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Rhombicosidodecahedral prism
In geometry, a rhombicosidodecahedral prism or small rhombicosidodecahedral prism is a convex uniform polychoron (four-dimensional polytope).
Rhombicosidodecahedral prism
Schlegel diagram
One rhombicosidodecahedron and triangular prisms show
TypePrismatic uniform polychoron
Uniform index61
Schläfli symbolt0,2,3{3,5,2} or rr{3,5}×{}
Coxeter-Dynkin
Cells64 total:
2 rr{5,3}
12 {}x{5}
20 {}x{3}
30 {4,3}
Faces244 total:40 {3}
180 {4}
24 {5}
Edges300
Vertices120
Vertex figure
Trapezoidal pyramid
Symmetry group[5,3,2], order 240
Propertiesconvex
It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes.
Alternative names
• (small) rhombicosidodecahedral dyadic prism (Norman W. Johnson)
• Sriddip (Jonathan Bowers: for small-rhombicosidodecahedral prism)
• (small) rhombicosidodecahedral hyperprism
External links
• 6. Convex uniform prismatic polychora - Model 61, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora) x x3o5x - sriddip".
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Wikipedia
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Rhombicuboctahedral prism
In geometry, a rhombicuboctahedral prism is a convex uniform polychoron (four-dimensional polytope).
Rhombicuboctahedral prism
TypePrismatic uniform polychoron
Uniform index53
Schläfli symbolt0,2,3{3,4,2} or rr{3,4}×{}
s2,3{3,4,2} or s2{3,4}×{}
Coxeter diagram
Cells28 total:
2 rr{4,3} or s2{3,4}
8 {}x{3}
18 {4,3}
Faces100 total:
16 {3}
84 {4}
Edges120
Vertices48
Vertex figure
Trapezoidal pyramid
Symmetry group[4,3,2], order 96
[3+,4,2], order 48
Propertiesconvex
It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes.
Images
Net
Schlegel diagram
One rhombicuboctahedron and triangular prisms show
Alternative names
• small rhombicuboctahedral prism
• (Small) rhombicuboctahedral dyadic prism (Norman W. Johnson)
• Sircope (Jonathan Bowers: for small-rhombicuboctahedral prism)
• (small) rhombicuboctahedral hyperprism
Related polytopes
Runcic snub cubic hosochoron
Runcic snub cubic hosochoron
Schläfli symbols3{2,4,3}
Coxeter diagram
Cells16 total:
2 t{3,3}
6 {3,3}
8 tricup
Faces52 total:
32 {3}
12{4}
8 {6}
Edges60
Vertices24
Vertex figure
Symmetry group[4,3,2+], order 48
Propertiesconvex
A related polychoron is the runcic snub cubic hosochoron, also known as a parabidiminished rectified tesseract, truncated tetrahedral alterprism, or truncated tetrahedral cupoliprism, s3{2,4,3}, . It is made from 2 truncated tetrahedra, 6 tetrahedra, and 8 triangular cupolae in the gaps, for a total of 16 cells, 52 faces, 60 edges, and 24 vertices. It is vertex-transitive, and equilateral, but not uniform, due to the cupolae. It has symmetry [2+,4,3], order 48.[1][2][3]
It is related to the 16-cell in its s{2,4,3}, construction.
It can also be seen as a prismatic polytope with two parallel truncated tetrahedra in dual positions, as seen in the compound of two truncated tetrahedra. Triangular cupolae connect the triangular and hexagonal faces, and the tetrahedral connect edge-wise between.
Projection
(triangular cupolae hidden)
Net
References
1. Klitzing, Richard. "4D tutcup".
2. Category S1: Simple Scaliforms Tutcup
3. http://bendwavy.org/klitzing/pdf/artConvSeg_8.pdf 4.55 truncated tetrahedron || inverse truncated tetrahedron
External links
• 6. Convex uniform prismatic polychora - Model 53, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora) x3o4x - sircope".
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Wikipedia
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Small rhombidodecacron
In geometry, the small rhombidodecacron is a nonconvex isohedral polyhedron. It is the dual of the small rhombidodecahedron. It is visually identical to the Small dodecacronic hexecontahedron. It has 60 intersecting antiparallelogram faces.
Small rhombidodecacron
TypeStar polyhedron
Face
ElementsF = 60, E = 120
V = 42 (χ = −18)
Symmetry groupIh, [5,3], *532
Index referencesDU39
dual polyhedronSmall rhombidodecahedron
Proportions
Each face has two angles of $\arccos({\frac {5}{8}}+{\frac {1}{8}}{\sqrt {5}})\approx 25.242\,832\,961\,52^{\circ }$ and two angles of $\arccos(-{\frac {1}{2}}+{\frac {1}{5}}{\sqrt {5}})\approx 93.025\,844\,508\,96^{\circ }$. The diagonals of each antiparallelogram intersect at an angle of $\arccos({\frac {1}{4}}+{\frac {1}{10}}{\sqrt {5}})\approx 61.731\,322\,529\,52^{\circ }$. The ratio between the lengths of the long edges and the short ones equals ${\frac {1}{2}}+{\frac {1}{2}}{\sqrt {5}}$, which is the golden ratio. The dihedral angle equals $\arccos({\frac {-19-8{\sqrt {5}}}{41}})\approx 154.121\,363\,125\,78^{\circ }$.
References
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208
External links
• Weisstein, Eric W. "Small rhombidodecacron". MathWorld.
• Uniform polyhedra and duals
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
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Small rhombidodecahedron
In geometry, the small rhombidodecahedron is a nonconvex uniform polyhedron, indexed as U39. It has 42 faces (30 squares and 12 decagons), 120 edges, and 60 vertices.[1] Its vertex figure is a crossed quadrilateral.
Small rhombidodecahedron
TypeUniform star polyhedron
ElementsF = 42, E = 120
V = 60 (χ = −18)
Faces by sides30{4}+12{10}
Coxeter diagram (with extra double-covered triangles)
(with extra double-covered pentagons)
Wythoff symbol2 5 (3/2 5/2) |
Symmetry groupIh, [5,3], *532
Index referencesU39, C46, W74
Dual polyhedronSmall rhombidodecacron
Vertex figure
4.10.4/3.10/9
Bowers acronymSird
Related polyhedra
It shares its vertex arrangement with the small stellated truncated dodecahedron and the uniform compounds of 6 or 12 pentagrammic prisms. It additionally shares its edge arrangement with the rhombicosidodecahedron (having the square faces in common), and with the small dodecicosidodecahedron (having the decagonal faces in common).
Rhombicosidodecahedron
Small dodecicosidodecahedron
Small rhombidodecahedron
Small stellated truncated dodecahedron
Compound of six pentagrammic prisms
Compound of twelve pentagrammic prisms
Small rhombidodecacron
Small rhombidodecacron
TypeStar polyhedron
Face
ElementsF = 60, E = 120
V = 42 (χ = −18)
Symmetry groupIh, [5,3], *532
Index referencesDU39
dual polyhedronSmall rhombidodecahedron
The small rhombidodecacron (or small dipteral ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the small rhombidodecahedron. It is visually identical to the Small dodecacronic hexecontahedron. It has 60 intersecting antiparallelogram faces.
References
1. Maeder, Roman. "39: small rhombidodecahedron". MathConsult.{{cite web}}: CS1 maint: url-status (link)
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208
External links
• Weisstein, Eric W. "Small rhombidodecahedron". MathWorld.
• Weisstein, Eric W. "Small rhombidodecacron". MathWorld.
• Uniform polyhedra and duals
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
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Small rhombihexacron
In geometry, the small rhombihexacron (or small dipteral disdodecahedron) is the dual of the small rhombihexahedron. It is visually identical to the small hexacronic icositetrahedron. Its faces are antiparallelograms formed by pairs of coplanar triangles.
Small rhombihexacron
TypeStar polyhedron
Face
ElementsF = 24, E = 48
V = 18 (χ = −6)
Symmetry groupOh, [4,3], *432
Index referencesDU18
dual polyhedronSmall rhombihexahedron
Proportions
Each antiparallelogram has two angles of $\arccos({\frac {1}{4}}+{\frac {1}{2}}{\sqrt {2}})\approx 16.842\,116\,236\,30^{\circ }$ and two angles of $\arccos(-{\frac {1}{2}}+{\frac {1}{4}}{\sqrt {2}})\approx 98.421\,058\,118\,15^{\circ }$. The diagonals of each antiparallelogram intersect at an angle of $\arccos({\frac {1}{4}}+{\frac {1}{8}}{\sqrt {2}})\approx 64.736\,825\,645\,55^{\circ }$. The dihedral angle equals $\arccos({\frac {-7-4{\sqrt {2}}}{17}})\approx 138.117\,959\,055\,51^{\circ }$. The ratio between the lengths of the long edges and the short ones equals ${\sqrt {2}}$.
References
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208
External links
Weisstein, Eric W. "Small rhombihexacron". MathWorld.
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Wikipedia
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Rhombitetrahexagonal tiling
In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{6,4}. It can be seen as constructed as a rectified tetrahexagonal tiling, r{6,4}, as well as an expanded order-4 hexagonal tiling or expanded order-6 square tiling.
Rhombitetrahexagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.4.6.4
Schläfli symbolrr{6,4} or $r{\begin{Bmatrix}6\\4\end{Bmatrix}}$
Wythoff symbol4 | 6 2
Coxeter diagram
Symmetry group[6,4], (*642)
DualDeltoidal tetrahexagonal tiling
PropertiesVertex-transitive
Constructions
There are two uniform constructions of this tiling, one from [6,4] or (*642) symmetry, and secondly removing the mirror middle, [6,1+,4], gives a rectangular fundamental domain [∞,3,∞], (*3222).
Two uniform constructions of 4.4.4.6
Name Rhombitetrahexagonal tiling
Image
Symmetry [6,4]
(*642)
[6,1+,4] = [∞,3,∞]
(*3222)
=
Schläfli symbol rr{6,4} t0,1,2,3{∞,3,∞}
Coxeter diagram =
There are 3 lower symmetry forms seen by including edge-colorings: sees the hexagons as truncated triangles, with two color edges, with [6,4+] (4*3) symmetry. sees the yellow squares as rectangles, with two color edges, with [6+,4] (6*2) symmetry. A final quarter symmetry combines these colorings, with [6+,4+] (32×) symmetry, with 2 and 3 fold gyration points and glide reflections.
Lower symmetry constructions
[6,4], (*632)
[6,4+], (4*3)
[6+,4], (6*2)
[6+,4+], (32×)
This four color tiling is related to a semiregular infinite skew polyhedron with the same vertex figure in Euclidean 3-space with a prismatic honeycomb construction of .
Symmetry
The dual tiling, called a deltoidal tetrahexagonal tiling, represents the fundamental domains of the *3222 orbifold, shown here from three different centers. Its fundamental domain is a Lambert quadrilateral, with 3 right angles. This symmetry can be seen from a [6,4], (*642) triangular symmetry with one mirror removed, constructed as [6,1+,4], (*3222). Removing half of the blue mirrors doubles the domain again into *3322 symmetry.
Related polyhedra and tiling
*n42 symmetry mutation of expanded tilings: n.4.4.4
Symmetry
[n,4], (*n42)
Spherical Euclidean Compact hyperbolic Paracomp.
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]
*∞42
[∞,4]
Expanded
figures
Config. 3.4.4.4 4.4.4.4 5.4.4.4 6.4.4.4 7.4.4.4 8.4.4.4 ∞.4.4.4
Rhombic
figures
config.
V3.4.4.4
V4.4.4.4
V5.4.4.4
V6.4.4.4
V7.4.4.4
V8.4.4.4
V∞.4.4.4
Uniform tetrahexagonal tilings
Symmetry: [6,4], (*642)
(with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries)
(And [(∞,3,∞,3)] (*3232) index 4 subsymmetry)
=
=
=
=
=
=
=
=
=
=
=
=
{6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4}
Uniform duals
V64 V4.12.12 V(4.6)2 V6.8.8 V46 V4.4.4.6 V4.8.12
Alternations
[1+,6,4]
(*443)
[6+,4]
(6*2)
[6,1+,4]
(*3222)
[6,4+]
(4*3)
[6,4,1+]
(*662)
[(6,4,2+)]
(2*32)
[6,4]+
(642)
=
=
=
=
=
=
h{6,4} s{6,4} hr{6,4} s{4,6} h{4,6} hrr{6,4} sr{6,4}
Uniform tilings in symmetry *3222
64
6.6.4.4
(3.4.4)2
4.3.4.3.3.3
6.6.4.4
6.4.4.4
3.4.4.4.4
(3.4.4)2
3.4.4.4.4
46
See also
Wikimedia Commons has media related to Uniform tiling 4-4-4-6.
• 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
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Wikipedia
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Rhombitetrapentagonal tiling
In geometry, the rhombitetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,2{4,5}.
Rhombitetrapentagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.4.5.4
Schläfli symbolrr{5,4} or $r{\begin{Bmatrix}5\\4\end{Bmatrix}}$
Wythoff symbol4 | 5 2
Coxeter diagram or
Symmetry group[5,4], (*542)
DualDeltoidal tetrapentagonal tiling
PropertiesVertex-transitive
Dual tiling
The dual is called the deltoidal tetrapentagonal tiling with face configuration V.4.4.4.5.
