. Then every graph on $n$ vertices and $m$ edges contains a (triangular) $B_{k}$.[9]
References
1. Weisstein, Eric W. "Book Graph". MathWorld.
2. Gallian, Joseph A. (1998). "A dynamic survey of graph labeling". Electronic Journal of Combinatorics. 5: Dynamic Survey 6. MR 1668059.
3. Lingsheng Shi; Zhipeng Song (2007). "Upper bounds on the spectral radius of book-free and/or K2,l-free graphs". Linear Algebra and Its Applications. 420 (2–3): 526–9. doi:10.1016/j.laa.2006.08.007.
4. Erdős, Paul (1963). "On the structure of linear graphs". Israel Journal of Mathematics. 1 (3): 156–160. doi:10.1007/BF02759702.
5. Gedeon, Katie R. (2017). "Kazhdan-Lusztig polynomials of thagomizer matroids". Electronic Journal of Combinatorics. 24 (3). Paper 3.12. arXiv:1610.05349. doi:10.37236/6567. MR 3691529. S2CID 23424650.; Xie, Matthew H. Y.; Zhang, Philip B. (2019). "Equivariant Kazhdan-Lusztig polynomials of thagomizer matroids". Proceedings of the American Mathematical Society. 147 (11): 4687–4695. doi:10.1090/proc/14608. MR 4011505.; Proudfoot, Nicholas; Ramos, Eric (2019). "Functorial invariants of trees and their cones". Selecta Mathematica. New Series. 25 (4). Paper 62. arXiv:1903.10592. doi:10.1007/s00029-019-0509-4. MR 4021848. S2CID 85517485.
6. Maffray, Frédéric (1992). "Kernels in perfect line-graphs". Journal of Combinatorial Theory. Series B. 55 (1): 1–8. doi:10.1016/0095-8956(92)90028-V. MR 1159851..
7. Barioli, Francesco (1998). "Completely positive matrices with a book-graph". Linear Algebra and Its Applications. 277 (1–3): 11–31. doi:10.1016/S0024-3795(97)10070-2.
8. Rousseau, C. C.; Sheehan, J. (1978). "On Ramsey numbers for books". Journal of Graph Theory. 2 (1): 77–87. doi:10.1002/jgt.3190020110. MR 0486186.
9. Erdős, P. (1962). "On a theorem of Rademacher-Turán". Illinois Journal of Mathematics. 6: 122–7. doi:10.1215/ijm/1255631811.
| Wikipedia |
Thaine's theorem
In mathematics, Thaine's theorem is an analogue of Stickelberger's theorem for real abelian fields, introduced by Thaine (1988). Thaine's method has been used to shorten the proof of the Mazur–Wiles theorem (Washington 1997), to prove that some Tate–Shafarevich groups are finite, and in the proof of Mihăilescu's theorem (Schoof 2008).
Formulation
Let $p$ and $q$ be distinct odd primes with $q$ not dividing $p-1$. Let $G^{+}$ be the Galois group of $F=\mathbb {Q} (\zeta _{p}^{+})$ over $\mathbb {Q} $, let $E$ be its group of units, let $C$ be the subgroup of cyclotomic units, and let $Cl^{+}$ be its class group. If $\theta \in \mathbb {Z} [G^{+}]$ annihilates $E/CE^{q}$ then it annihilates $Cl^{+}/Cl^{+q}$.
References
• Schoof, René (2008), Catalan's conjecture, Universitext, London: Springer-Verlag London, Ltd., ISBN 978-1-84800-184-8, MR 2459823 See in particular Chapter 14 (pp. 91–94) for the use of Thaine's theorem to prove Mihăilescu's theorem, and Chapter 16 "Thaine's Theorem" (pp. 107–115) for proof of a special case of Thaine's theorem.
• Thaine, Francisco (1988), "On the ideal class groups of real abelian number fields", Annals of Mathematics, 2nd ser., 128 (1): 1–18, doi:10.2307/1971460, JSTOR 1971460, MR 0951505
• Washington, Lawrence C. (1997), Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol. 83 (2nd ed.), New York: Springer-Verlag, ISBN 0-387-94762-0, MR 1421575 See in particular Chapter 15 (pp. 332–372) for Thaine's theorem (section 15.2) and its application to the Mazur–Wiles theorem.
| Wikipedia |
Thales's theorem
In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ∠ ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements.[1] It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras.
For the theorem sometimes called Thales' theorem and pertaining to similar triangles, see intercept theorem.
History
There is nothing extant of the writing of Thales. Work done in ancient Greece tended to be attributed to men of wisdom without respect to all the individuals involved in any particular intellectual constructions; this is true of Pythagoras especially. Attribution did tend to occur at a later time.[2] Reference to Thales was made by Proclus, and by Diogenes Laërtius documenting Pamphila's statement that Thales[3] "was the first to inscribe in a circle a right-angle triangle".
Babylonian mathematicians knew this for special cases before Thales proved it.[4] It is believed that Thales learned that an angle inscribed in a semicircle is a right angle during his travels to Babylon.[5] The theorem is named after Thales because he was said by ancient sources to have been the first to prove the theorem, using his own results that the base angles of an isosceles triangle are equal, and that the sum of angles of a triangle is equal to a straight angle (180°).
Dante's Paradiso (canto 13, lines 101–102) refers to Thales's theorem in the course of a speech.
Proof
First proof
The following facts are used: the sum of the angles in a triangle is equal to 180° and the base angles of an isosceles triangle are equal.
• Provided AC is a diameter, angle at B is constant right (90°).
• Figure for the proof.
Since OA = OB = OC, △OBA and △OBC are isosceles triangles, and by the equality of the base angles of an isosceles triangle, ∠ OBC = ∠ OCB and ∠ OBA = ∠ OAB.
Let α = ∠ BAO and β = ∠ OBC. The three internal angles of the ∆ABC triangle are α, (α + β), and β. Since the sum of the angles of a triangle is equal to 180°, we have
${\begin{aligned}\alpha +(\alpha +\beta )+\beta &=180^{\circ }\\2\alpha +2\beta &=180^{\circ }\\2(\alpha +\beta )&=180^{\circ }\\\therefore \alpha +\beta &=90^{\circ }.\end{aligned}}$
Q.E.D.
Second proof
The theorem may also be proven using trigonometry: Let O = (0, 0), A = (-1, 0), and C = (1, 0). Then B is a point on the unit circle (cos θ, sin θ). We will show that △ABC forms a right angle by proving that AB and BC are perpendicular — that is, the product of their slopes is equal to −1. We calculate the slopes for AB and BC:
${\begin{aligned}m_{AB}&={\frac {y_{B}-y_{A}}{x_{B}-x_{A}}}={\frac {\sin \theta }{\cos \theta +1}}\\[2pt]m_{BC}&={\frac {y_{B}-y_{C}}{x_{B}-x_{C}}}={\frac {\sin \theta }{\cos \theta -1}}\end{aligned}}$
Then we show that their product equals −1:
${\begin{aligned}&m_{AB}\cdot m_{BC}\\[4pt]={}&{\frac {\sin \theta }{\cos \theta +1}}\cdot {\frac {\sin \theta }{\cos \theta -1}}\\[4pt]={}&{\frac {\sin ^{2}\theta }{\cos ^{2}\theta -1}}\\[4pt]={}&{\frac {\sin ^{2}\theta }{-\sin ^{2}\theta }}\\[4pt]={}&{-1}\end{aligned}}$
Note the use of the Pythagorean trigonometric identity $\sin ^{2}\theta +\cos ^{2}\theta =1.$
Third proof
Let △ABC be a triangle in a circle where AB is a diameter in that circle. Then construct a new triangle △ABD by mirroring △ABC over the line AB and then mirroring it again over the line perpendicular to AB which goes through the center of the circle. Since lines AC and BD are parallel, likewise for AD and CB, the quadrilateral ACBD is a parallelogram. Since lines AB and CD, the diagonals of the parallelogram, are both diameters of the circle and therefore have equal length, the parallelogram must be a rectangle. All angles in a rectangle are right angles.
Converse
For any triangle, and, in particular, any right triangle, there is exactly one circle containing all three vertices of the triangle. (Sketch of proof. The locus of points equidistant from two given points is a straight line that is called the perpendicular bisector of the line segment connecting the points. The perpendicular bisectors of any two sides of a triangle intersect in exactly one point. This point must be equidistant from the vertices of the triangle.) This circle is called the circumcircle of the triangle.