Related polyhedra and tiling
Uniform pentagonal/square tilings
Symmetry: [5,4], (*542) [5,4]+, (542) [5+,4], (5*2) [5,4,1+], (*552)
{5,4} t{5,4} r{5,4} 2t{5,4}=t{4,5} 2r{5,4}={4,5} rr{5,4} tr{5,4} sr{5,4} s{5,4} h{4,5}
Uniform duals
V54 V4.10.10 V4.5.4.5 V5.8.8 V45 V4.4.5.4 V4.8.10 V3.3.4.3.5 V3.3.5.3.5 V55
*n42 symmetry mutation of expanded tilings: n.4.4.4
Symmetry
[n,4], (*n42)
Spherical Euclidean Compact hyperbolic Paracomp.
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]
*∞42
[∞,4]
Expanded
figures
Config. 3.4.4.4 4.4.4.4 5.4.4.4 6.4.4.4 7.4.4.4 8.4.4.4 ∞.4.4.4
Rhombic
figures
config.
V3.4.4.4
V4.4.4.4
V5.4.4.4
V6.4.4.4
V7.4.4.4
V8.4.4.4
V∞.4.4.4
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.
See also
Wikimedia Commons has media related to Uniform tiling 4-4-4-5.
• Uniform tilings in hyperbolic plane
• List of regular polytopes
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
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Small ditrigonal dodecacronic hexecontahedron
In geometry, the small ditrigonal dodecacronic hexecontahedron (or fat star) is a nonconvex isohedral polyhedron. It is the dual of the uniform small ditrigonal dodecicosidodecahedron. It is visually identical to the small dodecicosacron. Its faces are darts. A part of each dart lies inside the solid, hence is invisible in solid models.
Small ditrigonal dodecacronic hexecontahedron
TypeStar polyhedron
Face
ElementsF = 60, E = 120
V = 44 (χ = −16)
Symmetry groupIh, [5,3], *532
Index referencesDU43
dual polyhedronSmall ditrigonal dodecicosidodecahedron
Proportions
Faces have two angles of $\arccos({\frac {5}{12}}+{\frac {1}{4}}{\sqrt {5}})\approx 12.661\,078\,804\,43^{\circ }$, one of $\arccos(-{\frac {5}{12}}-{\frac {1}{60}}{\sqrt {5}})\approx 116.996\,396\,851\,70^{\circ }$ and one of $360^{\circ }-\arccos(-{\frac {1}{12}}-{\frac {19}{60}}{\sqrt {5}})\approx 217.681\,445\,539\,45^{\circ }$. Its dihedral angles equal $\arccos({\frac {-44-3{\sqrt {5}}}{61}})\approx 146.230\,659\,755\,53^{\circ }$. The ratio between the lengths of the long and short edges is ${\frac {31+5{\sqrt {5}}}{38}}\approx 1.110\,008\,944\,41$.
References
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208
External links
• Weisstein, Eric W. "Small ditrigonal dodecacronic hexecontahedron". MathWorld.
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Large set (combinatorics)
In combinatorial mathematics, a large set of positive integers
$S=\{s_{0},s_{1},s_{2},s_{3},\dots \}$
is one such that the infinite sum of the reciprocals
${\frac {1}{s_{0}}}+{\frac {1}{s_{1}}}+{\frac {1}{s_{2}}}+{\frac {1}{s_{3}}}+\cdots $
diverges. A small set is any subset of the positive integers that is not large; that is, one whose sum of reciprocals converges.
Large sets appear in the Müntz–Szász theorem and in the Erdős conjecture on arithmetic progressions.
Examples
• Every finite subset of the positive integers is small.
• The set $\{1,2,3,4,5,\dots \}$ of all positive integers is a large set; this statement is equivalent to the divergence of the harmonic series. More generally, any arithmetic progression (i.e., a set of all integers of the form an + b with a ≥ 1, b ≥ 1 and n = 0, 1, 2, 3, ...) is a large set.
• The set of square numbers is small (see Basel problem). So is the set of cube numbers, the set of 4th powers, and so on. More generally, the set of positive integer values of any polynomial of degree 2 or larger forms a small set.
• The set {1, 2, 4, 8, ...} of powers of 2 is a small set, and so is any geometric progression (i.e., a set of numbers of the form of the form abn with a ≥ 1, b ≥ 2 and n = 0, 1, 2, 3, ...).
• The set of prime numbers is large. The set of twin primes is small (see Brun's constant).
• The set of prime powers which are not prime (i.e., all numbers of the form pn with n ≥ 2 and p prime) is small although the primes are large. This property is frequently used in analytic number theory. More generally, the set of perfect powers is small; even the set of powerful numbers is small.
• The set of numbers whose expansions in a given base exclude a given digit is small. For example, the set
$\{1,2,\dots ,5,6,8,9,\dots ,15,16,18,19,\dots ,65,66,68,69,80,81,\dots \}$
of integers whose decimal expansion does not include the digit 7 is small. Such series are called Kempner series.
• Any set whose upper asymptotic density is nonzero, is large.
Properties
• Every subset of a small set is small.
• The union of finitely many small sets is small, because the sum of two convergent series is a convergent series. (In set theoretic terminology, the small sets form an ideal.)
• The complement of every small set is large.
• The Müntz–Szász theorem states that a set $S=\{s_{1},s_{2},s_{3},\dots \}$ is large if and only if the set of polynomials spanned by
$\{1,x^{s_{1}},x^{s_{2}},x^{s_{3}},\dots \}$
is dense in the uniform norm topology of continuous functions on a closed interval in the positive real numbers. This is a generalization of the Stone–Weierstrass theorem.
Open problems involving large sets
Paul Erdős conjectured that all large sets contain arbitrarily long arithmetic progressions. He offered a prize of $3000 for a proof, more than for any of his other conjectures, and joked that this prize offer violated the minimum wage law.[1] The question is still open.
It is not known how to identify whether a given set is large or small in general. As a result, there are many sets which are not known to be either large or small.
See also
• List of sums of reciprocals
Notes
1. Carl Pomerance, Paul Erdős, Number Theorist Extraordinaire. (Part of the article The Mathematics of Paul Erdős), in Notices of the AMS, January, 1998.
References
• A. D. Wadhwa (1975). An interesting subseries of the harmonic series. American Mathematical Monthly 82 (9) 931–933. JSTOR 2318503
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Small set expansion hypothesis
The small set expansion hypothesis or small set expansion conjecture in computational complexity theory is an unproven computational hardness assumption related to the unique games conjecture. Under the small set expansion hypothesis it is assumed to be computationally infeasible to distinguish between a certain class of expander graphs called "small set expanders" and other graphs that are very far from being small set expanders.
Statement
The edge expansion of a set $X$ of vertices in a graph $G$ is defined as
${\frac {|\partial X|}{|X|}},$
where the vertical bars denote the number of elements of a set, and $\partial X$ denotes the set of edges that have one endpoint in $X$ and the other endpoint in its complement.[lower-alpha 1] This number can be as low as zero, when $X$ is a connected component of the graph, because in this case there are no edges connecting $X$ to other parts of the graph. For a regular graph with $d$ edges incident to each vertex, the edge expansion can be as high as $d$, for a subset $X$ that induces an independent set, as in this case all of the edges that touch vertices in $X$ belong to $\partial X$.[1][2]
The edge expansion of a graph with $n$ vertices is defined to be the minimum edge expansion among its subsets of at most $n/2$ vertices.[lower-alpha 2] Instead, the small set expansion is defined as the same minimum, but only over smaller subsets, of at most $n/\log _{2}n$ vertices. A small set expander is a graph whose small set expansion is large. The small set expansion hypothesis asserts that, for every $\varepsilon >0$, it is NP-hard to distinguish between $d$-regular graphs with small set expansion at least $(1-\varepsilon )d$ (good small set expanders), and $d$-regular graphs with small set expansion at most $\varepsilon d$ (very far from being a small set expander).[1][lower-alpha 3]
Consequences
The small set expansion hypothesis implies the NP-hardness of several other computational problems. Although this does not prove that these problems actually are NP-hard, it nevertheless suggests that it would be difficult to find an efficient solution for them, because this would also solve other problems whose solution has so far been elusive (including the small set expansion problem itself). In the other direction, this opens the door to disproving the small set expansion hypothesis, by providing other problems through which it could be attacked.[1]
In particular, there exists a polynomial-time reduction from the recognition of small set expanders to the problem of determining the approximate value of unique games, showing that the small set expansion hypothesis implies the unique games conjecture.[1][2] Boaz Barak has suggested more strongly that these two hypotheses are equivalent.[1] In fact, the small set expansion hypothesis is equivalent to a restricted form of the unique games conjecture, asserting the hardness of unique games instances whose underlying graphs are small set expanders.[3] On the other hand, it is possible to quickly solve unique games instances whose graph is "certifiably" a small set expander, in the sense that this can be verified by sum-of-squares optimization.[4]
Another application of the small set expansion hypothesis concerns the computational problem of approximating the treewidth of graphs, a structural parameter closely related to expansion. For graphs of treewidth $w$, the best approximation ratio known for a polynomial time approximation algorithm is $O({\sqrt {\log w}})$.[5] The small set expansion hypothesis, if true, implies that there does not exist an approximation algorithm for this problem with constant approximation ratio.[6] It also can be used to imply the inapproximability of finding a complete bipartite graph with the maximum number of edges (possibly restricted to having equal numbers of vertices on each side of its bipartition) in a larger graph.[7]
The small set expansion hypothesis implies the optimality of known approximation ratios for certain variants of the edge cover problem, in which one must choose as few vertices as possible to cover a given number of edges in a graph.[8]
History and partial results
The small set expansion hypothesis was formulated, and connected to the unique games conjecture, by Prasad Raghavendra and David Steurer in 2010.[2]
One approach to resolving the small set expansion hypothesis is to seek approximation algorithms for the edge expansion of small vertex sets that would be good enough to distinguish the two classes of graphs in the hypothesis. In this light, the best approximation known, for the edge expansion of subsets of at most $n/\log n$ vertices in a $d$-regular graph, has an approximation ratio of $O({\sqrt {\log n\log \log n}})$. This is not strong enough to refute the hypothesis; doing so would require finding an algorithm with a bounded approximation ratio.[9]
Notes
1. This definition follows the notation used in the expander graph article; some sources, such as Raghavendra & Steurer (2010), instead normalize the edge expansion by dividing it by the degree of the graph.
2. This definition avoids using subsets whose number of vertices is close to $n$, because these subsets would have small expansion even in graphs that otherwise have high expansion.
3. This formulation is from Barak (2016), who notes that it eliminates some unimportant parameters appearing in other formulations of the same hypothesis, such as that in Raghavendra & Steurer (2010).
References
1. Barak, Boaz (2016), "SOS Lecture 6: The SOS approach to refuting the UGC" (PDF), Lecture notes on "Proofs, beliefs and algorithms through the lens of Sum of Squares", retrieved 2023-03-14
2. Raghavendra, Prasad; Steurer, David (2010), "Graph expansion and the unique games conjecture", in Schulman, Leonard J. (ed.), Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, Association for Computing Machinery, pp. 755–764, doi:10.1145/1806689.1806792
3. Raghavendra, Prasad; Steurer, David; Tulsiani, Madhur (2012), "Reductions between expansion problems", Proceedings of the 27th Conference on Computational Complexity, CCC 2012, Porto, Portugal, June 26-29, 2012, IEEE Computer Society, pp. 64–73, arXiv:1011.2586, doi:10.1109/CCC.2012.43
4. Bafna, Mitali; Barak, Boaz; Kothari, Pravesh K.; Schramm, Tselil; Steurer, David (2021), "Playing unique games on certified small-set expanders", in Khuller, Samir; Williams, Virginia Vassilevska (eds.), STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, Association for Computing Machinery, pp. 1629–1642, arXiv:2006.09969, doi:10.1145/3406325.3451099
5. Feige, Uriel; Hajiaghayi, Mohammadtaghi; Lee, James R. (2008), "Improved approximation algorithms for minimum weight vertex separators", SIAM Journal on Computing, 38 (2): 629–657, doi:10.1137/05064299X, MR 2411037
6. Wu, Yu; Austrin, Per; Pitassi, Toniann; Liu, David (2014), "Inapproximability of treewidth, one-shot pebbling, and related layout problems", Journal of Artificial Intelligence Research, 49: 569–600, doi:10.1613/jair.4030, MR 3195329
7. Manurangsi, Pasin (2018), "Inapproximability of maximum biclique problems, minimum k-cut and densest at-least-k-subgraph from the small set expansion hypothesis", Algorithms, 11 (1): P10:1–P10:22, arXiv:1705.03581, doi:10.3390/a11010010, MR 3758880
8. Gandhi, Rajiv; Kortsarz, Guy (2015), "On set expansion problems and the small set expansion conjecture" (PDF), Discrete Applied Mathematics, 194: 93–101, doi:10.1016/j.dam.2015.05.028, MR 3391764
9. Bansal, Nikhil; Feige, Uriel; Krauthgamer, Robert; Makarychev, Konstantin; Nagarajan, Viswanath; Naor, Joseph; Schwartz, Roy (2014), "Min-max graph partitioning and small set expansion" (PDF), SIAM Journal on Computing, 43 (2): 872–904, doi:10.1137/120873996, MR 3504685
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Small snub icosicosidodecahedron
In geometry, the small snub icosicosidodecahedron or snub disicosidodecahedron is a uniform star polyhedron, indexed as U32. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices. Its stellation core is a truncated pentakis dodecahedron. It also called a holosnub icosahedron, ß{3,5}.