One way of formulating Thales's theorem is: if the center of a triangle's circumcircle lies on the triangle then the triangle is right, and the center of its circumcircle lies on its hypotenuse.
The converse of Thales's theorem is then: the center of the circumcircle of a right triangle lies on its hypotenuse. (Equivalently, a right triangle's hypotenuse is a diameter of its circumcircle.)
Proof of the converse using geometry
This proof consists of 'completing' the right triangle to form a rectangle and noticing that the center of that rectangle is equidistant from the vertices and so is the center of the circumscribing circle of the original triangle, it utilizes two facts:
• adjacent angles in a parallelogram are supplementary (add to 180°) and,
• the diagonals of a rectangle are equal and cross each other in their median point.
Let there be a right angle ∠ ABC, r a line parallel to BC passing by A, and s a line parallel to AB passing by C. Let D be the point of intersection of lines r and s. (It has not been proven that D lies on the circle.)
The quadrilateral ABCD forms a parallelogram by construction (as opposite sides are parallel). Since in a parallelogram adjacent angles are supplementary (add to 180°) and ∠ ABC is a right angle (90°) then angles ∠ BAD, ∠ BCD, ∠ ADC are also right (90°); consequently ABCD is a rectangle.
Let O be the point of intersection of the diagonals AC and BD. Then the point O, by the second fact above, is equidistant from A, B, and C. And so O is center of the circumscribing circle, and the hypotenuse of the triangle (AC) is a diameter of the circle.
Alternate proof of the converse using geometry
Given a right triangle ABC with hypotenuse AC, construct a circle Ω whose diameter is AC. Let O be the center of Ω. Let D be the intersection of Ω and the ray OB. By Thales's theorem, ∠ ADC is right. But then D must equal B. (If D lies inside △ABC, ∠ ADC would be obtuse, and if D lies outside △ABC, ∠ ADC would be acute.)
Proof of the converse using linear algebra
This proof utilizes two facts:
• two lines form a right angle if and only if the dot product of their directional vectors is zero, and
• the square of the length of a vector is given by the dot product of the vector with itself.
Let there be a right angle ∠ ABC and circle M with AC as a diameter. Let M's center lie on the origin, for easier calculation. Then we know
• A = −C, because the circle centered at the origin has AC as diameter, and
• (A – B) · (B – C) = 0, because ∠ ABC is a right angle.
It follows
${\begin{aligned}0&=(A-B)\cdot (B-C)\\&=(A-B)\cdot (B+A)\\&=|A|^{2}-|B|^{2}.\\[4pt]\therefore \ |A|&=|B|.\end{aligned}}$
This means that A and B are equidistant from the origin, i.e. from the center of M. Since A lies on M, so does B, and the circle M is therefore the triangle's circumcircle.
The above calculations in fact establish that both directions of Thales's theorem are valid in any inner product space.
Generalizations and related results
Thales's theorem is a special case of the following theorem:
Given three points A, B and C on a circle with center O, the angle ∠ AOC is twice as large as the angle ∠ ABC.
See inscribed angle, the proof of this theorem is quite similar to the proof of Thales's theorem given above.
A related result to Thales's theorem is the following:
• If AC is a diameter of a circle, then:
• If B is inside the circle, then ∠ ABC > 90°
• If B is on the circle, then ∠ ABC = 90°
• If B is outside the circle, then ∠ ABC < 90°.
Application
Thales's theorem can be used to construct the tangent to a given circle that passes through a given point. In the figure at right, given circle k with centre O and the point P outside k, bisect OP at H and draw the circle of radius OH with centre H. OP is a diameter of this circle, so the triangles connecting OP to the points T and T′ where the circles intersect are both right triangles.
Thales's theorem can also be used to find the centre of a circle using an object with a right angle, such as a set square or rectangular sheet of paper larger than the circle.[6] The angle is placed anywhere on its circumference (figure 1). The intersections of the two sides with the circumference define a diameter (figure 2). Repeating this with a different set of intersections yields another diameter (figure 3). The centre is at the intersection of the diameters.
See also
• Synthetic geometry
• Inverse Pythagorean theorem
Notes
1. Heath, Thomas L. (1956). The Thirteen Books of Euclid's Elements. Vol. 2 (Books 3–9) (2nd ed.). Dover. p. 61. ISBN 0486600890. Originally published by Cambridge University Press. 1st edition 1908, 2nd edition 1926.
2. Allen, G. Donald (2000). "Thales of Miletus" (PDF). Retrieved 2012-02-12.
3. Patronis, Tasos; Patsopoulos, Dimitris (January 2006). "The Theorem of Thales: A Study of the Naming of Theorems in School Geometry Textbooks". The International Journal for the History of Mathematics Education: 57–68. ISSN 1932-8826. Archived from the original on 2018-04-25.
4. de Laet, Siegfried J. (1996). History of Humanity: Scientific and Cultural Development. UNESCO, Volume 3, p. 14. ISBN 92-3-102812-X
5. Boyer, Carl B. and Merzbach, Uta C. (2010). A History of Mathematics. John Wiley and Sons, Chapter IV. ISBN 0-470-63056-6
6. Resources for Teaching Mathematics: 14–16 Colin Foster
References
• Agricola, Ilka; Friedrich, Thomas (2008). Elementary Geometry. AMS. p. 50. ISBN 978-0-8218-4347-5.
• Heath, T.L. (1921). A History of Greek Mathematics: From Thales to Euclid. Vol. I. Oxford. pp. 131ff.
External links
• Weisstein, Eric W. "Thales' Theorem". MathWorld.
• Munching on Inscribed Angles
• Thales's theorem explained, with interactive animation
• Demos of Thales's theorem by Michael Schreiber, The Wolfram Demonstrations Project.
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| Wikipedia |
Thales of Miletus
Thales of Miletus (/ˈθeɪliːz/ THAY-leez; Greek: Θαλῆς; c. 626/623 – c. 548/545 BC) was an Ancient Greek pre-Socratic philosopher from Miletus in Ionia, Asia Minor. Thales was one of the Seven Sages, founding figures of Ancient Greece, and credited with the saying "know thyself" which was inscribed on the Temple of Apollo at Delphi.
Thales of Miletus
Posthumous portrait of Thales by Wilhelm Meyer, based on a bust from the 4th century
Bornc. 626/623 BC
Miletus, Ionian League
(modern-day Balat, Didim, Aydın, Turkey)
Diedc. 548/545 BC (aged c. 78)
EraPre-Socratic philosophy
RegionWestern philosophy
SchoolIonian / Milesian
Main interests
• metaphysics
• mathematics
• astronomy
Notable ideas
• Philosophical inquiry
• Water is the arche
• Thales's theorem
• Intercept theorem
• Know thyself
• Static electricity
Influences
• Babylonian astronomy
• Ancient Egyptian mathematics
• Ancient Egyptian religion
Influenced
• Anaximander
• Anaximenes
• Pythagoras
• Xenophanes
• Cleostratus
Many regard him as the first philosopher in the Greek tradition, breaking from the prior use of mythology to explain the world and instead using natural philosophy. He is thus otherwise credited as the first to have engaged in mathematics, science, and deductive reasoning.
The first philosophers followed him in explaining all of nature as based on the existence of a single ultimate substance. Thales theorized that this single substance was water. Thales thought the Earth floated in water.
In mathematics, Thales is the namesake of Thales's theorem, and the intercept theorem can also be known as Thales's theorem. Thales was said to have calculated the heights of the pyramids and the distance of ships from the shore. In science, Thales was an astronomer who reportedly predicted the weather and a solar eclipse. He was also credited with discovering the position of the constellation Ursa Major as well as the timings of the solstices and equinoxes. Thales was also an engineer; credited with diverting the Halys River.
Life
The main source concerning the details of Thales's life and career is the doxographer Diogenes Laërtius, in his third century AD work Lives and Opinions of the Eminent Philosophers.[1] While it is all we have, Diogenes wrote some eight centuries after Thales's death and his sources often contained "unreliable or even fabricated information."[2][lower-alpha 1] It is known Thales was from Miletus, a mercantile city settled at the mouth of the Maeander river.