Small snub icosicosidodecahedron
TypeUniform star polyhedron
ElementsF = 112, E = 180
V = 60 (χ = −8)
Faces by sides(40+60){3}+12{5/2}
Coxeter diagram
Wythoff symbol| 5/2 3 3
Symmetry groupIh, [5,3], *532
Index referencesU32, C41, W110
Dual polyhedronSmall hexagonal hexecontahedron
Vertex figure
35.5/2
Bowers acronymSeside
The 40 non-snub triangular faces form 20 coplanar pairs, forming star hexagons that are not quite regular. Unlike most snub polyhedra, it has reflection symmetries.
Convex hull
Its convex hull is a nonuniform truncated icosahedron.
Truncated icosahedron
(regular faces)
Convex hull
(isogonal hexagons)
Small snub icosicosidodecahedron
Cartesian coordinates
Cartesian coordinates for the vertices of a small snub icosicosidodecahedron are all the even permutations of
(±(1-ϕ+α), 0, ±(3+ϕα))
(±(ϕ-1+α), ±2, ±(2ϕ-1+ϕα))
(±(ϕ+1+α), ±2(ϕ-1), ±(1+ϕα))
where ϕ = (1+√5)/2 is the golden ratio and α = √3ϕ−2.
See also
• List of uniform polyhedra
• Small retrosnub icosicosidodecahedron
External links
• Weisstein, Eric W. "Small snub icosicosidodecahedron". MathWorld.
• Klitzing, Richard. "3D star small snub icosicosidodecahedron".
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Small stellapentakis dodecahedron
In geometry, the small stellapentakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the truncated great dodecahedron. It has 60 intersecting triangular faces.
Small stellapentakis dodecahedron
TypeStar polyhedron
Face
ElementsF = 60, E = 90
V = 24 (χ = −6)
Symmetry groupIh, [5,3], *532
Index referencesDU37
dual polyhedronTruncated great dodecahedron
Proportions
The triangles have two acute angles of $\arccos({\frac {1}{2}}+{\frac {1}{5}}{\sqrt {5}})\approx 18.699\,407\,085\,149^{\circ }$ and one obtuse angle of $\arccos({\frac {1}{10}}-{\frac {2}{5}}{\sqrt {5}})\approx 142.601\,185\,829\,70^{\circ }$. The dihedral angle equals $\arccos({\frac {-24-5{\sqrt {5}}}{41}})\approx 149.099\,125\,827\,35^{\circ }$. Part of each triangle lies within the solid, hence is invisible in solid models.
References
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208
External links
• Weisstein, Eric W. "Small stellapentakis dodecahedron". MathWorld.
• Uniform polyhedra and duals
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
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Small stellated 120-cell honeycomb
In the geometry of hyperbolic 4-space, the small stellated 120-cell honeycomb is one of four regular star-honeycombs. With Schläfli symbol {5/2,5,3,3}, it has three small stellated 120-cells around each face. It is dual to the pentagrammic-order 600-cell honeycomb.
Small stellated 120-cell honeycomb
(No image)
TypeHyperbolic regular honeycomb
Schläfli symbol{5/2,5,3,3}
Coxeter diagram
4-faces {5/2,5,3}
Cells {5/2,5}
Faces {5/2}
Face figure {3}
Edge figure {3,3}
Vertex figure {5,3,3}
DualPentagrammic-order 600-cell honeycomb
Coxeter groupH4, [5,3,3,3]
PropertiesRegular
It can be seen as a stellation of the 120-cell honeycomb, and is thus analogous to the three-dimensional small stellated dodecahedron {5/2,5} and four-dimensional small stellated 120-cell {5/2,5,3}. It has density 5.
See also
• 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)
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Small stellated 120-cell
In geometry, the small stellated 120-cell or stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol {5/2,5,3}. It is one of 10 regular Schläfli-Hess polytopes.
Small stellated 120-cell
Orthogonal projection
TypeSchläfli-Hess polytope
Cells120 {5/2,5}
Faces720 {5/2}
Edges1200
Vertices120
Vertex figure{5,3}
Schläfli symbol{5/2,5,3}
Coxeter-Dynkin diagram
Symmetry groupH4, [3,3,5]
DualIcosahedral 120-cell
PropertiesRegular
Related polytopes
It has the same edge arrangement as the great grand 120-cell, and also shares its 120 vertices with the 600-cell and eight other regular star 4-polytopes. It may also be seen as the first stellation of the 120-cell. In this sense it could be seen as analogous to the three-dimensional small stellated dodecahedron, which is the first stellation of the dodecahedron. Indeed, the small stellated 120-cell is dual to the icosahedral 120-cell, which could be taken as a 4D analogue of the great dodecahedron, dual of the small stellated dodecahedron.
The edges of the small stellated 120-cell are τ2 as long as those of the 120-cell core inside the 4-polytope.
Orthographic projections by Coxeter planes
H3 A2 / B3 / D4 A3 / B2
See also
• List of regular polytopes
• Convex regular 4-polytope - Set of convex regular 4-polytope
• Kepler-Poinsot solids - regular star polyhedron
• Star polygon - regular star polygons
References
• Edmund Hess, (1883) Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder .
• H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26, Regular Star-polytopes, pp. 404–408)
• Klitzing, Richard. "4D uniform polytopes (polychora) o3o5o5/2x - sishi".
External links
• Regular polychora
• Discussion on names
• Reguläre Polytope
• The Regular Star Polychora
• Zome Model of the Final Stellation of the 120-cell
• The First Stellation of the 120-cell, A Zome Model
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
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Dodecadodecahedron
In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36.[1] It is the rectification of the great dodecahedron (and that of its dual, the small stellated dodecahedron). It was discovered independently by Hess (1878), Badoureau (1881) and Pitsch (1882).
Dodecadodecahedron
TypeUniform star polyhedron
ElementsF = 24, E = 60
V = 30 (χ = −6)
Faces by sides12{5}+12{5/2}
Coxeter diagram
Wythoff symbol2 | 5 5/2
2 | 5 5/3
2 | 5/2 5/4
2 | 5/3 5/4
Symmetry groupIh, [5,3], *532
Index referencesU36, C45, W73
Dual polyhedronMedial rhombic triacontahedron
Vertex figure
5.5/2.5.5/2
Bowers acronymDid
The edges of this model form 10 central hexagons, and these, projected onto a sphere, become 10 great circles. These 10, along with the great circles from projections of two other polyhedra, form the 31 great circles of the spherical icosahedron used in construction of geodesic domes.
Wythoff constructions
It has four Wythoff constructions between four Schwarz triangle families: 2 | 5 5/2, 2 | 5 5/3, 2 | 5/2 5/4, 2 | 5/3 5/4, but represent identical results. Similarly it can be given four extended Schläfli symbols: r{5/2,5}, r{5/3,5}, r{5/2,5/4}, and r{5/3,5/4} or as Coxeter-Dynkin diagrams: , , , and .
Net
A shape with the same exterior appearance as the dodecadodecahedron can be constructed by folding up these nets:
12 pentagrams and 20 rhombic clusters are necessary. However, this construction replaces the crossing pentagonal faces of the dodecadodecahedron with non-crossing sets of rhombi, so it does not produce the same internal structure.
Related polyhedra
Its convex hull is the icosidodecahedron. It also shares its edge arrangement with the small dodecahemicosahedron (having the pentagrammic faces in common), and with the great dodecahemicosahedron (having the pentagonal faces in common).
Dodecadodecahedron
Small dodecahemicosahedron
Great dodecahemicosahedron
Icosidodecahedron (convex hull)
This polyhedron can be considered a rectified great dodecahedron. It is center of a truncation sequence between a small stellated dodecahedron and great dodecahedron:
The truncated small stellated dodecahedron looks like a dodecahedron on the surface, but it has 24 faces: 12 pentagons from the truncated vertices and 12 overlapping as (truncated pentagrams). The truncation of the dodecadodecahedron itself is not uniform and attempting to make it uniform results in a degenerate polyhedron (that looks like a small rhombidodecahedron with {10/2} polygons filling up the dodecahedral set of holes), but it has a uniform quasitruncation, the truncated dodecadodecahedron.
Name Small stellated dodecahedron Truncated small stellated dodecahedron Dodecadodecahedron Truncated
great
dodecahedron
Great
dodecahedron
Coxeter-Dynkin
diagram
Picture
It is topologically equivalent to a quotient space of the hyperbolic order-4 pentagonal tiling, by distorting the pentagrams back into regular pentagons. As such, it is topologically a regular polyhedron of index two:[2][3]
The colours in the above image correspond to the red pentagrams and yellow pentagons of the dodecadodecahedron at the top of this article.
Medial rhombic triacontahedron
Medial rhombic triacontahedron
TypeStar polyhedron
Face
ElementsF = 30, E = 60
V = 24 (χ = −6)
Symmetry groupIh, [5,3], *532
Index referencesDU36
dual polyhedronDodecadodecahedron
The medial rhombic triacontahedron is a nonconvex isohedral polyhedron. It is the dual of the dodecadodecahedron. It has 30 intersecting rhombic faces.
It can also be called the small stellated triacontahedron.
Stellation
The medial rhombic triacontahedron is a stellation of the rhombic triacontahedron, which is the dual of the icosidodecahedron, the convex hull of the dodecadodecahedron (dual to the original medial rhombic triacontahedron).
Related hyperbolic tiling
It is topologically equivalent to a quotient space of the hyperbolic order-5 square tiling, by distorting the rhombi into squares. As such, it is topologically a regular polyhedron of index two:[4]
Note that the order-5 square tiling is dual to the order-4 pentagonal tiling, and a quotient space of the order-4 pentagonal tiling is topologically equivalent to the dual of the medial rhombic triacontahedron, the dodecadodecahedron.
See also
• List of uniform polyhedra
References
1. Maeder, Roman. "36: dodecadodecahedron". www.mathconsult.ch. Retrieved 2020-02-03.
2. The Regular Polyhedra (of index two), David A. Richter
3. The Golay Code on the Dodecadodecahedron, David A. Richter
4. The Regular Polyhedra (of index two), David A. Richter
• Badoureau (1881), "Mémoire sur les figures isoscèles", Journal de l'École Polytechnique, 49: 47–172
• Hess, Edmund (1878), Vier archimedeische Polyeder höherer Art, Cassel. Th. Kay, JFM 10.0346.03
• Pitsch (1882), "Über halbreguläre Sternpolyheder", Zeitschrift für das Realschulwesen, 7, JFM 14.0448.01
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208
External links
• Weisstein, Eric W. "Dodecadodecahedron". MathWorld.
• Weisstein, Eric W. "Medial Rhombic Triacontahedron". MathWorld.
• Uniform polyhedra and duals
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
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Triakis octahedron
In geometry, a triakis octahedron (or trigonal trisoctahedron[1] or kisoctahedron[2]) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube.
Triakis octahedron
(Click here for rotating model)
TypeCatalan solid
Coxeter diagram
Conway notationkO
Face typeV3.8.8
isosceles triangle
Faces24
Edges36
Vertices14
Vertices by type8{3}+6{8}
Symmetry groupOh, B3, [4,3], (*432)
Rotation groupO, [4,3]+, (432)
Dihedral angle147°21′00″
arccos(−3 + 8√2/17)
Propertiesconvex, face-transitive
Truncated cube
(dual polyhedron)
Net
It can be seen as an octahedron with triangular pyramids added to each face; that is, it is the Kleetope of the octahedron. It is also sometimes called a trisoctahedron, or, more fully, trigonal trisoctahedron. Both names reflect that it has three triangular faces for every face of an octahedron. The tetragonal trisoctahedron is another name for the deltoidal icositetrahedron, a different polyhedron with three quadrilateral faces for every face of an octahedron.
This convex polyhedron is topologically similar to the concave stellated octahedron. They have the same face connectivity, but the vertices are in different relative distances from the center.
If its shorter edges have length 1, its surface area and volume are:
${\begin{aligned}A&=3{\sqrt {7+4{\sqrt {2}}}}\\V&={\frac {3+2{\sqrt {2}}}{2}}\end{aligned}}$
Cartesian coordinates
Let α = √2 − 1, then the 14 points (±α, ±α, ±α) and (±1, 0, 0), (0, ±1, 0) and (0, 0, ±1) are the vertices of a triakis octahedron centered at the origin.
The length of the long edges equals √2, and that of the short edges 2√2 − 2.
The faces are isosceles triangles with one obtuse and two acute angles. The obtuse angle equals arccos(1/4 − √2/2) ≈ 117.20057038016° and the acute ones equal arccos(1/2 + √2/4) ≈ 31.39971480992°.
Orthogonal projections
The triakis octahedron has three symmetry positions, two located on vertices, and one mid-edge:
Orthogonal projections
Projective
symmetry
[2] [4] [6]
Triakis
octahedron
Truncated
cube
Cultural references
• A triakis octahedron is a vital element in the plot of cult author Hugh Cook's novel The Wishstone and the Wonderworkers.
Related polyhedra
The triakis octahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.
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
The triakis octahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.
*n32 symmetry mutation of truncated tilings: t{n,3}
Symmetry
*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]
Truncated
figures
Symbol t{2,3} t{3,3} t{4,3} t{5,3} t{6,3} t{7,3} t{8,3} t{∞,3} t{12i,3} t{9i,3} t{6i,3}
Triakis
figures
Config. V3.4.4 V3.6.6 V3.8.8 V3.10.10 V3.12.12 V3.14.14 V3.16.16 V3.∞.∞
The triakis octahedron is also a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have (*n42) reflectional symmetry.
*n42 symmetry mutation of truncated tilings: n.8.8
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
Config. 2.8.8 3.8.8 4.8.8 5.8.8 6.8.8 7.8.8 8.8.8 ∞.8.8
n-kis
figures
Config. V2.8.8 V3.8.8 V4.8.8 V5.8.8 V6.8.8 V7.8.8 V8.8.8 V∞.8.8
References
1. "Clipart tagged: 'forms'". etc.usf.edu.
2. Conway, Symmetries of things, p. 284
• 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 17, Triakisoctahedron)
• 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 284, Triakis octahedron)
External links
• Eric W. Weisstein, Triakis octahedron (Catalan solid) at MathWorld.
• Triakis Octahedron – Interactive Polyhedron Model
• Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
• VRML model
• Conway Notation for Polyhedra Try: "dtC"
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.
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Small triambic icosahedron
In geometry, the small triambic icosahedron is a star polyhedron composed of 20 intersecting non-regular hexagon faces. It has 60 edges and 32 vertices, and Euler characteristic of −8. It is an isohedron, meaning that all of its faces are symmetric to each other. Branko Grünbaum has conjectured that it is the only Euclidean isohedron with convex faces of six or more sides,[1] but the small hexagonal hexecontahedron is another example.
Small triambic icosahedron
TypeDual uniform polyhedron
IndexDU30, 2/59, W26
Elements
(As a star polyhedron)
F = 20, E = 60
V = 32 (χ = −8)
Symmetry groupicosahedral (Ih)
Dual polyhedronsmall ditrigonal icosidodecahedron
Stellation diagramStellation coreConvex hull
Icosahedron
Pentakis dodecahedron
Geometry
The faces are equilateral hexagons, with alternating angles of $\arccos(-{\frac {1}{4}})\approx 104.477\,512\,185\,93^{\circ }$ and $\arccos({\frac {1}{4}})+60^{\circ }\approx 135.522\,487\,814\,07^{\circ }$. The dihedral angle equals $\arccos(-{\frac {1}{3}})\approx 109.471\,220\,634\,49$.
Related shapes
The external surface of the small triambic icosahedron (removing the parts of each hexagonal face that are surrounded by other faces, but interpreting the resulting disconnected plane figures as still being faces) coincides with one of the stellations of the icosahedron.[2] If instead, after removing the surrounded parts of each face, each resulting triple of coplanar triangles is considered to be three separate faces, then the result is one form of the triakis icosahedron, formed by adding a triangular pyramid to each face of an icosahedron.
The dual polyhedron of the small triambic icosahedron is the small ditrigonal icosidodecahedron. As this is a uniform polyhedron, the small triambic icosahedron is a uniform dual. Other uniform duals whose exterior surfaces are stellations of the icosahedron are the medial triambic icosahedron and the great triambic icosahedron.
References
1. Grünbaum, Branko (2008). "Can every face of a polyhedron have many sides?". Geometry, games, graphs and education: the Joe Malkevitch Festschrift. Bedford, Massachusetts: Comap, Inc. pp. 9–26. hdl:1773/4593. MR 2512345.
2. 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))
Further reading
• Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. (p. 46, Model W26, triakis icosahedron)
• Wenninger, Magnus (1983). Dual Models. Cambridge University Press. ISBN 0-521-54325-8. (pp. 42–46, dual to uniform polyhedron W70)
• H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, 3.6 6.2 Stellating the Platonic solids, pp.96-104
External links
• Weisstein, Eric W. "Small triambic icosahedron". MathWorld.
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Small-world routing
In network theory, small-world routing refers to routing methods for small-world networks. Networks of this type are peculiar in that relatively short paths exist between any two nodes. Determining these paths, however, can be a difficult problem from the perspective of an individual routing node in the network if no further information is known about the network as a whole.
Greedy routing
Nearly every solution to the problem of routing in small world involves the application of greedy routing. This sort of routing depends on a relative reference point by which any node in the path can choose the next node it believes is closest to the destination. That is, there must be something to be greedy about. For example, this could be geographic location, IP address, etc. In the case of Milgram's original small-world experiment, participants knew the location and occupation of the final recipient and could therefore forward messages based on those parameters.
Constructing a reference base
Greedy routing will not readily work when there is no obvious reference base. This can occur, for example, in overlay networks where information about the destination's location in the underlying network is not available. Friend-to-friend networks are a particular example of this problem. In such networks, trust is ensured by the fact that you only know underlying information about nodes with whom you are already a neighbor.
One solution in this case, is to impose some sort of artificial addressing on the nodes in such a way that this addressing can be effectively used by greedy routing methods. A 2005 paper by a developer of the Freenet Project discusses how this can be accomplished in friend to friend networks. Given the assumption that these networks exhibit small world properties, often as the result of real-world or acquaintance relationships, it should be possible to recover an embedded Kleinberg small-world graph. This is accomplished by selecting random pairs of nodes and potentially swapping them based on an objective function that minimizes the product of all the distances between any given node and its neighbors.
An important problem involved with this solution is the possibility of local minima. This can occur if nodes are in a situation that is optimal only considering a local neighborhood, while ignoring the possibility of a higher optimality resulting from swaps with distant nodes. In the above paper, the authors proposed a simulated annealing method where less-than-optimal swaps were made with a small probability. This probability was proportional to the value of making the switches. Another possible metaheuristic optimization method is a tabu search, which adds a memory to the swap decision. In its most simplistic form, a limited history of past swaps is remembered so that they will be excluded from the list of possible swapping nodes.
This method for constructing a reference base can also be adapted to distributed settings, where decisions can only be made at the level of individual nodes who have no knowledge of the overall network. It turns out that the only modification necessary is in the method for selecting pairs of random nodes. In a distributed setting, this is done by having each node periodically send out a random walker terminating at a node to be considered for swapping.
The Kleinberg model
The Kleinberg model of a network is effective at demonstrating the effectiveness of greedy small world routing. The model uses an n x n grid of nodes to represent a network, where each node is connected with an undirected edge to its neighbors. To give it the "small world" effect, a number of long range edges are added to the network that tend to favor nodes closer in distance rather than farther. When adding edges, the probability of connecting some random vertex $v$ to another random vertex w is proportional to $1/d(v,w)^{q}$, where $q$ is the clustering exponent.[1]
Greedy routing in the Kleinberg model
It is easy to see that a greedy algorithm, without using the long range edges, can navigate from random vertices $v\rightarrow w$ on the grid in $O(n)$ time. By following the guaranteed connections to our neighbors, we can move one unit at a time in the direction of our destination. This is also the case when the clustering component $q$ is large and the "long range" edges end up staying very close; we simply do not take advantage of the weaker ties in this model. When $q=0$, the long range edges are uniformly connected at random which means the long range edges are "too random" to be used efficiently for decentralized search. Kleinberg has shown that the optimal clustering coefficient for this model is $q=2$, or an inverse square distribution.[2]
To reason why this is the case, if a circle of radius r is drawn around the initial node it will have nodal density $n/(\pi r^{2})$ where n is the number of nodes in the circular area. As this circle gets expanded further out, the number of nodes in the given area increases proportional to $r^{2}$ as the probability of having a random link with any node remains proportional $1/r^{2}$, meaning the probability of the original node having a weak tie with any node a given distance away is effectively independent of distance. Therefore, it is concluded that with $q=2$, long-range edges are evenly distributed over all distances, which is effective for letting us funnel to our final destination.
Some structured Peer-to-peer systems based on DHTs often are implementing variants of Kleinberg's Small-World topology to enable efficient routing within Peer-to-peer network with limited node degrees.[3]
See also
• Social network – Social structure made up of a set of social actors
• Small-world network – Graph where most nodes are reachable in a small number of steps
• Watts-Strogatz model – Method of generating random small-world graphsPages displaying short descriptions of redirect targets
References
1. Kleinberg, Jon. "Networks, Crowds, and Markets: Reasoning about a Highly Connected World" (PDF). Retrieved 10 May 2011.
2. Kleinberg, Jon M. (August 2000). "Navigation in a small world". Nature. 406 (6798): 845. Bibcode:2000Natur.406..845K. doi:10.1038/35022643. ISSN 1476-4687. PMID 10972276.
3. Manku, Gurmeet Singh Manku. "Symphony: Distributed Hashing in a Small World" (PDF). usenix.org.{{cite web}}: CS1 maint: url-status (link)
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Least common multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm(a, b), is the smallest positive integer that is divisible by both a and b.[1][2] Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero.[3] However, some authors define lcm(a, 0) as 0 for all a, since 0 is the only common multiple of a and 0.
The least common multiple of the denominators of two fractions is the "lowest common denominator" (lcd), and can be used for adding, subtracting or comparing the fractions.
The least common multiple of more than two integers a, b, c, . . . , usually denoted by lcm(a, b, c, . . .), is defined as the smallest positive integer that is divisible by each of a, b, c, . . .[1]
Overview
A multiple of a number is the product of that number and an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2. By the same principle, 10 is the least common multiple of −5 and −2 as well.
Notation
The least common multiple of two integers a and b is denoted as lcm(a, b).[1] Some older textbooks use [a, b].[3][4]
Example
$\operatorname {lcm} (4,6)$
Multiples of 4 are:
$4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,...$
Multiples of 6 are:
$6,12,18,24,30,36,42,48,54,60,66,72,...$
Common multiples of 4 and 6 are the numbers that are in both lists:
$12,24,36,48,60,72,...$
In this list, the smallest number is 12. Hence, the least common multiple is 12.
Applications
When adding, subtracting, or comparing simple fractions, the least common multiple of the denominators (often called the lowest common denominator) is used, because each of the fractions can be expressed as a fraction with this denominator. For example,
${2 \over 21}+{1 \over 6}={4 \over 42}+{7 \over 42}={11 \over 42}$
where the denominator 42 was used, because it is the least common multiple of 21 and 6.
Gears problem
Suppose there are two meshing gears in a machine, having m and n teeth, respectively, and the gears are marked by a line segment drawn from the center of the first gear to the center of the second gear. When the gears begin rotating, the number of rotations the first gear must complete to realign the line segment can be calculated by using $\operatorname {lcm} (m,n)$. The first gear must complete $\operatorname {lcm} (m,n) \over m$ rotations for the realignment. By that time, the second gear will have made $\operatorname {lcm} (m,n) \over n$ rotations.
Planetary alignment
Suppose there are three planets revolving around a star which take l, m and n units of time, respectively, to complete their orbits. Assume that l, m and n are integers. Assuming the planets started moving around the star after an initial linear alignment, all the planets attain a linear alignment again after $\operatorname {lcm} (l,m,n)$ units of time. At this time, the first, second and third planet will have completed $\operatorname {lcm} (l,m,n) \over l$, $\operatorname {lcm} (l,m,n) \over m$ and $\operatorname {lcm} (l,m,n) \over n$ orbits, respectively, around the star.[5]
Calculation
There are several ways to compute least common multiples.
Using the greatest common divisor
The least common multiple can be computed from the greatest common divisor (gcd) with the formula
$\operatorname {lcm} (a,b)={\frac {|ab|}{\gcd(a,b)}}.$
To avoid introducing integers that are larger than the result, it is convenient to use the equivalent formulas
$\operatorname {lcm} (a,b)=|a|\,{\frac {|b|}{\gcd(a,b)}}=|b|\,{\frac {|a|}{\gcd(a,b)}},$
where the result of the division is always an integer.
These formulas are also valid when exactly one of a and b is 0, since gcd(a, 0) = |a|. However, if both a and b are 0, these formulas would cause division by zero; so, lcm(0, 0) = 0 must be considered as a special case.
To return to the example above,
$\operatorname {lcm} (21,6)=6\times {\frac {21}{\gcd(21,6)}}=6\times {\frac {21}{3}}=6\times 7=42.$
There are fast algorithms, such as the Euclidean algorithm for computing the gcd that do not require the numbers to be factored. For very large integers, there are even faster algorithms for the three involved operations (multiplication, gcd, and division); see Fast multiplication. As these algorithms are more efficient with factors of similar size, it is more efficient to divide the largest argument of the lcm by the gcd of the arguments, as in the example above.
Using prime factorization
The unique factorization theorem indicates that every positive integer greater than 1 can be written in only one way as a product of prime numbers. The prime numbers can be considered as the atomic elements which, when combined, make up a composite number.
For example:
$90=2^{1}\cdot 3^{2}\cdot 5^{1}=2\cdot 3\cdot 3\cdot 5.$
Here, the composite number 90 is made up of one atom of the prime number 2, two atoms of the prime number 3, and one atom of the prime number 5.
This fact can be used to find the lcm of a set of numbers.
Example: lcm(8,9,21)
Factor each number and express it as a product of prime number powers.
${\begin{aligned}8&=2^{3}\\9&=3^{2}\\21&=3^{1}\cdot 7^{1}\end{aligned}}$
The lcm will be the product of multiplying the highest power of each prime number together. The highest power of the three prime numbers 2, 3, and 7 is 23, 32, and 71, respectively. Thus,
$\operatorname {lcm} (8,9,21)=2^{3}\cdot 3^{2}\cdot 7^{1}=8\cdot 9\cdot 7=504.$
This method is not as efficient as reducing to the greatest common divisor, since there is no known general efficient algorithm for integer factorization.