The dates of Thales's life are not exactly known, but are roughly established by a few datable events mentioned in the sources. According to the historian Herodotus, writing in the 5th century BC, Thales predicted a solar eclipse in 585 BC.[4] Assuming one's acme occurred at the age of 40, the chronicle of Apollodorus of Athens, written during the 2nd century BC, therefore placed Thales's birth about the year 625 BC.[5][6]
Ancestry and family
While the probability is that Thales was as Greek as most Milesians,[7] Herodotus described Thales as "a Phoenician by remote descent".[8] Diogenes Laërtius references Herodotus, Duris, and Democritus, who all agree "that Thales was the son of Examyas and Cleobulina, and belonged to the Thelidae who are Phoenicians and amongst the noblest descendants of Cadmus and Agenor" who had been banished from Phoenicia and that Thales was enrolled as a citizen in Miletus along with Neleus.[9][10]
However, Friedrich Nietzsche and others interpret this quote as meaning only that his ancestors were seafaring Cadmeians from Boeotia.[11][12] It is also possible that he was of mixed ancestry, given his father had a Carian name and his mother had a Greek name.[12][13][14] Diogenes Laërtius seems to also reference an alternative account: "Most writers, however, represent him as a genuine Milesian and of a distinguished family".[15] Encyclopedia Britannica (1952) concluded that Thales was most likely a native Milesian of noble birth and that he was certainly a Greek.[13]
Diogenes continues, by delivering more conflicting reports: one that Thales married and either fathered a son (Cybisthus or Cybisthon) or adopted his nephew of the same name; the second that he never married, telling his mother as a young man that it was too early to marry, and as an older man that it was too late.[lower-alpha 2] Plutarch had earlier told this version: Solon visited Thales and asked him why he remained single; Thales answered that he did not like the idea of having to worry about children. Nevertheless, several years later, anxious for family, he adopted his nephew Cybisthus.[17]
Travels
The culture of Archaic Greece was heavily influenced by those of the Levant and Mesopotamia.[18] It is said Thales was engaged in trade and visited either Egypt or Babylonia.[19] However, there is no strong evidence that Thales ever visited countries in the Near East, and the issue is disputed among scholars.[20] Visits to such places were a commonplace attribution to various philosophers by later writers, especially when these writers tried to explain the origin of their mathematical knowledge, such as with Thales or Pythagoras or Eudoxus.[21]
Egypt
Several ancient authors assume that Thales, at one point in his life, visited Egypt, where he learned about geometry.[22] It is considered possible that Thales visited Egypt, since Miletus had a permanent colony there (namely Naucratis). It is also said Thales had close contacts with the priests of Thebes who instructed him, or even that he instructed them in geometry.[23] It is also possible Thales knew about Egypt from accounts of others, without actually visiting it.[24]
Babylon
Aside from Egypt, the other mathematically advanced, ancient civilization before the Greeks was Babylonia, another commonplace attribution of travel for a mathematically-minded philosopher.[25] At least one ancient historian, Josephus, claims Thales visited Babylonia.
Historians Roger L. Cooke and B.L. Van der Waerden come down on the side of Babylonian mathematics influencing the Greeks, citing the use of e. g. the sexagesimal system (or base 60).[25] Cooke notes "This relation, however, is controversial."[25] Others historians, such as D. R. Dicks, take issue with the idea of Babylonian influence on Greek mathematics. For until around the time of Hipparchus (c. 190–120 BC) their sexagesimal system was unknown.[26]
Herodotus wrote the Greeks learnt the gnomon from the Babylonians. Thales's follower Anaximander is credited with introducing the gnomon to the Greeks.[27] Herodotus also wrote that the practice of dividing the day into 12 parts, and the polos, came to the Greeks from the Babylonians.[lower-alpha 3] Yet this too is disputed, for example by historian L. Zhmud, who points out the gnomon was known to both Egyptians and Babylonians, the division of the day into twelve parts (and by analogy the year) was known to the Egyptians already in the 2nd millennium BC, and the idea of the polos was not used outside of Greece at this time.[29]
Sagacity
Thales is recognized as one of the Seven Sages of Greece, semi-legendary wise statesman and founding figures of Ancient Greece. While which seven one chooses may change, the seven has a canonical four which includes Thales, Solon of Athens, Pittacus of Mytilene, and Bias of Priene. Diogenes Laërtius tells us that the Seven Sages were created in the archonship of Damasius at Athens about 582 BC and that Thales was the first sage.[30][lower-alpha 4]
Each sage has a quote or maxim attributed to him, which was inscribed on the Temple of Apollo at Delphi. Thales has arguably the most famous of all, gnothi seauton or know thyself. According to the 10th-century Byzantine encyclopedia the Suda, the proverb is both "applied to those whose boasts exceed what they are" and "a warning to pay no attention to the opinion of the multitude."[31][lower-alpha 5]
Golden tripod
Diogenes Laërtius relates several stories of an expensive, gold tripod or bowl that is to go to the most wise. In one version (that Laërtius credits to Callimachus in his Iambics) Bathycles of Arcadia states in his will that an expensive bowl "'should be given to him who had done most good by his wisdom.' So it was given to Thales, went the round of all the sages, and came back to Thales again. And he sent it to Apollo at Didyma, with this dedication...'Thales the Milesian, son of Examyas [dedicates this] to Delphinian Apollo after twice winning the prize from all the Greeks.'"[37]
Diplomacy
According to Diogenes Laërtius, Thales gained fame as a counselor when he advised the Milesians not to engage in a symmachia, a "fighting together", with the Lydians. This has sometimes been interpreted as an alliance.[38]
Croesus was defeated before the city of Sardis by Cyrus the Great, who subsequently spared Miletus because it had taken no action. Cyrus was so impressed by Croesus’ wisdom and his connection with the sages that he spared him and took his advice on various matters. The Ionian cities should be demoi, or "districts".
He counselled them to establish a single seat of government, and pointed out Teos as the fittest place for it; "for that," he said, "was the centre of Ionia. Their other cities might still continue to enjoy their own laws, just as if they were independent states."[39]
Miletus, however, received favorable terms from Cyrus. The others remained in an Ionian League of twelve cities (excluding Miletus), and were subjugated by the Persians.
Theories and studies
Early Greeks, and other civilizations before them, often invoked idiosyncratic explanations of natural phenomena with reference to the will of anthropomorphic gods and heroes. Instead, Thales aimed to explain natural phenomena via rational hypotheses that referenced natural processes themselves.[40] Logos rather than mythos. Many, most notably Aristotle, regard him as the first philosopher in the Greek tradition,[41][42] Rather than theologoi or mythologoi, Aristotle referred to the first philosophers as physiologoi, or natural philosophers, and Thales as the first among them. Also, while the other Seven Sages were strictly law-givers and statesmen and not speculative philosophers, Plutarch noted "it would seem that Thales was the only wise man of the time who carried his speculations beyond the realm of the practical."[43]
Water is the arche
Thales's most famous idea was his philosophical and cosmological thesis that all is water, which comes down to us through a passage from Aristotle's Metaphysics.[44] In the work, Aristotle reported Thales's theory that the arche or originating principle of nature was a single material substance: water.[45] Aristotle then proceeded to proffer a number of conjectures based on his own observations to lend some credence to why Thales may have advanced this idea (though Aristotle did not hold it himself).[46][lower-alpha 6]
While Aristotle's conjecture on why Thales held water as the originating principle of matter is his own thinking, his statement that Thales held it as water is generally accepted as genuinely originating with Thales. Writing centuries later, Diogenes Laërtius also states that Thales taught "Water constituted (ὑπεστήσατο, 'stood under') the principle of all things."[47][lower-alpha 7]
According to Aristotle:[49]
That from which is everything that exists and from which it first becomes and into which it is rendered at last, its substance remaining under it, but transforming in qualities, that they say is the element and principle of things that are. …For it is necessary that there be some nature (φύσις), either one or more than one, from which become the other things of the object being saved... [The first philosophers] do not all agree as to the number and the nature of these principles. Thales the founder of this type of philosophy says that it is water.
Aristotle further adds:
Presumably he derived this assumption from seeing that the nutriment of everything is moist, and that heat itself is generated from moisture and depends upon it for its existence (and that from which a thing is generated is always its first principle). He derived his assumption from this; and also from the fact that the seeds of everything have a moist nature, whereas water is the first principle of the nature of moist things."[45][lower-alpha 8]
The 1870 book Dictionary of Greek and Roman Biography and Mythology noted:[42]
In his dogma that water is the origin of things, that is, that it is that out of which every thing arises, and into which every thing resolves itself, Thales may have followed Orphic cosmogonies, while, unlike them, he sought to establish the truth of the assertion. Hence, Aristotle, immediately after he has called him the originator of philosophy brings forward the reasons which Thales was believed to have adduced in confirmation of that assertion; for that no written development of it, or indeed any book by Thales, was extant, is proved by the expressions which Aristotle uses when he brings forward the doctrines and proofs of the Milesian. (p. 1016)
Most agree that Thales's stamp on thought is the unity of substance. Not merely the empirical claim that all is water, but the deeper philosophical claim that all is one. For example, Friedrich Nietzsche, in his Philosophy in the Tragic Age of the Greeks, wrote:[50]
Greek philosophy seems to begin with an absurd notion, with the proposition that water is the primal origin and the womb of all things. Is it really necessary for us to take serious notice of this proposition? It is, and for three reasons. First, because it tells us something about the primal origin of all things; second, because it does so in language devoid of image or fable, and finally, because contained in it, if only embryonically, is the thought, "all things are one."
Mathematics
Thales was known for his introducting the theoretical and practical use of geometry to Greece, and is often considered the first person in the western world to have applied deductive reasoning to geometry, and by extension is often considered the West's "first mathematician."[6][51][52] He also proved skilled in arithmetic, and is credited with the West's oldest definition of number: a "collection of units," "following the Egyptian view".[53][54]
The evidence for the primacy of Thales comes to us from a book by Proclus, who wrote a thousand years after Thales. but is believed to have had a copy of Eudemus's lost book History of Geometry.[lower-alpha 9] Proclus wrote "Thales was the first to go to Egypt and bring back to Greece this study."[52] He goes on to tell us that in addition to applying the knowledge he gained in Egypt "He himself discovered many propositions and disclosed the underlying principles of many others to his successors, in some case his method being more general, in others more empirical."[52] In addition to Proclus, Hieronymus of Rhodes also cites Thales as the first Greek mathematician.
Geometry
Proclus attributed to Thales the discovery that a circle is bisected by its diameter, that the base angles of an isosceles triangle are equal and that vertical angles are equal. According to one author, while visiting Egypt,[22] Thales observed that when the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal.[55] Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as: all straight angles are equal, equals added to equals are equal, and equals subtracted from equals are equal.
Several quotes on geometry are attributed to him. For example, he said:
Megiston topos: apanta gar chorei (Μέγιστον τόπος· ἄπαντα γὰρ χωρεῖ.)
The greatest is space, for it holds all things.[56]
Thales's theorems
There are two theorems named after Thales in elementary geometry, one known as Thales's theorem has to do with a triangle inscribed in a circle and having the circle's diameter as one leg, the other theorem being also called the intercept theorem and is equivalent to the theorem about ratios in similar triangles.
Right triangle inscribed in a circle
Main article: Thales's theorem
Pamphila says that, having learnt geometry from the Egyptians, he [Thales] was the first to inscribe in a circle a right-angled triangle, whereupon he sacrificed an ox.[52] This is sometimes cited as history's first mathematical discovery.[57] Due to the variations among testimonies, such as the story of the ox sacrifice being accredited to Pythagoras upon discovery of the Pythagorean theorem rather than Thales, some historians (such as D. R. Dicks) question whether such anecdotes have any historical worth whatsoever.[26]
It is believed the Babylonians knew the theorem for special cases before Thales proved it.[58][59] The theorem is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements.[60] Dante's Paradiso refers to Thales's theorem in the course of a speech.[61]
Similar triangles
Main article: Intercept theorem
The story is told in Diogenes Laërtius, Pliny the Elder, and Plutarch,[52][62] sourced from Hieronymus of Rhodes, that when Thales visited Egypt,[22] he measured the height of the pyramids by their shadows at the moment when his own shadow was equal to his height.[lower-alpha 10] According to Plutarch, it pleased the pharoah Amasis. More practically, Thales had the ability to measure the distances of ships at sea.
These stories illustrate Thales's familiarity with the intercept theorem, and for this reason the 26th proposition in the first book of Euclid's Elements was attributed to Thales.[64] They also indicate that he was familiar with the Egyptian seked, or seqed, the ratio of the run to the rise of a slope (cotangent).[65][lower-alpha 11] According to Kirk & Raven,[7] all you need for this feat is three straight sticks pinned at one end and knowledge of your altitude. One stick goes vertically into the ground. A second is made level. With the third you sight the ship and calculate the seked from the height of the stick and its distance from the point of insertion to the line of sight.[66]
Astronomy
Thales was also a noted astronomer credited in antiquity with describing the position of Ursa Minor, and he thought the constellation might be useful as a guide for navigation at sea. He calculated the duration of the year and the timings of the equinoxes and solstices. He is additionally attribute with calculating the position of the Pleiades.[7] Plutarch indicates that in his day (c. AD 100) there was an extant work, the Astronomy, composed in verse and attributed to Thales.[67] While some say he left no writings, others say that he wrote On the Solstice and On the Equinox. The Nautical Star-guide has also been attributed to him, but this was disputed even in ancient times.[7][lower-alpha 12] No writing attributed to him has survived. Lobon of Argus asserted that the writings of Thales amounted to two hundred lines.[68]
Cosmological model
Thales thought the Earth must be a flat disk or mound of land and dirt which is floating in an expanse of water.[69] Heraclitus Homericus states that Thales drew his conclusion from seeing moist substance turn into air, slime and earth. It seems likely that Thales viewed the land as coming from the water on which it floated and the oceans that surround it, perhaps inspired by observing silt deposits.[70]
He thought the stars were balls of dirt on fire.[71] He seemed to correctly gather that the moon reflects the Sun's light.[72] A crater on the Moon is named in his honor.
Meteorology
Rather than assuming that earthquakes were the result of supernatural whims, Thales explained them by theorizing that the Earth floats on water and that earthquakes occur when the Earth is rocked by waves.[73][40] He is attributed with the first observation of the Hyades, supposed by the ancients to indicate the approach of rain when they rose with the Sun.[74] According to Seneca, Thales explained the flooding of the Nile as due to the river being beaten back by the etesian wind.[75]
Olive presses
A story, with different versions, recounts how Thales achieved riches from an olive harvest by prediction of the weather. In one version, he bought all the olive presses in Miletus after predicting the weather and a good harvest for a particular year. Another version of the story has Aristotle explain that Thales had reserved presses in advance, at a discount, and could rent them out at a high price when demand peaked, following his prediction of a particularly good harvest. This first version of the story would constitute the first historically known creation and use of futures, whereas the second version would be the first historically known creation and use of options.[76]
Aristotle explains that Thales's objective in doing this was not to enrich himself but to prove to his fellow Milesians that philosophy could be useful, contrary to what they thought,[77] or alternatively, Thales had made his foray into enterprise because of a personal challenge put to him by an individual who had asked why, if Thales was an intelligent famous philosopher, he had yet to attain wealth.