The same method can also be illustrated with a Venn diagram as follows, with the prime factorization of each of the two numbers demonstrated in each circle and all factors they share in common in the intersection. The lcm then can be found by multiplying all of the prime numbers in the diagram.
Here is an example:
48 = 2 × 2 × 2 × 2 × 3,
180 = 2 × 2 × 3 × 3 × 5,
sharing two "2"s and a "3" in common:
Least common multiple = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 720
Greatest common divisor = 2 × 2 × 3 = 12
Product = 2 × 2 × 2 × 2 × 3 × 2 × 2 × 3 × 3 × 5 = 8640
This also works for the greatest common divisor (gcd), except that instead of multiplying all of the numbers in the Venn diagram, one multiplies only the prime factors that are in the intersection. Thus the gcd of 48 and 180 is 2 × 2 × 3 = 12.
Using a simple algorithm
This method works easily for finding the lcm of several integers.
Let there be a finite sequence of positive integers X = (x1, x2, ..., xn), n > 1. The algorithm proceeds in steps as follows: on each step m it examines and updates the sequence X(m) = (x1(m), x2(m), ..., xn(m)), X(1) = X, where X(m) is the mth iteration of X, that is, X at step m of the algorithm, etc. The purpose of the examination is to pick the least (perhaps, one of many) element of the sequence X(m). Assuming xk0(m) is the selected element, the sequence X(m+1) is defined as
xk(m+1) = xk(m), k ≠ k0
xk0(m+1) = xk0(m) + xk0(1).
In other words, the least element is increased by the corresponding x whereas the rest of the elements pass from X(m) to X(m+1) unchanged.
The algorithm stops when all elements in sequence X(m) are equal. Their common value L is exactly lcm(X).
For example, if X = X(1) = (3, 4, 6), the steps in the algorithm produce:
X(2) = (6, 4, 6)
X(3) = (6, 8, 6)
X(4) = (6, 8, 12) - by choosing the second 6
X(5) = (9, 8, 12)
X(6) = (9, 12, 12)
X(7) = (12, 12, 12) so lcm = 12.
Using the table-method
This method works for any number of numbers. One begins by listing all of the numbers vertically in a table (in this example 4, 7, 12, 21, and 42):
4
7
12
21
42
The process begins by dividing all of the numbers by 2. If 2 divides any of them evenly, write 2 in a new column at the top of the table, and the result of division by 2 of each number in the space to the right in this new column. If a number is not evenly divisible, just rewrite the number again. If 2 does not divide evenly into any of the numbers, repeat this procedure with the next smallest prime number, 3 (see below).
× 2
4 2
7 7
12 6
21 21
42 21
Now, assuming that 2 did divide at least one number (as in this example), check if 2 divides again:
× 2 2
4 2 1
7 7 7
12 6 3
21 21 21
42 21 21
Once 2 no longer divides any number in the current column, repeat the procedure by dividing by the next larger prime, 3. Once 3 no longer divides, try the next larger primes, 5 then 7, etc. The process ends when all of the numbers have been reduced to 1 (the column under the last prime divisor consists only of 1's).
× 2 2 3 7
4 2 1 1 1
7 7 7 7 1
12 6 3 1 1
21 21 21 7 1
42 21 21 7 1
Now, multiply the numbers in the top row to obtain the lcm. In this case, it is 2 × 2 × 3 × 7 = 84.
As a general computational algorithm, the above is quite inefficient. One would never want to implement it in software: it takes too many steps and requires too much storage space. A far more efficient numerical algorithm can be obtained by using Euclid's algorithm to compute the gcd first, and then obtaining the lcm by division.
Formulas
Fundamental theorem of arithmetic
According to the fundamental theorem of arithmetic, every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors:
$n=2^{n_{2}}3^{n_{3}}5^{n_{5}}7^{n_{7}}\cdots =\prod _{p}p^{n_{p}},$
where the exponents n2, n3, ... are non-negative integers; for example, 84 = 22 31 50 71 110 130 ...
Given two positive integers $ a=\prod _{p}p^{a_{p}}$ and $ b=\prod _{p}p^{b_{p}}$, their least common multiple and greatest common divisor are given by the formulas
$\gcd(a,b)=\prod _{p}p^{\min(a_{p},b_{p})}$
and
$\operatorname {lcm} (a,b)=\prod _{p}p^{\max(a_{p},b_{p})}.$
Since
$\min(x,y)+\max(x,y)=x+y,$
this gives
$\gcd(a,b)\operatorname {lcm} (a,b)=ab.$
In fact, every rational number can be written uniquely as the product of primes, if negative exponents are allowed. When this is done, the above formulas remain valid. For example:
${\begin{aligned}4&=2^{2}3^{0},&6&=2^{1}3^{1},&\gcd(4,6)&=2^{1}3^{0}=2,&\operatorname {lcm} (4,6)&=2^{2}3^{1}=12.\\[8pt]{\tfrac {1}{3}}&=2^{0}3^{-1}5^{0},&{\tfrac {2}{5}}&=2^{1}3^{0}5^{-1},&\gcd \left({\tfrac {1}{3}},{\tfrac {2}{5}}\right)&=2^{0}3^{-1}5^{-1}={\tfrac {1}{15}},&\operatorname {lcm} \left({\tfrac {1}{3}},{\tfrac {2}{5}}\right)&=2^{1}3^{0}5^{0}=2,\\[8pt]{\tfrac {1}{6}}&=2^{-1}3^{-1},&{\tfrac {3}{4}}&=2^{-2}3^{1},&\gcd \left({\tfrac {1}{6}},{\tfrac {3}{4}}\right)&=2^{-2}3^{-1}={\tfrac {1}{12}},&\operatorname {lcm} \left({\tfrac {1}{6}},{\tfrac {3}{4}}\right)&=2^{-1}3^{1}={\tfrac {3}{2}}.\end{aligned}}$
Lattice-theoretic
The positive integers may be partially ordered by divisibility: if a divides b (that is, if b is an integer multiple of a) write a ≤ b (or equivalently, b ≥ a). (Note that the usual magnitude-based definition of ≤ is not used here.)
Under this ordering, the positive integers become a lattice, with meet given by the gcd and join given by the lcm. The proof is straightforward, if a bit tedious; it amounts to checking that lcm and gcd satisfy the axioms for meet and join. Putting the lcm and gcd into this more general context establishes a duality between them:
If a formula involving integer variables, gcd, lcm, ≤ and ≥ is true, then the formula obtained by switching gcd with lcm and switching ≥ with ≤ is also true. (Remember ≤ is defined as divides).
The following pairs of dual formulas are special cases of general lattice-theoretic identities.
Commutative laws
$\operatorname {lcm} (a,b)=\operatorname {lcm} (b,a),$
$\gcd(a,b)=\gcd(b,a).$
Associative laws
$\operatorname {lcm} (a,\operatorname {lcm} (b,c))=\operatorname {lcm} (\operatorname {lcm} (a,b),c),$
$\gcd(a,\gcd(b,c))=\gcd(\gcd(a,b),c).$
Absorption laws
$\operatorname {lcm} (a,\gcd(a,b))=a,$
$\gcd(a,\operatorname {lcm} (a,b))=a.$
Idempotent laws
$\operatorname {lcm} (a,a)=a,$
$\gcd(a,a)=a.$
Define divides in terms of lcm and gcd
$a\geq b\iff a=\operatorname {lcm} (a,b),$
$a\leq b\iff a=\gcd(a,b).$
It can also be shown[6] that this lattice is distributive; that is, lcm distributes over gcd and gcd distributes over lcm:
$\operatorname {lcm} (a,\gcd(b,c))=\gcd(\operatorname {lcm} (a,b),\operatorname {lcm} (a,c)),$
$\gcd(a,\operatorname {lcm} (b,c))=\operatorname {lcm} (\gcd(a,b),\gcd(a,c)).$
This identity is self-dual:
$\gcd(\operatorname {lcm} (a,b),\operatorname {lcm} (b,c),\operatorname {lcm} (a,c))=\operatorname {lcm} (\gcd(a,b),\gcd(b,c),\gcd(a,c)).$
Other
• Let D be the product of ω(D) distinct prime numbers (that is, D is squarefree).
Then[7]
$|\{(x,y)\;:\;\operatorname {lcm} (x,y)=D\}|=3^{\omega (D)},$
where the absolute bars || denote the cardinality of a set.
• If none of $a_{1},a_{2},\ldots ,a_{r}$ is zero, then
$\operatorname {lcm} (a_{1},a_{2},\ldots ,a_{r})=\operatorname {lcm} (\operatorname {lcm} (a_{1},a_{2},\ldots ,a_{r-1}),a_{r}).$[8][9]
In commutative rings
The least common multiple can be defined generally over commutative rings as follows:
Let a and b be elements of a commutative ring R. A common multiple of a and b is an element m of R such that both a and b divide m (that is, there exist elements x and y of R such that ax = m and by = m). A least common multiple of a and b is a common multiple that is minimal, in the sense that for any other common multiple n of a and b, m divides n.
In general, two elements in a commutative ring can have no least common multiple or more than one. However, any two least common multiples of the same pair of elements are associates.[10] In a unique factorization domain, any two elements have a least common multiple.[11] In a principal ideal domain, the least common multiple of a and b can be characterised as a generator of the intersection of the ideals generated by a and b[10] (the intersection of a collection of ideals is always an ideal).
See also
• Anomalous cancellation
• Coprime integers
• Chebyshev function
Notes
1. Weisstein, Eric W. "Least Common Multiple". mathworld.wolfram.com. Retrieved 2020-08-30.
2. Hardy & Wright, § 5.1, p. 48
3. Long (1972, p. 39)
4. Pettofrezzo & Byrkit (1970, p. 56)
5. "nasa spacemath" (PDF).
6. The next three formulas are from Landau, Ex. III.3, p. 254
7. Crandall & Pomerance, ex. 2.4, p. 101.
8. Long (1972, p. 41)
9. Pettofrezzo & Byrkit (1970, p. 58)
10. Burton 1970, p. 94.
11. Grillet 2007, p. 142.
References
• Burton, David M. (1970). A First Course in Rings and Ideals. Reading, MA: Addison-Wesley. ISBN 978-0-201-00731-2.
• Crandall, Richard; Pomerance, Carl (2001), Prime Numbers: A Computational Perspective, New York: Springer, ISBN 0-387-94777-9
• Grillet, Pierre Antoine (2007). Abstract Algebra (2nd ed.). New York, NY: Springer. ISBN 978-0-387-71568-1.
• Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (Fifth edition), Oxford: Oxford University Press, ISBN 978-0-19-853171-5
• Landau, Edmund (1966), Elementary Number Theory, New York: Chelsea
• Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77-171950
• Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766
Fractions and ratios
Division and ratio
• Dividend ÷ Divisor = Quotient
Fraction
• Numerator/Denominator = Quotient
• Algebraic
• Aspect
• Binary
• Continued
• Decimal
• Dyadic
• Egyptian
• Golden
• Silver
• Integer
• Irreducible
• Reduction
• Just intonation
• LCD
• Musical interval
• Paper size
• Percentage
• Unit
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Wikipedia
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Smallest grammar problem
In data compression and the theory of formal languages, the smallest grammar problem is the problem of finding the smallest context-free grammar that generates a given string of characters (but no other string). The size of a grammar is defined by some authors as the number of symbols on the right side of the production rules.[1] Others also add the number of rules to that.[2] The (decision version of the) problem is NP-complete.[1] The smallest context-free grammar that generates a given string is always a straight-line grammar without useless rules.
See also
• Grammar-based code
• Kolmogorov Complexity
• Lossless data compression
References
1. Charikar, Moses; Lehman, Eric; Liu, Ding; Panigrahy, Rina; Prabhakaran, Manoj; Sahai, Amit; Shelat, Abhi (2005). "The Smallest Grammar Problem" (PDF). IEEE Transactions on Information Theory. 51 (7): 2554–2576. CiteSeerX 10.1.1.185.2130. doi:10.1109/TIT.2005.850116. S2CID 6900082. Zbl 1296.68086.
2. Florian Benz and Timo Kötzing, “An effective heuristic for the smallest grammar problem,” Proceedings of the fifteenth annual conference on Genetic and evolutionary computation conference - GECCO ’13, 2013. ISBN 978-1-4503-1963-8 doi:10.1145/2463372.2463441
• Charikar, Moses; Lehman, Eric; Liu, Ding; Panigrahy, Rina; Prabhakaran, Manoj; Rasala, April; Sahai, Amit; Shelat, Abhi (2002). "Approximating the Smallest Grammar: Kolmogorov Complexity in Natural Models" (PDF). Proceedings of the thirty-fourth annual ACM symposium on theory of computing (STOC 2002), Montreal, Quebec, Canada, May 19–21, 2002. New York, NY: ACM Press. pp. 792–801. doi:10.1145/509907.510021. ISBN 978-1-581-13495-7. S2CID 282489. Zbl 1192.68397.
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Wikipedia
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Sample maximum and minimum
In statistics, the sample maximum and sample minimum, also called the largest observation and smallest observation, are the values of the greatest and least elements of a sample.[1] They are basic summary statistics, used in descriptive statistics such as the five-number summary and Bowley's seven-figure summary and the associated box plot.