Prediction of solar eclipse
As mentioned above, according to Herodotus, Thales predicted a solar eclipse in 585 BC.[4] Only the eclipse of May 28, 585 BC matches the conditions of visibility necessary to explain it. American writer Isaac Asimov described this battle as the earliest historical event whose date is known with precision to the day, and called the prediction "the birth of science". As well as first mathematician and first philosopher, Thales is often given the label of the first western scientist and the "father of science".[78][79] but some contemporary scholars reject this interpretation.[80]
Herodotus writes that in the sixth year of the war, the Lydians under King Alyattes and the Medes under Cyaxares were engaged in an indecisive battle when suddenly day turned into night, leading to both parties halting the fighting and negotiating a peace agreement. Herodotus also mentions that the loss of daylight had been predicted by Thales. He does not, however, mention the location of the battle.[81]
Afterwards, on the refusal of Alyattes to give up his suppliants when Cyaxares sent to demand them of him, war broke out between the Lydians and the Medes, and continued for five years, with various success. In the course of it the Medes gained many victories over the Lydians, and the Lydians also gained many victories over the Medes. Among their other battles there was one night engagement. As, however, the balance had not inclined in favour of either nation, another combat took place in the sixth year, in the course of which, just as the battle was growing warm, day was on a sudden changed into night. This event had been foretold by Thales, the Milesian, who forewarned the Ionians of it, fixing for it the very year in which it actually took place. The Medes and Lydians, when they observed the change, ceased fighting, and were alike anxious to have terms of peace agreed on.[39]
However, based on the list of Median kings and the duration of their reign reported elsewhere by Herodotus, Cyaxares died 10 years before the eclipse.[82][83]
D. R. Dicks joins other historians (F. Martini, J. L. E. Dreyer, O. Neugebauer) in rejecting the historicity of the eclipse story.[26] Dicks links the story of Thales discovering the cause for a solar eclipse with Herodotus' claim that Thales discovered the cycle of the sun with relation to the solstices, and concludes "he could not possibly have possessed this knowledge which neither the Egyptians nor the Babylonians nor his immediate successors possessed."[26]
Falling into a well
Plato, Diogenes Laertius, and Hippolytus all relay the story that Thales was so intent upon watching the stars that he failed to watch where he was walking, and fell into a well.[84][85][lower-alpha 13]
"Thales was studying the stars and gazing into the sky, when he fell into a well, and a jolly and witty Thracian servant girl made fun of him, saying that he was crazy to know about what was up in the heavens while he could not see what was in front of him beneath his feet."[87]
Engineering
As well as astronomy, Thales involved himself in other practical applications of mathematics, including engineering.[88] Another story by Herodotus is that Croesus sent his army to the Persian territory. He was stopped by the river Halys, then unbridged. Thales then got the army across the river by digging a diversion upstream so as to reduce the flow, making it possible to cross the river.[89] While Herodotus reported that most of his fellow Greeks believe that Thales did divert the river Halys to assist King Croesus' military endeavors, he himself finds it doubtful.[26] Plato praises Thales along with Anacharsis, who is credited as the originator of the potter's wheel and the anchor.[90]
Divinity
According to Aristotle, Thales thought "all things are full of gods",[7][91] i. e. lodestones had souls, because iron is attracted to them (by the force of magnetism).[92] The same applied to amber for its capacity to generate static electricity. The reasoning for such hylozoism or organicism seems to be if something moved, then it was alive, and if it was alive, then it must have a soul.[93][94]
As well as gods seen in the movement caused by what came to be known as magnetism and electricity, it seems Thales also had a supreme God which structured the universe:
"Thales", says Cicero,[95] "assures that water is the principle of all things; and that God is that Mind which shaped and created all things from water."
According to Henry Fielding (1775), Diogenes Laërtius (1.35) affirmed that Thales posed "the independent pre-existence of God from all eternity, stating "that God was the oldest of all beings, for he existed without a previous cause even in the way of generation; that the world was the most beautiful of all things; for it was created by God."[96]
Nicholas Molinari has recently argued that Thales was influenced by the archaic water deity Acheloios, who was equated with water and worshipped in Miletus during Thales's life. For evidence, he points to the fact that hydor meant specifically "fresh water," and also that Acheloios was seen as a shape-shifter in myth and art, so able to become anything. He also points out that the rivers of the world were seen as the "sinews of Acheloios" in antiquity, and this multiplicity of deities is reflected in Thales's idea that "all things are full of gods."[97]
Death and legacy
Diogenes Laërtius quotes Apollodorus as saying that Thales died at the age of 78 during the 58th Olympiad (548–545 BC) and attributes his death to heat stroke and thirst while watching the games.[98]
Influence
As the first philosopher and mathematician, Thales had a profound influence on other Greek thinkers and therefore on Western history. However, due to the scarcity of sources concerning Thales and the discrepancies between the accounts given in the sources that have survived, there is a scholarly debate over the extent of the influence of Thales had and on which of the Greek philosophers and mathematicians that came after him.[lower-alpha 14]
The first three philosophers in the Western tradition were all cosmologists from Miletus, and Thales was the very first, followed by Anaximander, who was followed in turn by Anaximenes. They have been dubbed the Milesian school. According to the Suda, Thales had been the "teacher and kineman" of Anaximander.[100] Rather than water, Anaximander held all was made of apeiron or the unlimited; while Anaximenes, the successor of Anaximander, perhaps more like Thales with water, held that everything was composed of air.[101]
John Burnet (1892) noted[102]
Lastly, we have one admitted instance of a philosophic guild, that of the Pythagoreans. And it will be found that the hypothesis, if it is to be called by that name, of a regular organisation of scientific activity will alone explain all the facts. The development of doctrine in the hands of Thales, Anaximander, and Anaximenes, for instance, can only be understood as the elaboration of a single idea in a school with a continuous tradition.
As two of the first Greek mathematicians, Thales is also considered an influence on Pythagoras. According to Iamblichus, Pythagoras "had benefited by the instruction of Thales in many respects, but his greatest lesson had been to learn the value of saving time."[103] Early sources report that Pythagoras, in this story a pupil of Anaximander, visited Thales as a young man, and that Thales advised him to travel to Egypt to further his philosophical and mathematical studies.
Thales was also considered the teacher of the astronomer Mandrolytus of Priene.[104] It is possible he was also the teacher of Cleostratus of Tenedos.[105]
Notes
1. This use of hearsay and a lack of citing original sources leads some historians, like Dicks and Werner Jaeger, to view the whole idea of pre-Socratic philosophy as a construct from a later age, "fashioned during the two or three generations from Plato to the immediate pupils of Aristotle".[3]
2. In addition, his supposed mother, Cleobulina, has also been described as his companion instead of his mother.[16]
3. The exact meaning of this use of the word polos is unknown, current theories include: "the heavenly dome", "the tip of the axis of the celestial sphere", or a spherical concave sundial.[28]
4. The same story, however, asserts that Thales emigrated to Miletus; and that he did not become a student of nature until after his political career. This story has to be rejected if one is to believe that Thales was a native of Miletus, and other typical things about him like his prediction of the eclipse.
5. The aphorism has also been attributed to various other philosophers. Diogenes Laërtius attributes it to Thales[32][33] but notes that Antisthenes in his Successions of Philosophers attributes it instead to Phemonoe, a mythical Greek poet. The Roman poet Juvenal quotes the phrase in Greek and states that the precept descended e caelo (from heaven).[34] Other names of potential include Pythagoras[35] and Heraclitus.[36]
6. Geoffrey Kirk and John Raven, English compilers of the fragments of the Pre-Socratics, assert that Aristotle's "judgments are often distorted by his view of earlier philosophy as a stumbling progress toward the truth that Aristotle himself revealed in his physical doctrines."[7]
7. Historian Abraham Feldman wrote that for Thales "...water united all things...all whatness is wetness".[48]
8. Feldman notes "The social significance of water in the time of Thales induced him to discern through hardware and dry-goods, through soil and sperm, blood, sweat and tears, one fundamental fluid stuff...water, the most commonplace and powerful material known to him."[48]
9. While some historians, such as Colin R. Fletcher, note there could have been a precursor to Thales named by Eudemus, without the work "the question becomes mere speculation."[52] Fletcher grants there is no other viable contender to the title of first Greek mathematician; and that Thales qualifies as a practitioner in the field. "Thales had at his command the techniques of observation, experimentation, superposition and deduction...he has proved himself mathematician."[52]
10. A right triangle with two equal legs is a 45-degree right triangle, all of which are similar. The length of the pyramid's shadow measured from the center of the pyramid at that moment must have been equal to its height.[63]
11. The seked is at the base of problems 56, 57, 58, 59 and 60 of the Rhind papyrus — an ancient Egyptian mathematical document.
12. According to Diogenes Laertius, the Nautical Astronomy attributed to Thales was written by Phocus of Samos.
13. The Scottish philosopher Adam Smith once similarly, absent-mindedly fell into a tannery pit.[86]
14. Edmund Husserl[99] attempts to capture the new movement as follows. Philosophical man is a "new cultural configuration" based in stepping back from "pregiven tradition" and taking up a rational "inquiry into what is true in itself;" that is, an ideal of truth.
References
1. Translation of his biography on Thales: Thales Archived 9 February 2008 at the Wayback Machine, classicpersuasion site; original Greek text, under ΘΑΛΗΣ, the Library of Ancient Texts Online site.
2. See McKirahan, Richard D. Jr. (1994). Philosophy Before Socrates. Indianapolis: Hackett. p. 5. ISBN 978-0-87220-176-7.
3. Jaeger, Werner (1948). Aristotle (2nd ed.). p. 454.
4. Herodotus, 1.74.2, and A. D. Godley's footnote 1; Pliny, 2.9 (12) and Bostock's footnote 2.
5. Cohen, Mark S.; Curd, Patricia; Reeve, C. D. C. (2011). Readings in Ancient Greek Philosophy (Fourth Edition): From Thales to Aristotle. Indianapolis, Indiana: Hackett Publishing. p. 10. ISBN 978-1603846073.
6. Frank N. Magill, The Ancient World: Dictionary of World Biography, Volume 1, Routledge, 2003 ISBN 1135457395
7. Kirk, G. S.; Raven, J. E. (1957). "Chapter II: Thales of Miletus". The Presocratic Philosophers. Cambridge University Press. pp. 74–98.
8. Freely, John (2012). The Flame of Miletus: The Birth of Science in Ancient Greece (And How It Changed the World). London: I. B. Tauris & Co. Ltd. p. 7. ISBN 978-1-78076-051-3. Retrieved 1 October 2017.
9. Lawson, Russell M. (2004). Science in the Ancient World: An Encyclopedia. Santa Barbara, California; Denver, Colorado; and Oxford, England: ABC CLIO. pp. 234–235. ISBN 978-1-85109-534-6.
10. Thatcher, Oliver J. (2004). The Library Of Original Sources: The Greek World. The Minerva Group, Inc. p. 138. ISBN 978-1-4102-1402-7.
11. Nietzsche, Friedrich (2001). The Pre-Platonic Philosophers. p. 23. ISBN 978-0252025594.
12. Alexander Herda. Burying a sage: the heroon of Thales in the agora of Miletos: With remarks on some other excavated Heroa and on cults and graves of the mythical founders of the city. 2èmes Rencontres d'archéologie de l'IFEA : Le Mort dans la ville Pratiques, contextes et impacts des inhumations intra-muros en Anatolie, du début de l'Age du Bronze à l'époque romaine., Nov 2011, Istanbul, Turkey. pp. 67–122
13. Yust, Walter (1952). Encyclopaedia Britannica: A New Survey of Universal Knowledge. Encyclopaedia Britannica. p. 13.
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19. Plutarch, Life of Solon § 2.4
20. O'Grady, Patricia F. (2017). Thales of Miletus: The Beginnings of Western Science and Philosophy. Taylor & Francis. p. 263. ISBN 978-1-351-89537-8.
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27. Diogenes Laertius (II, 1)
28. "LacusCurtius • Ancient Astronomy: Polus (Smith's Dictionary, 1875)". penelope.uchicago.edu.
29. Zhmud, Leonid (2006). The Origin of the History of Science in Classical Antiquity. Die Deutsche Bibliothek.
30. Diogenes Laërtius 1.22
31. "SOL Search". www.cs.uky.edu.
32. Lives I.40
33. "SOL Search". www.cs.uky.edu.
34. Satires 11.27
35. Vico, Giambattista; Visconti, Gian Galeazzo (1993). On humanistic education: (six inaugural orations, 1699–1707). Six Inaugural Orations, 1699–1707 From the Definitive Latin Text, Introduction, and Notes of Gian Galeazzo Visconti. Cornell University Press. p. 4. ISBN 0801480876.
36. Doctoral thesis, "Know Thyself in Greek and Latin Literature," Eliza G. Wilkens, U. Chi, 1917, p. 12 ( online).
37. Laërtius 1925, § 28
38. Diogenes Laërtius 1.25
39. Herodotus translated by George Rawlinson. "The Histories".
40. Patricia F. O'Grady (2017). Thales of Miletus: The Beginnings of Western Science and Philosophy. Taylor & Francis. p. 102. ISBN 978-1-351-89536-1.
41. Aristotle, Metaphysics Alpha, 983b18.
42. Smith, William, ed. (1870). "Thales". Dictionary of Greek and Roman Biography and Mythology. p. 1016.
43. Plutarch, Life of Solon, 3.5
44. Aristotle. Metaphysics. 983 b6 8–11.
45. Aristotle. "Book I 983b". Aristotle, Metaphysics. Perseus Project.
46. See Aristotle, Metaphysics Alpha, 983b 1–27.
47. Diogenes Laërtius. Lives of the Eminent Philosophers. Book 1, paragraph 27.
48. Feldman, Abraham (October 1945). "Thoughts on Thales". The Classical Journal. 41 (1): 4–6. ISSN 0009-8353. JSTOR 3292119.
49. Metaphysics 983 b6 8–11, 17–21.
50. § 3
51. Boyer 1989, p. 43 (3rd ed.)
52. Fletcher, Colin R. (December 1982). "Thales – our founder?". The Mathematical Gazette. 66 (438): 267. doi:10.2307/3615512. JSTOR 3615512. S2CID 125626522.
53. Gerasa.), Nicomachus (of (8 May 1926). "Introduction to Arithmetic". Macmillan – via Google Books.
54. A History of Greek Mathematics, Heath, p. 70
55. Shute, William George; Shirk, William W.; Porter, George F. (1960). Plane and Solid Geometry. American Book Company. pp. 25–27.
56. Laërtius 1925, § 35
57. Boyer 1989, p. "Ionia and the Pythagoreans" p. 43.
58. de Laet, Siegfried J. (1996). History of Humanity: Scientific and Cultural Development. UNESCO, Volume 3, p. 14. ISBN 92-3-102812-X
59. Boyer, Carl B. and Merzbach, Uta C. (2010). A History of Mathematics. John Wiley and Sons, Chapter IV. ISBN 0-470-63056-6
60. Heath, Thomas L. (1956). The Thirteen Books of Euclid's Elements. Vol. 2 (Books 3–9) (2nd ed.). Dover. p. 61. ISBN 0486600890. Originally published by Cambridge University Press. 1st edition 1908, 2nd edition 1926.
61. canto 13, lines 101–102
62. Plutarch, Moralia, The Dinner of the Seven Wise Men, 147A
63. J J O'Connor and E F Robertson
64. "Ars Quatuor Coronatorum: Being the Transactions of the Quatuor Coronati Lodge No. 2076, London". W. J. Parre H, Limited. 10 June 1897 – via Google Books.
65. History of Astronomy, by Richard Perason, p. 65
66. Proclus, In Euclidem, 352
67. Plutarch, De Pythiae oraculis, 18.
68. D.L. I.34
69. Allman, George Johnston (1911). "Thales of Miletus" . In Chisholm, Hugh (ed.). Encyclopædia Britannica. Vol. 26 (11th ed.). Cambridge University Press. p. 721.
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71. Pseudo-Plutarch, Placita Philosopharum § 2.13
72. Pseudo-Plutarch, Placita Philosopharum § 2.28
73. Krech III, Shepard; Merchant, Carolyn; McNeill, John Robert, eds. (2003). "Earthquakes". Encyclopedia of World Environmental History. Vol. 1: A–G. Routledge. pp. 358–364.
74. History of Meteorology to 1800 by H. Howard Frisinger p. 3
75. Ibid, p. 4
76. George Crawford, Bidyut Sen – Derivatives for Decision Makers: Strategic Management Issues, John Wiley & Sons, 1996 ISBN 978-0471129943
77. Aristotle, Politics 1259a
78. Singer, C. (2008). A Short History of Science to the 19th century. Streeter Press. p. 35.
79. Needham, C. W. (1978). Cerebral Logic: Solving the Problem of Mind and Brain. Loose Leaf. p. 75. ISBN 978-0-398-03754-3.
80. Finkelberg, Aryeh (2017). Heraclitus and Thales' Conceptual Scheme: A Historical Study. Brill. p. 318, fn. 38. ISBN 978-9004338210.
81. Herodotus: Histories 1,74,2 (online)
82. Alden A. Mosshammer: Thales' Eclipse. Transactions of the American Philological Association, Vol. 111, 1981, pp. 145–155 (JSTOR)
83. Otta Wenskus (2016). "Die angebliche Vorhersage einer Sonnenfinsternis durch Thales von Milet. Warum sich diese Legende so hartnäckig hält und warum es wichtig ist, ihr nicht zu glauben" (PDF) (in German). pp. 2–17.
84. Theaetetus (174 A)
85. D.L. II.4–5
86. Powell, Jim (1 March 1995). "Brilliant but Absent-Minded Adam Smith | Jim Powell". fee.org.
87. Theaetetus 174a
88. O'Connor, John J.; Robertson, Edmund F. "Thales of Miletus". MacTutor History of Mathematics Archive. University of St Andrews.