The minimum and the maximum value are the first and last order statistics (often denoted X(1) and X(n) respectively, for a sample size of n).
If the sample has outliers, they necessarily include the sample maximum or sample minimum, or both, depending on whether they are extremely high or low. However, the sample maximum and minimum need not be outliers, if they are not unusually far from other observations.
Robustness
The sample maximum and minimum are the least robust statistics: they are maximally sensitive to outliers.
This can either be an advantage or a drawback: if extreme values are real (not measurement errors), and of real consequence, as in applications of extreme value theory such as building dikes or financial loss, then outliers (as reflected in sample extrema) are important. On the other hand, if outliers have little or no impact on actual outcomes, then using non-robust statistics such as the sample extrema simply cloud the statistics, and robust alternatives should be used, such as other quantiles: the 10th and 90th percentiles (first and last decile) are more robust alternatives.
Derived statistics
In addition to being a component of every statistic that uses all elements of the sample, the sample extrema are important parts of the range, a measure of dispersion, and mid-range, a measure of location. They also realize the maximum absolute deviation: one of them is the furthest point from any given point, particularly a measure of center such as the median or mean.
Applications
Smooth maximum
For a sample set, the maximum function is non-smooth and thus non-differentiable. For optimization problems that occur in statistics it often needs to be approximated by a smooth function that is close to the maximum of the set.
A smooth maximum, for example,
g(x1, x2, …, xn) = log( exp(x1) + exp(x2) + … + exp(xn) )
is a good approximation of the sample maximum.
Summary statistics
The sample maximum and minimum are basic summary statistics, showing the most extreme observations, and are used in the five-number summary and a version of the seven-number summary and the associated box plot.
Prediction interval
Further information: Prediction interval § Non-parametric
The sample maximum and minimum provide a non-parametric prediction interval: in a sample from a population, or more generally an exchangeable sequence of random variables, each observation is equally likely to be the maximum or minimum.
Thus if one has a sample $\{X_{1},\dots ,X_{n}\},$ and one picks another observation $X_{n+1},$ then this has $1/(n+1)$ probability of being the largest value seen so far, $1/(n+1)$ probability of being the smallest value seen so far, and thus the other $(n-1)/(n+1)$ of the time, $X_{n+1}$ falls between the sample maximum and sample minimum of $\{X_{1},\dots ,X_{n}\}.$ Thus, denoting the sample maximum and minimum by M and m, this yields an $(n-1)/(n+1)$ prediction interval of [m,M].
For example, if n = 19, then [m,M] gives an 18/20 = 90% prediction interval – 90% of the time, the 20th observation falls between the smallest and largest observation seen heretofore. Likewise, n = 39 gives a 95% prediction interval, and n = 199 gives a 99% prediction interval.
Estimation
Due to their sensitivity to outliers, the sample extrema cannot reliably be used as estimators unless data is clean – robust alternatives include the first and last deciles.
However, with clean data or in theoretical settings, they can sometimes prove very good estimators, particularly for platykurtic distributions, where for small data sets the mid-range is the most efficient estimator.
They are inefficient estimators of location for mesokurtic distributions, such as the normal distribution, and leptokurtic distributions, however.
Uniform distribution
Further information: German tank problem
For sampling without replacement from a uniform distribution with one or two unknown endpoints (so $1,2,\dots ,N$ with N unknown, or $M,M+1,\dots ,N$ with both M and N unknown), the sample maximum, or respectively the sample maximum and sample minimum, are sufficient and complete statistics for the unknown endpoints; thus an unbiased estimator derived from these will be UMVU estimator.
If only the top endpoint is unknown, the sample maximum is a biased estimator for the population maximum, but the unbiased estimator ${\frac {k+1}{k}}m-1$ (where m is the sample maximum and k is the sample size) is the UMVU estimator; see German tank problem for details.
If both endpoints are unknown, then the sample range is a biased estimator for the population range, but correcting as for maximum above yields the UMVU estimator.
If both endpoints are unknown, then the mid-range is an unbiased (and hence UMVU) estimator of the midpoint of the interval (here equivalently the population median, average, or mid-range).
The reason the sample extrema are sufficient statistics is that the conditional distribution of the non-extreme samples is just the distribution for the uniform interval between the sample maximum and minimum – once the endpoints are fixed, the values of the interior points add no additional information.
Normality testing
The sample extrema can be used for a simple normality test, specifically of kurtosis: one computes the t-statistic of the sample maximum and minimum (subtracts sample mean and divides by the sample standard deviation), and if they are unusually large for the sample size (as per the three sigma rule and table therein, or more precisely a Student's t-distribution), then the kurtosis of the sample distribution deviates significantly from that of the normal distribution.
For instance, a daily process should expect a 3σ event once per year (of calendar days; once every year and a half of business days), while a 4σ event happens on average every 40 years of calendar days, 60 years of business days (once in a lifetime), 5σ events happen every 5,000 years (once in recorded history), and 6σ events happen every 1.5 million years (essentially never). Thus if the sample extrema are 6 sigmas from the mean, one has a significant failure of normality.
Further, this test is very easy to communicate without involved statistics.
These tests of normality can be applied if one faces kurtosis risk, for instance.
Extreme value theory
Sample extrema play two main roles in extreme value theory:
• first, they give a lower bound on extreme events – events can be at least this extreme, and for this size sample;
• second, they can sometimes be used in estimators of probability of more extreme events.
However, caution must be used in using sample extrema as guidelines: in heavy-tailed distributions or for non-stationary processes, extreme events can be significantly more extreme than any previously observed event. This is elaborated in black swan theory.
See also
• Maxima and minima
References
1. "NEDARC - Min, Max, and Range". www.nedarc.org. Retrieved 2023-02-17.
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Wikipedia
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Smarandache–Wellin number
In mathematics, a Smarandache–Wellin number is an integer that in a given base is the concatenation of the first n prime numbers written in that base. Smarandache–Wellin numbers are named after Florentin Smarandache and Paul R. Wellin.
The first decimal Smarandache–Wellin numbers are:
2, 23, 235, 2357, 235711, 23571113, 2357111317, 235711131719, 23571113171923, 2357111317192329, ... (sequence A019518 in the OEIS).
Smarandache–Wellin prime
A Smarandache–Wellin number that is also prime is called a Smarandache–Wellin prime. The first three are 2, 23 and 2357 (sequence A069151 in the OEIS). The fourth is 355 digits long: it is the result of concatenating the first 128 prime numbers, through 719.[1]
The primes at the end of the concatenation in the Smarandache–Wellin primes are
2, 3, 7, 719, 1033, 2297, 3037, 11927, ... (sequence A046284 in the OEIS).
The indices of the Smarandache–Wellin primes in the sequence of Smarandache–Wellin numbers are:
1, 2, 4, 128, 174, 342, 435, 1429, ... (sequence A046035 in the OEIS).
The 1429th Smarandache–Wellin number is a probable prime with 5719 digits ending in 11927, discovered by Eric W. Weisstein in 1998.[2] If it is proven prime, it will be the eighth Smarandache–Wellin prime. In March 2009, Weisstein's search showed the index of the next Smarandache–Wellin prime (if one exists) is at least 22077.[3]
See also
• Copeland–Erdős constant
• Champernowne constant, another example of a number obtained by concatenating a representation in a given base.
References
1. Pomerance, Carl B.; Crandall, Richard E. (2001). Prime Numbers: a computational perspective. Springer. pp. 78 Ex 1.86. ISBN 0-387-25282-7.
2. Rivera, Carlos, Primes by Listing
3. Weisstein, Eric W. "Integer Sequence Primes". MathWorld. Retrieved 2011-07-28.
External links
• Weisstein, Eric W. "Smarandache–Wellin number". MathWorld.
• Weisstein, Eric W. "Smarandache–Wellin prime". MathWorld.
• "Smarandache-Wellin number". PlanetMath.
• List of first 54 Smarandache–Wellin numbers with factorizations
• Smarandache–Wellin primes at The Prime Glossary
• Smith, S. "A Set of Conjectures on Smarandache Sequences." Bull. Pure Appl. Sci. 15E, 101–107, 1996.
Prime number classes
By formula
• Fermat (22n + 1)
• Mersenne (2p − 1)
• Double Mersenne (22p−1 − 1)
• Wagstaff (2p + 1)/3
• Proth (k·2n + 1)
• Factorial (n! ± 1)
• Primorial (pn# ± 1)
• Euclid (pn# + 1)
• Pythagorean (4n + 1)
• Pierpont (2m·3n + 1)
• Quartan (x4 + y4)
• Solinas (2m ± 2n ± 1)
• Cullen (n·2n + 1)
• Woodall (n·2n − 1)
• Cuban (x3 − y3)/(x − y)
• Leyland (xy + yx)
• Thabit (3·2n − 1)
• Williams ((b−1)·bn − 1)
• Mills (⌊A3n⌋)
By integer sequence
• Fibonacci
• Lucas
• Pell
• Newman–Shanks–Williams
• Perrin
• Partitions
• Bell
• Motzkin
By property
• Wieferich (pair)
• Wall–Sun–Sun
• Wolstenholme
• Wilson
• Lucky
• Fortunate
• Ramanujan
• Pillai
• Regular
• Strong
• Stern
• Supersingular (elliptic curve)
• Supersingular (moonshine theory)
• Good
• Super
• Higgs
• Highly cototient
• Unique
Base-dependent
• Palindromic
• Emirp
• Repunit (10n − 1)/9
• Permutable
• Circular
• Truncatable
• Minimal
• Delicate
• Primeval
• Full reptend
• Unique
• Happy
• Self
• Smarandache–Wellin
• Strobogrammatic
• Dihedral
• Tetradic
Patterns
• Twin (p, p + 2)
• Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …)
• Triplet (p, p + 2 or p + 4, p + 6)
• Quadruplet (p, p + 2, p + 6, p + 8)
• k-tuple
• Cousin (p, p + 4)
• Sexy (p, p + 6)
• Chen
• Sophie Germain/Safe (p, 2p + 1)
• Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...)
• Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...)
• Balanced (consecutive p − n, p, p + n)
By size
• Mega (1,000,000+ digits)
• Largest known
• list
Complex numbers
• Eisenstein prime
• Gaussian prime
Composite numbers
• Pseudoprime
• Catalan
• Elliptic
• Euler
• Euler–Jacobi
• Fermat
• Frobenius
• Lucas
• Somer–Lucas
• Strong
• Carmichael number
• Almost prime
• Semiprime
• Sphenic number
• Interprime
• Pernicious
Related topics
• Probable prime
• Industrial-grade prime
• Illegal prime
• Formula for primes
• Prime gap
First 60 primes
• 2
• 3
• 5
• 7
• 11
• 13
• 17
• 19
• 23
• 29
• 31
• 37
• 41
• 43
• 47
• 53
• 59
• 61
• 67
• 71
• 73
• 79
• 83
• 89
• 97
• 101
• 103
• 107
• 109
• 113
• 127
• 131
• 137
• 139
• 149
• 151
• 157
• 163
• 167
• 173
• 179
• 181
• 191
• 193
• 197
• 199
• 211
• 223
• 227
• 229
• 233
• 239
• 241
• 251
• 257
• 263
• 269
• 271
• 277
• 281
List of prime numbers
Classes of natural numbers
Powers and related numbers
• Achilles
• Power of 2
• Power of 3
• Power of 10
• Square
• Cube
• Fourth power
• Fifth power
• Sixth power
• Seventh power
• Eighth power
• Perfect power
• Powerful
• Prime power
Of the form a × 2b ± 1
• Cullen
• Double Mersenne
• Fermat
• Mersenne
• Proth
• Thabit
• Woodall
Other polynomial numbers
• Hilbert
• Idoneal
• Leyland
• Loeschian
• Lucky numbers of Euler
Recursively defined numbers
• Fibonacci
• Jacobsthal
• Leonardo
• Lucas
• Padovan
• Pell
• Perrin
Possessing a specific set of other numbers
• Amenable
• Congruent
• Knödel
• Riesel
• Sierpiński
Expressible via specific sums
• Nonhypotenuse
• Polite
• Practical
• Primary pseudoperfect
• Ulam
• Wolstenholme
Figurate numbers
2-dimensional
centered
• Centered triangular
• Centered square
• Centered pentagonal
• Centered hexagonal
• Centered heptagonal
• Centered octagonal
• Centered nonagonal
• Centered decagonal
• Star
non-centered
• Triangular
• Square
• Square triangular
• Pentagonal
• Hexagonal
• Heptagonal
• Octagonal
• Nonagonal
• Decagonal
• Dodecagonal
3-dimensional
centered
• Centered tetrahedral
• Centered cube
• Centered octahedral
• Centered dodecahedral
• Centered icosahedral
non-centered
• Tetrahedral
• Cubic
• Octahedral
• Dodecahedral
• Icosahedral
• Stella octangula
pyramidal
• Square pyramidal
4-dimensional
non-centered
• Pentatope
• Squared triangular
• Tesseractic
Combinatorial numbers
• Bell
• Cake
• Catalan
• Dedekind
• Delannoy
• Euler
• Eulerian
• Fuss–Catalan
• Lah
• Lazy caterer's sequence
• Lobb
• Motzkin
• Narayana
• Ordered Bell
• Schröder
• Schröder–Hipparchus
• Stirling first
• Stirling second
• Telephone number
• Wedderburn–Etherington
Primes
• Wieferich
• Wall–Sun–Sun
• Wolstenholme prime
• Wilson
Pseudoprimes
• Carmichael number
• Catalan pseudoprime
• Elliptic pseudoprime
• Euler pseudoprime
• Euler–Jacobi pseudoprime
• Fermat pseudoprime
• Frobenius pseudoprime
• Lucas pseudoprime
• Lucas–Carmichael number
• Somer–Lucas pseudoprime
• Strong pseudoprime
Arithmetic functions and dynamics
Divisor functions
• Abundant
• Almost perfect
• Arithmetic
• Betrothed
• Colossally abundant
• Deficient
• Descartes
• Hemiperfect
• Highly abundant
• Highly composite
• Hyperperfect
• Multiply perfect
• Perfect
• Practical
• Primitive abundant
• Quasiperfect
• Refactorable
• Semiperfect
• Sublime
• Superabundant
• Superior highly composite
• Superperfect
Prime omega functions
• Almost prime
• Semiprime
Euler's totient function
• Highly cototient
• Highly totient
• Noncototient
• Nontotient
• Perfect totient
• Sparsely totient
Aliquot sequences
• Amicable
• Perfect
• Sociable
• Untouchable
Primorial
• Euclid
• Fortunate
Other prime factor or divisor related numbers
• Blum
• Cyclic
• Erdős–Nicolas
• Erdős–Woods
• Friendly
• Giuga
• Harmonic divisor
• Jordan–Pólya
• Lucas–Carmichael
• Pronic
• Regular
• Rough
• Smooth
• Sphenic
• Størmer
• Super-Poulet
• Zeisel
Numeral system-dependent numbers
Arithmetic functions
and dynamics
• Persistence
• Additive
• Multiplicative
Digit sum
• Digit sum
• Digital root
• Self
• Sum-product
Digit product
• Multiplicative digital root
• Sum-product
Coding-related
• Meertens
Other
• Dudeney
• Factorion
• Kaprekar
• Kaprekar's constant
• Keith
• Lychrel
• Narcissistic
• Perfect digit-to-digit invariant
• Perfect digital invariant
• Happy
P-adic numbers-related
• Automorphic
• Trimorphic
Digit-composition related
• Palindromic
• Pandigital
• Repdigit
• Repunit
• Self-descriptive
• Smarandache–Wellin
• Undulating
Digit-permutation related
• Cyclic
• Digit-reassembly
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Smith conjecture
In mathematics, the Smith conjecture states that if f is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of f cannot be a nontrivial knot.
Paul A. Smith (1939, remark after theorem 4) showed that a non-trivial orientation-preserving diffeomorphism of finite order with fixed points must have a fixed point set equal to a circle, and asked in (Eilenberg 1949, Problem 36) if the fixed point set could be knotted. Friedhelm Waldhausen (1969) proved the Smith conjecture for the special case of diffeomorphisms of order 2 (and hence any even order). The proof of the general case was described by John Morgan and Hyman Bass (1984) and depended on several major advances in 3-manifold theory, In particular the work of William Thurston on hyperbolic structures on 3-manifolds, and results by William Meeks and Shing-Tung Yau on minimal surfaces in 3-manifolds, with some additional help from Bass, Cameron Gordon, Peter Shalen, and Rick Litherland.
Deane Montgomery and Leo Zippin (1954) gave an example of a continuous involution of the 3-sphere whose fixed point set is a wildly embedded circle, so the Smith conjecture is false in the topological (rather than the smooth or PL) category. Charles Giffen (1966) showed that the analogue of the Smith conjecture in higher dimensions is false: the fixed point set of a periodic diffeomorphism of a sphere of dimension at least 4 can be a knotted sphere of codimension 2.
See also
• Hilbert–Smith conjecture
References
• Eilenberg, Samuel (1949), "On the Problems of Topology", Annals of Mathematics, Second Series, 50 (2): 247–260, doi:10.2307/1969448, ISSN 0003-486X, JSTOR 1969448, MR 0030189
• Giffen, Charles H. (1966), "The generalized Smith conjecture", American Journal of Mathematics, 88 (1): 187–198, doi:10.2307/2373054, ISSN 0002-9327, JSTOR 2373054, MR 0198462
• Montgomery, Deane; Zippin, Leo (1954), "Examples of transformation groups", Proceedings of the American Mathematical Society, 5 (3): 460–465, doi:10.2307/2031959, ISSN 0002-9939, JSTOR 2031959, MR 0062436
• Morgan, John W.; Bass, Hyman, eds. (1984), The Smith conjecture, Pure and Applied Mathematics, vol. 112, Boston, MA: Academic Press, ISBN 978-0-12-506980-9, MR 0758459
• Smith, Paul A. (1939), "Transformations of finite period. II", Annals of Mathematics, Second Series, 40 (3): 690–711, Bibcode:1939AnMat..40..690S, doi:10.2307/1968950, ISSN 0003-486X, JSTOR 1968950, MR 0000177
• Waldhausen, Friedhelm (1969), "Über Involutionen der 3-Sphäre", Topology, 8: 81–91, doi:10.1016/0040-9383(69)90033-0, ISSN 0040-9383, MR 0236916
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Smith graph
In the mathematical field of graph theory, a Smith graph is either of two kinds of graph.
• It is a graph whose adjacency matrix has largest eigenvalue at most 2,[1] or has spectral radius 2[2] or at most 2.[3] The graphs with spectral radius 2 form two infinite families and three sporadic examples; if we ask for spectral radius at most 2 then there are two additional infinite families and three more sporadic examples. The infinite families with spectral radius less than 2 are the paths and the paths with one extra edge attached to the vertex next to an endpoint; the infinite families with spectral radius exactly 2 are the cycles and the paths with an extra edge attached to each of the vertices next to an endpoint.
For the nomogram technique in radio frequency engineering, see Smith chart. For the Biggs–Smith distance-regular graph, see Biggs–Smith graph.
These are also the simply laced affine (and finite, if the spectral radius may be less than 2) Dynkin diagrams.
• It is a strongly regular graph with certain kinds of parameter values.[4]
References
1. John H. Smith (June 2–14, 1969). "Some properties of the spectrum of a graph". In Richard Guy (ed.). Combinatorial Structures and Their Applications. Proceedings of the Calgary International Conference on Combinatorial Structures and Their Applications. University of Calgary, Calgary, Alberta, Canada: Gordon and Breach. pp. 403–406.
2. Radosavljević, Z.; Mihailović, B.; Rašajski, M. (2008). "Decomposition of Smith graphs in maximal reflexive cacti". Discrete Mathematics. 308 (2–3): 355–366. doi:10.1016/j.disc.2006.11.049.
3. Cvetković, Dragoš (2017). "Spectral Theory of Smith Graphs". Bulletin (Académie Serbe des Sciences et des Arts. Classe des Sciences Mathématiques et Naturelles. Sciences Mathématiques) (42): 19–40. JSTOR 26359061.
4. Cameron, P.J; MacPherson, H.D (1985). "Rank three permutation groups with rank three subconstituents". Journal of Combinatorial Theory, Series B. 39: 1–16. doi:10.1016/0095-8956(85)90034-6.
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Wikipedia
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Smith number
In number theory, a Smith number is a composite number for which, in a given number base, the sum of its digits is equal to the sum of the digits in its prime factorization in the same base. In the case of numbers that are not square-free, the factorization is written without exponents, writing the repeated factor as many times as needed.
Smith number
Named afterHarold Smith (brother-in-law of Albert Wilansky)
Author of publicationAlbert Wilansky
Total no. of termsinfinity
First terms4, 22, 27, 58, 85, 94, 121
OEIS indexA006753
Smith numbers were named by Albert Wilansky of Lehigh University, as he noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith:
4937775 = 3 · 5 · 5 · 65837
while
4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 + 5 + 5 + (6 + 5 + 8 + 3 + 7)
in base 10.[1]
Mathematical definition
Let $n$ be a natural number. For base $b>1$, let the function $F_{b}(n)$ be the digit sum of $n$ in base $b$. A natural number $n$ with prime factorisation
$n=\prod _{\stackrel {p\mid n,}{p{\text{ prime}}}}p^{v_{p}(n)}$
is a Smith number if
$F_{b}(n)=\sum _{\stackrel {p\mid n,}{p{\text{ prime}}}}v_{p}(n)F_{b}(p).$
Here the exponent $v_{p}(n)$ is the multiplicity of $p$ as a prime factor of $n$ (also known as the p-adic valuation of $n$).
For example, in base 10, 378 = 21 · 33 · 71 is a Smith number since 3 + 7 + 8 = 2 · 1 + 3 · 3 + 7 · 1, and 22 = 21 · 111 is a Smith number, because 2 + 2 = 2 · 1 + (1 + 1) · 1.
The first few Smith numbers in base 10 are
4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985. (sequence A006753 in the OEIS)
Properties
W.L. McDaniel in 1987 proved that there are infinitely many Smith numbers.[1][2] The number of Smith numbers in base 10 below 10n for n = 1, 2, ... is given by
1, 6, 49, 376, 3294, 29928, 278411, 2632758, 25154060, 241882509, ... (sequence A104170 in the OEIS).
Two consecutive Smith numbers (for example, 728 and 729, or 2964 and 2965) are called Smith brothers.[3] It is not known how many Smith brothers there are. The starting elements of the smallest Smith n-tuple (meaning n consecutive Smith numbers) in base 10 for n = 1, 2, ... are[4]
4, 728, 73615, 4463535, 15966114, 2050918644, 164736913905, ... (sequence A059754 in the OEIS).
Smith numbers can be constructed from factored repunits.[5] As of 2010, the largest known Smith number in base 10 is
9 × R1031 × (104594 + 3×102297 + 1)1476 ×103913210
where R1031 is the base 10 repunit (101031 − 1)/9.
See also
• Equidigital number
Notes
1. Sándor & Crstici (2004) p.383
2. McDaniel, Wayne (1987). "The existence of infinitely many k-Smith numbers". Fibonacci Quarterly. 25 (1): 76–80. Zbl 0608.10012.
3. Sándor & Crstici (2004) p.384
4. Shyam Sunder Gupta. "Fascinating Smith Numbers".
5. Hoffman (1998), pp. 205–6
References
• Gardner, Martin (1988). Penrose Tiles to Trapdoor Ciphers. pp. 299–300.
• Hoffman, Paul (1998). The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion.
• Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 32–36. ISBN 1-4020-2546-7. Zbl 1079.11001.
External links
• Weisstein, Eric W. "Smith Number". MathWorld.
• Copeland, Ed. "4937775 – Smith Numbers". Numberphile. Brady Haran. Archived from the original on 2021-12-21.
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Smith space
In functional analysis and related areas of mathematics, a Smith space is a complete compactly generated locally convex topological vector space $X$ having a universal compact set, i.e. a compact set $K$ which absorbs every other compact set $T\subseteq X$ (i.e. $T\subseteq \lambda \cdot K$ for some $\lambda >0$).
Smith spaces are named after Marianne Ruth Freundlich Smith, who introduced them[1] as duals to Banach spaces in some versions of duality theory for topological vector spaces. All Smith spaces are stereotype and are in the stereotype duality relations with Banach spaces:[2][3]
• for any Banach space $X$ its stereotype dual space[4] $X^{\star }$ is a Smith space,
• and vice versa, for any Smith space $X$ its stereotype dual space $X^{\star }$ is a Banach space.
Smith spaces are special cases of Brauner spaces.
Examples
• As follows from the duality theorems, for any Banach space $X$ its stereotype dual space $X^{\star }$ is a Smith space. The polar $K=B^{\circ }$ of the unit ball $B$ in $X$ is the universal compact set in $X^{\star }$. If $X^{*}$ denotes the normed dual space for $X$, and $X'$ the space $X^{*}$ endowed with the $X$-weak topology, then the topology of $X^{\star }$ lies between the topology of $X^{*}$ and the topology of $X'$, so there are natural (linear continuous) bijections
$X^{*}\to X^{\star }\to X'.$
If $X$ is infinite-dimensional, then no two of these topologies coincide. At the same time, for infinite dimensional $X$ the space $X^{\star }$ is not barreled (and even is not a Mackey space if $X$ is reflexive as a Banach space[5]).
• If $K$ is a convex balanced compact set in a locally convex space $Y$, then its linear span ${\mathbb {C} }K=\operatorname {span} (K)$ possesses a unique structure of a Smith space with $K$ as the universal compact set (and with the same topology on $K$).[6]
• If $M$ is a (Hausdorff) compact topological space, and ${\mathcal {C}}(M)$ the Banach space of continuous functions on $M$ (with the usual sup-norm), then the stereotype dual space ${\mathcal {C}}^{\star }(M)$ (of Radon measures on $M$ with the topology of uniform convergence on compact sets in ${\mathcal {C}}(M)$) is a Smith space. In the special case when $M=G$ is endowed with a structure of a topological group the space ${\mathcal {C}}^{\star }(G)$ becomes a natural example of a stereotype group algebra.[7]
• A Banach space $X$ is a Smith space if and only if $X$ is finite-dimensional.
See also
• Stereotype space
• Brauner space
Notes
1. Smith 1952.
2. Akbarov 2003, p. 220.
3. Akbarov 2009, p. 467.
4. The stereotype dual space to a locally convex space $X$ is the space $X^{\star }$ of all linear continuous functionals $f:X\to \mathbb {C} $ endowed with the topology of uniform convergence on totally bounded sets in $X$.