89. Bill Thayer. "75". Herodotus. Retrieved 19 January 2019. {{cite book}}: |work= ignored (help)
90. Plato, Republic, Book 10, section 600a
91. Aristotle. De Anima. p. 411a7.
92. Nathan Ida, Engineering Electromagnetics, Springer, 2015 ISBN 3319078062
93. "Preocratic Reflexitivity, p. 97" (Document). {{cite document}}: Cite document requires |publisher= (help)
94. Farrington, B., 1944 Greek Science. Pelican
95. Cicero. De Natura Deorum. p. i.,10.
96. Fielding, Henry (1775). An essay on conversation. John Bell. p. 346.
97. Nicholas J. Molinari, Acheloios, Thales, and the Origin of Philosophy: A Response to the Neo-Marxians. Oxford: Archaeopress, 2022 https://www.archaeopress.com/Archaeopress/Products/9781803270869; cf. also Nicholas J. Molinari, Concerning Water as the Archai: Acheloios, Thales, and the Origin of Philosophy. A Dissertation Providing Philosophical, Mythological, and Archaeological Responses to the Neo-Marxians, Doctoral Dissertation, Newport, RI: Salve Regina University, 2020 https://philpapers.org/rec/MOLCWA-2
98. Diogenes Laërtius. "Lives of Eminent Philosophers".
99. The Vienna Lecture
100. G. S. Kirk (1 July 1960). Popper on science and presocratics. p. 330. doi:10.1093/mind/LXIX.275.318. ISSN 0026-4423. JSTOR 2251995. OCLC 4649661606. Archived from the original (PDF) on 17 September 2020. {{cite book}}: |journal= ignored (help)
101. Daniel W. Graham. "Anaximenes (d. 528 B.C.E.)". IEP. Retrieved 20 July 2019.
102. Burnet, John (1892). Early Greek Philosophy. A. and C. Black. p. 29.
103. Life of Pythagoras 3.13
104. Curnow, Trevor (22 June 2006). The Philosophers of the Ancient World: An A–Z Guide. A&C Black. ISBN 9780715634974 – via Google Books.
105. Webb, E. J. (1921). "Cleostratus Redivivus". The Journal of Hellenic Studies. 41: 70–85. doi:10.2307/624797. JSTOR 624797. S2CID 250254883 – via JSTOR.
Works cited
• Boyer, C.B. (1989). A History of Mathematics (2nd ed.). New York: Wiley. ISBN 978-0-471-09763-1. (1991 pbk ed. ISBN 0-471-54397-7; 2011 3rd edition)
• Burnet, John (1957) [1892]. Early Greek Philosophy. The Meridian Library. Third Edition
• Laërtius, Diogenes (1925). "The Seven Sages: Thales" . Lives of the Eminent Philosophers. Vol. 1:1. Translated by Hicks, Robert Drew (Two volume ed.). Loeb Classical Library.
• Herodotus, Histories, A. D. Godley (translator), Cambridge: Harvard University Press, 1920; ISBN 0-674-99133-8. Online version at Perseus
• Hans Joachim Störig, Kleine Weltgeschichte der Philosophie. Fischer, Frankfurt/M. 2004, ISBN 3-596-50832-0.
• Lloyd, G. E. R. Early Greek Science: Thales to Aristotle.
• Nahm, Milton C. (1962) [1934]. Selections from Early Greek Philosophy. Appleton-Century-Crofts.
• Pliny the Elder, The Natural History (eds. John Bostock, M.D., F.R.S. H.T. Riley, Esq., B.A.) London. Taylor and Francis. (1855). Online version at the Perseus Digital Library.
• William, Turner (1913). "Ionian School of Philosophy" . In Herbermann, Charles (ed.). Catholic Encyclopedia. New York: Robert Appleton Company.
• Allman, George Johnston (1911). "Thales of Miletus" . In Chisholm, Hugh (ed.). Encyclopædia Britannica. Vol. 26 (11th ed.). Cambridge University Press. p. 721.
Further reading
• Couprie, Dirk L. (2011). Heaven and Earth in Ancient Greek Cosmology: from Thales to Heraclides Ponticus. Springer. ISBN 978-1441981158.
• Luchte, James (2011). Early Greek Thought: Before the Dawn. London: Bloomsbury Publishing. ISBN 978-0567353313.
• O'Grady, Patricia F. (2002). Thales of Miletus: The Beginnings of Western Science and Philosophy. Western Philosophy Series. Vol. 58. Ashgate. ISBN 978-0754605331.
• Mazzeo, Pietro (2010). Talete, il primo filosofo. Bari: Editrice Tipografica.
• Priou, Alex (2016). "The Origin and Foundations of Milesian Thought." The Review of Metaphysics 70, 3–31.
• Wöhrle, Georg., ed. (2014). The Milesians: Thales. Translation and additional material by Richard McKirahan. Traditio Praesocratica. Vol. 1. Walter de Gruyter. ISBN 978-3-11-031525-7.
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| Wikipedia |
3x + 1 semigroup
In algebra, the 3x + 1 semigroup is a special subsemigroup of the multiplicative semigroup of all positive rational numbers.[1] The elements of a generating set of this semigroup are related to the sequence of numbers involved in the still open Collatz conjecture or the "3x + 1 problem". The 3x + 1 semigroup has been used to prove a weaker form of the Collatz conjecture. In fact, it was in such context the concept of the 3x + 1 semigroup was introduced by H. Farkas in 2005.[2] Various generalizations of the 3x + 1 semigroup have been constructed and their properties have been investigated.[3]
Definition
The 3x + 1 semigroup is the multiplicative semigroup of positive rational numbers generated by the set
$\{2\}\cup \left\{{\frac {2k+1}{3k+2}}:k\geq 0\right\}=\left\{2,{\frac {1}{2}},{\frac {3}{5}},{\frac {5}{8}},{\frac {7}{11}},\ldots \right\}.$
The function T : Z → Z, where Z is the set of all integers, as defined below is used in the "shortcut" definition of the Collatz conjecture:
$T(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n{\text{ is even}}\\[4px]{\frac {3n+1}{2}}&{\text{if }}n{\text{ is odd}}\end{cases}}$
The Collatz conjecture asserts that for each positive integer n, there is some iterate of T with itself which maps n to 1, that is, there is some integer k such that T(k)(n) = 1. For example if n = 7 then the values of T(k)(n) for k = 1, 2, 3,... are 11, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1 and T(11)(7) = 1.
The relation between the 3x + 1 semigroup and the Collatz conjecture is that the 3x + 1 semigroup is also generated by the set
$\left\{{\dfrac {n}{T(n)}}:n>0\right\}.$
The weak Collatz conjecture
The weak Collatz conjecture asserts the following: "The 3x + 1 semigroup contains every positive integer." This was formulated by Farkas and it has been proved to be true as a consequence of the following property of the 3x + 1 semigroup:[1]
The 3x + 1 semigroup S equals the set of all positive rationals a/b in lowest terms having the property that b ≠ 0 (mod 3). In particular, S contains every positive integer.
The wild semigroup
The semigroup generated by the set
$\left\{{\frac {1}{2}}\right\}\cup \left\{{\frac {3k+2}{2k+1}}:k\geq 0\right\},$
which is also generated by the set
$\left\{{\frac {T(n)}{n}}:n>0\right\},$
is called the wild semigroup. The integers in the wild semigroup consists of all integers m such that m ≠ 0 (mod 3).[4]
See also
• Wild number
References
1. Applegate, David; Lagarias, Jeffrey C. (2006). "The 3x + 1 semigroup". Journal of Number Theory. 117 (1): 146–159. doi:10.1016/j.jnt.2005.06.010. MR 2204740.
2. H. Farkas (2005). "Variants of the 3 N + 1 problem and multiplicative semigroups", Geometry, Spectral Theory, Groups and Dynamics: Proceedings in Memor y of Robert Brooks. Springer.
3. Ana Caraiani. "Multiplicative Semigroups Related to the 3x+1 Problem" (PDF). Princeton University. Retrieved 17 March 2016.
4. J.C. Lagarias (2006). "Wild and Wooley numbers" (PDF). American Mathematical Monthly. 113 (2): 97–108. doi:10.2307/27641862. JSTOR 27641862. Retrieved 18 March 2016.
| Wikipedia |
The Algorithm Auction
The Algorithm Auction is the world's first auction of computer algorithms.[1] Created by Ruse Laboratories, the initial auction featured seven lots and was held at the Cooper Hewitt, Smithsonian Design Museum on March 27, 2015.[2]
Not to be confused with Auction algorithm.
Part of a series on
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• Software
Five lots were physical representations of famous code or algorithms, including a signed, handwritten copy of the original Hello, World! C program by its creator Brian Kernighan on dot-matrix printer paper, a printed copy of 5,000 lines of Assembly code comprising the earliest known version of Turtle Graphics, signed by its creator Hal Abelson, a necktie containing the six-line qrpff algorithm capable of decrypting content on a commercially produced DVD video disc, and a pair of drawings representing OkCupid's original Compatibility Calculation algorithm, signed by the company founders.[3] The qrpff lot sold for $2,500.[4]
Two other lots were “living algorithms,” including a set of JavaScript tools for building applications that are accessible to the visually impaired and the other is for a program that converts lines of software code into music.[5] Winning bidders received, along with artifacts related to the algorithms, a full intellectual property license to use, modify, or open-source the code.[6] All lots were sold, with Hello World receiving the most bids.[7]
Exhibited alongside the auction lots were a facsimile of the Plimpton 322 tablet on loan from Columbia University, and Nigella, an art-world facing computer virus named after Nigella Lawson and created by cypherpunk and hacktivist Richard Jones.[8]
Sebastian Chan, Director of Digital & Emerging Media at the Cooper–Hewitt,[9] attended the event remotely from Milan, Italy via a Beam Pro telepresence robot.[10]
Effects
Following the auction, the Museum of Modern Art held a salon titled The Way of the Algorithm highlighting algorithms as "a ubiquitous and indispensable component of our lives."[11]
References
1. "The Algorithm Auction". Artsy. Artsy. Retrieved 22 April 2015.
2. Turner, Zeke (23 March 2015). "Beautiful Code". The New Yorker. Condé Nast. Retrieved 22 April 2015.
3. Johnson, Phil. "Coming to an art gallery near you: Software code". ITworld. IDG Enterprise. Archived from the original on 2016-03-04. Retrieved 22 April 2015.
4. Hotz, Robert Lee (27 May 2015). "What's Hot in the Art World? Algorithms". The Wall Street Journal. Dow Jones & Company. Retrieved 19 June 2015.
5. Stinson, Liz. "The First Auction for Algorithms Is Attracting $1,000 Bids". Wired. Condé Nast. Retrieved 22 April 2015.
6. "Anthony Ferraro – Hypothetical Beats". Artsy. Artsy. Retrieved 22 April 2015.
7. Spilka, Simone (27 March 2015). "Algorithm Auction Proves Code is Art". PSFK. Retrieved 22 April 2015.
8. "The Algorithm Auction Press Release". Artsy. Ruse Laboratories. 19 March 2015. Retrieved 22 April 2015.
9. simonsc (9 September 2013). "Meet the Staff: Sebastian Chan". Cooper Hewitt. Smithsonian. Retrieved 22 April 2015.
10. "Beam Pro Robot-mosphere". BFA. Retrieved 22 April 2015.
11. Antonelli, Paola. "The Way of the Algorithm". MoMA R&D. Retrieved 19 June 2015.
| Wikipedia |
The Ancient Tradition of Geometric Problems
The Ancient Tradition of Geometric Problems is a book on ancient Greek mathematics, focusing on three problems now known to be impossible if one uses only the straightedge and compass constructions favored by the Greek mathematicians: squaring the circle, doubling the cube, and trisecting the angle. It was written by Wilbur Knorr (1945–1997), a historian of mathematics, and published in 1986 by Birkhäuser. Dover Publications reprinted it in 1993.
Topics
The Ancient Tradition of Geometric Problems studies the three classical problems of circle-squaring, cube-doubling, and angle trisection throughout the history of Greek mathematics,[1][2] also considering several other problems studied by the Greeks in which a geometric object with certain properties is to be constructed, in many cases through transformations to other construction problems.[2] The study runs from Plato and the story of the Delian oracle to the second century BC, when Archimedes and Apollonius of Perga flourished;[1][3] Knorr suggests that the decline in Greek geometry after that time represented a shift in interest to other topics in mathematics rather than a decline in mathematics as a whole.[3] Unlike the earlier work on this material by Thomas Heath, Knorr sticks to the source material as it is, reconstructing the motivation and lines of reasoning followed by the Greek mathematicians and their connections to each other, rather than adding justifications for the correctness of the constructions based on modern mathematical techniques.[4]
In modern times, the impossibility of solving the three classical problems by straightedge and compass, finally proven in the 19th century,[5] has often been viewed as analogous to the foundational crisis of mathematics of the early 20th century, in which David Hilbert's program of reducing mathematics to a system of axioms and calculational rules struggled against logical inconsistencies in its axiom systems, intuitionist rejection of formalism and dualism, and Gödel's incompleteness theorems showing that no such axiom system could formalize all mathematical truths and remain consistent. However, Knorr argues in The Ancient Tradition of Geometric Problems that this point of view is anachronistic,[1] and that the Greek mathematicians themselves were more interested in finding and classifying the mathematical tools that could solve these problems than they were in imposing artificial limitations on themselves and in the philosophical consequences of these limitations.[1][2][3][4]
When a geometric construction problem does not admit a compass-and-straightedge solution, then either the constraints on the problem or on the solution techniques can be relaxed, and Knorr argues that the Greeks did both. Constructions described by the book include the solution by Menaechmus of doubling the cube by finding the intersection points of two conic sections, several neusis constructions involving fitting a segment of a given length between two points or curves, and the use of the Quadratrix of Hippias for trisecting angles and squaring circles.[5] Some specific theories on the authorship of Greek mathematics, put forward by the book, include the legitimacy of a letter on square-doubling from Eratosthenes to Ptolemy III Euergetes,[6] a distinction between Socratic-era sophist Hippias and the Hippias who invented the quadratrix, and a similar distinction between Aristaeus the Elder, a mathematician of the time of Euclid, and the Aristaeus who authored a book on solids (mentioned by Pappus of Alexandria), and whom Knorr places at the time of Apollonius.[4][6]
The book is heavily illustrated, and many endnotes provide sources for quotations, additional discussion, and references to related research.[7]
Audience and reception
The book is written for a general audience, unlike a follow-up work published by Knorr, Textual Studies in Ancient and Medieval Geometry (1989), which is aimed at other experts in the close reading of Greek mathematical texts.[1] Nevertheless, reviewer Alan Stenger calls The Ancient Tradition of Geometric Problems "very specialized and scholarly".[7] Reviewer Colin R. Fletcher calls it "essential reading" for understanding the background and content of the Greek mathematical problem-solving tradition.[2] In its historical scholarship, historian of mathematics Tom Whiteside writes that the book's occasionally speculative nature is justified by its fresh interpretations, well-founded conjectures, and deep knowledge of the subject.[5]
References
1. Drucker, Thomas (December 1991), "Review of The Ancient Tradition of Geometric Problems", Isis, 82 (4): 718–720, JSTOR 233339
2. Fletcher, C. R. (1988), "Review of The Ancient Tradition of Geometric Problems", Mathematical Reviews, MR 0884893
3. Neuenschwander, E., "Review of The Ancient Tradition of Geometric Problems", zbMATH (in German), Zbl 0588.01002
4. Caveing, Maurice (July–December 1991), "Review of The Ancient Tradition of Geometric Problems", Revue d'histoire des sciences (in French), 44 (3/4): 487–489, JSTOR 23632881
5. Whiteside, D. T. (September 1990), "Review of The Ancient Tradition of Geometric Problems", The British Journal for the History of Science, 23 (3): 373–375, JSTOR 4026791
6. Bulmer-Thomas, Ivor (1989), "Ancient geometry (review of The Ancient Tradition of Geometric Problems)", The Classical Review, New Series, 39 (2): 364–365, JSTOR 711650
7. Stenger, Allen (February 2013), "Review of The Ancient Tradition of Geometric Problems", MAA Reviews, Mathematical Association of America
External links
• The Ancient Tradition of Geometric Problems at the Internet Archive
| Wikipedia |
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