5. Akbarov 2003, p. 221, Example 4.8.
6. Akbarov 2009, p. 468.
7. Akbarov 2003, p. 272.
References
• Smith, M.F. (1952). "The Pontrjagin duality theorem in linear spaces". Annals of Mathematics. 56 (2): 248–253. doi:10.2307/1969798. JSTOR 1969798.
• Akbarov, S.S. (2003). "Pontryagin duality in the theory of topological vector spaces and in topological algebra". Journal of Mathematical Sciences. 113 (2): 179–349. doi:10.1023/A:1020929201133. S2CID 115297067.
• Akbarov, S.S. (2009). "Holomorphic functions of exponential type and duality for Stein groups with algebraic connected component of identity". Journal of Mathematical Sciences. 162 (4): 459–586. arXiv:0806.3205. doi:10.1007/s10958-009-9646-1. S2CID 115153766.
• Furber, R.W.J. (2017). Categorical Duality in Probability and Quantum Foundations (PDF) (PhD). Radboud University.
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Smith–Minkowski–Siegel mass formula
In mathematics, the Smith–Minkowski–Siegel mass formula (or Minkowski–Siegel mass formula) is a formula for the sum of the weights of the lattices (quadratic forms) in a genus, weighted by the reciprocals of the orders of their automorphism groups. The mass formula is often given for integral quadratic forms, though it can be generalized to quadratic forms over any algebraic number field.
In 0 and 1 dimensions the mass formula is trivial, in 2 dimensions it is essentially equivalent to Dirichlet's class number formulas for imaginary quadratic fields, and in 3 dimensions some partial results were given by Gotthold Eisenstein. The mass formula in higher dimensions was first given by H. J. S. Smith (1867), though his results were forgotten for many years. It was rediscovered by H. Minkowski (1885), and an error in Minkowski's paper was found and corrected by C. L. Siegel (1935).
Many published versions of the mass formula have errors; in particular the 2-adic densities are difficult to get right, and it is sometimes forgotten that the trivial cases of dimensions 0 and 1 are different from the cases of dimension at least 2. Conway & Sloane (1988) give an expository account and precise statement of the mass formula for integral quadratic forms, which is reliable because they check it on a large number of explicit cases.
For recent proofs of the mass formula see (Kitaoka 1999) and (Eskin, Rudnick & Sarnak 1991).
The Smith–Minkowski–Siegel mass formula is essentially the constant term of the Weil–Siegel formula.
Statement of the mass formula
If f is an n-dimensional positive definite integral quadratic form (or lattice) then the mass of its genus is defined to be
$m(f)=\sum _{\Lambda }{1 \over |{\operatorname {Aut} (\Lambda )}|}$
where the sum is over all integrally inequivalent forms in the same genus as f, and Aut(Λ) is the automorphism group of Λ. The form of the mass formula given by Conway & Sloane (1988) states that for n ≥ 2 the mass is given by
$m(f)=2\pi ^{-n(n+1)/4}\prod _{j=1}^{n}\Gamma (j/2)\prod _{p{\text{ prime}}}2m_{p}(f)$
where mp(f) is the p-mass of f, given by
$m_{p}(f)={p^{(rn(n-1)+s(n+1))/2} \over N(p^{r})}$
for sufficiently large r, where ps is the highest power of p dividing the determinant of f. The number N(pr) is the number of n by n matrices X with coefficients that are integers mod p r such that
$X^{\text{tr}}AX\equiv A\ {\bmod {\ }}p^{r}$
where A is the Gram matrix of f, or in other words the order of the automorphism group of the form reduced mod p r.
Some authors state the mass formula in terms of the p-adic density
$\alpha _{p}(f)={N(p^{r}) \over p^{rn(n-1)/2}}={p^{s(n+1)/2} \over m_{p}(f)}$
instead of the p-mass. The p-mass is invariant under rescaling f but the p-density is not.
In the (trivial) cases of dimension 0 or 1 the mass formula needs some modifications. The factor of 2 in front represents the Tamagawa number of the special orthogonal group, which is only 1 in dimensions 0 and 1. Also the factor of 2 in front of mp(f) represents the index of the special orthogonal group in the orthogonal group, which is only 1 in 0 dimensions.
Evaluation of the mass
The mass formula gives the mass as an infinite product over all primes. This can be rewritten as a finite product as follows. For all but a finite number of primes (those not dividing 2 det(ƒ)) the p-mass mp(ƒ) is equal to the standard p-mass stdp(ƒ), given by
$\operatorname {std} _{p}(f)={1 \over 2(1-p^{-2})(1-p^{-4})\dots (1-p^{2-n})(1-{(-1)^{n/2}\det(f) \choose p}p^{-n/2})}\quad $ (for n = dim(ƒ) even)
$\operatorname {std} _{p}(f)={1 \over 2(1-p^{-2})(1-p^{-4})\dots (1-p^{1-n})}$ (for n = dim(ƒ) odd)
where the Legendre symbol in the second line is interpreted as 0 if p divides 2 det(ƒ).
If all the p-masses have their standard value, then the total mass is the standard mass
$\operatorname {std} (f)=2\pi ^{-n(n+1)/4}\left(\prod _{j=1}^{n}\Gamma (j/2)\right)\zeta (2)\zeta (4)\dots \zeta (n-1)$ (For n odd)
$\operatorname {std} (f)=2\pi ^{-n(n+1)/4}\left(\prod _{j=1}^{n}\Gamma (j/2)\right)\zeta (2)\zeta (4)\dots \zeta (n-2)\zeta _{D}(n/2)$ (For n even)
where
$\zeta _{D}(s)=\prod _{p}{1 \over 1-{{\big (}{\frac {D}{p}}{\big )}}p^{-s}}$
D = (−1)n/2 det(ƒ)
The values of the Riemann zeta function for an even integers s are given in terms of Bernoulli numbers by
$\zeta (s)={(2\pi )^{s} \over 2\times s!}|B_{s}|.$
So the mass of ƒ is given as a finite product of rational numbers as
$m(f)=\operatorname {std} (f)\prod _{p|2\det(f)}{m_{p}(f) \over \operatorname {std} _{p}(f)}.$
Evaluation of the p-mass
If the form f has a p-adic Jordan decomposition
$f=\sum qf_{q}$
where q runs through powers of p and fq has determinant prime to p and dimension n(q), then the p-mass is given by
$m_{p}(f)=\prod _{q}M_{p}(f_{q})\times \prod _{q<q'}(q'/q)^{n(q)n(q')/2}\times 2^{n(I,I)-n(II)}$
Here n(II) is the sum of the dimensions of all Jordan components of type 2 and p = 2, and n(I,I) is the total number of pairs of adjacent constituents fq, f2q that are both of type I.
The factor Mp(fq) is called a diagonal factor and is a power of p times the order of a certain orthogonal group over the field with p elements. For odd p its value is given by
${1 \over 2(1-p^{-2})(1-p^{-4})\dots (1-p^{1-n})}$
when n is odd, or
${1 \over 2(1-p^{-2})(1-p^{-4})\dots (1-p^{2-n})(1-p^{-n/2})}$
when n is even and (−1)n/2dq is a quadratic residue, or
${1 \over 2(1-p^{-2})(1-p^{-4})\dots (1-p^{2-n})(1+p^{-n/2})}$
when n is even and (−1)n/2dq is a quadratic nonresidue.
For p = 2 the diagonal factor Mp(fq) is notoriously tricky to calculate. (The notation is misleading as it depends not only on fq but also on f2q and fq/2.)
• We say that fq is odd if it represents an odd 2-adic integer, and even otherwise.
• The octane value of fq is an integer mod 8; if fq is even its octane value is 0 if the determinant is +1 or −1 mod 8, and is 4 if the determinant is +3 or −3 mod 8, while if fq is odd it can be diagonalized and its octane value is then the number of diagonal entries that are 1 mod 4 minus the number that are 3 mod 4.
• We say that fq is bound if at least one of f2q and fq/2 is odd, and say it is free otherwise.
• The integer t is defined so that the dimension of fq is 2t if fq is even, and 2t + 1 or 2t + 2 if fq is odd.
Then the diagonal factor Mp(fq) is given as follows.
${1 \over 2(1-p^{-2})(1-p^{-4})\dots (1-p^{-2t})}$
when the form is bound or has octane value +2 or −2 mod 8 or
${1 \over 2(1-p^{-2})(1-p^{-4})\dots (1-p^{2-2t})(1-p^{-t})}$
when the form is free and has octane value −1 or 0 or 1 mod 8 or
${1 \over 2(1-p^{-2})(1-p^{-4})\dots (1-p^{2-2t})(1+p^{-t})}$
when the form is free and has octane value −3 or 3 or 4 mod 8.
Evaluation of ζD(s)
The required values of the Dirichlet series ζD(s) can be evaluated as follows. We write χ for the Dirichlet character with χ(m) given by 0 if m is even, and the Jacobi symbol ${\left({\frac {D}{m}}\right)}$ if m is odd. We write k for the modulus of this character and k1 for its conductor, and put χ = χ1ψ where χ1 is the principal character mod k and ψ is a primitive character mod k1. Then
$\zeta _{D}(s)=L(s,\chi )=L(s,\psi )\prod _{p|k}\left(1-{\psi (p) \over p^{s}}\right)$
The functional equation for the L-series is
$L(1-s,\psi )={k_{1}^{s-1}\Gamma (s) \over (2\pi )^{s}}(i^{-s}+\psi (-1)i^{s})G(\psi )L(s,\psi )$
where G is the Gauss sum
$G(\psi )=\sum _{r=1}^{k_{1}}\psi (r)e^{2\pi ir/k_{1}}.$
If s is a positive integer then
$L(1-s,\psi )=-{k_{1}^{s-1} \over s}\sum _{r=1}^{k_{1}}\psi (r)B_{s}(r/k_{1})$
where Bs(x) is a Bernoulli polynomial.
Examples
For the case of even unimodular lattices Λ of dimension n > 0 divisible by 8 the mass formula is
$\sum _{\Lambda }{1 \over |\operatorname {Aut} (\Lambda )|}={|B_{n/2}| \over n}\prod _{1\leq j<n/2}{|B_{2j}| \over 4j}$
where Bk is a Bernoulli number.
Dimension n = 0
The formula above fails for n = 0, and in general the mass formula needs to be modified in the trivial cases when the dimension is at most 1. For n = 0 there is just one lattice, the zero lattice, of weight 1, so the total mass is 1.
Dimension n = 8
The mass formula gives the total mass as
${|B_{4}| \over 8}{|B_{2}| \over 4}{|B_{4}| \over 8}{|B_{6}| \over 12}={1/30 \over 8}\;{1/6 \over 4}\;{1/30 \over 8}\;{1/42 \over 12}={1 \over 696729600}.$
There is exactly one even unimodular lattice of dimension 8, the E8 lattice, whose automorphism group is the Weyl group of E8 of order 696729600, so this verifies the mass formula in this case. Smith originally gave a nonconstructive proof of the existence of an even unimodular lattice of dimension 8 using the fact that the mass is non-zero.
Dimension n = 16
The mass formula gives the total mass as
${|B_{8}| \over 16}{|B_{2}| \over 4}{|B_{4}| \over 8}{|B_{6}| \over 12}{|B_{8}| \over 16}{|B_{10}| \over 20}{|B_{12}| \over 24}{|B_{14}| \over 28}={691 \over 277667181515243520000}.$
There are two even unimodular lattices of dimension 16, one with root system E82 and automorphism group of order 2×6967296002 = 970864271032320000, and one with root system D16 and automorphism group of order 21516! = 685597979049984000.
So the mass formula is
${1 \over 970864271032320000}+{1 \over 685597979049984000}={691 \over 277667181515243520000}.$
Dimension n = 24
There are 24 even unimodular lattices of dimension 24, called the Niemeier lattices. The mass formula for them is checked in (Conway & Sloane 1998, pp. 410–413).
Dimension n = 32
The mass in this case is large, more than 40 million. This implies that there are more than 80 million even unimodular lattices of dimension 32, as each has automorphism group of order at least 2 so contributes at most 1/2 to the mass. By refining this argument, King (2003) showed that there are more than a billion such lattices. In higher dimensions the mass, and hence the number of lattices, increases very rapidly.
Generalizations
Siegel gave a more general formula that counts the weighted number of representations of one quadratic form by forms in some genus; the Smith–Minkowski–Siegel mass formula is the special case when one form is the zero form.
Tamagawa showed that the mass formula was equivalent to the statement that the Tamagawa number of the orthogonal group is 2, which is equivalent to saying that the Tamagawa number of its simply connected cover the spin group is 1. André Weil conjectured more generally that the Tamagawa number of any simply connected semisimple group is 1, and this conjecture was proved by Kottwitz in 1988.
King (2003) gave a mass formula for unimodular lattices without roots (or with given root system).
See also
• Siegel identity
References
• Conway, J. H.; Sloane, N. J. A. (1998), Sphere packings, lattices, and groups, Berlin: Springer-Verlag, ISBN 978-0-387-98585-5
• Conway, J. H.; Sloane, N. J. A. (1988), "Low-Dimensional Lattices. IV. The Mass Formula", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 419 (1988): 259–286, Bibcode:1988RSPSA.419..259C, CiteSeerX 10.1.1.24.2955, doi:10.1098/rspa.1988.0107, JSTOR 2398465
• Eskin, Alex; Rudnick, Zeév; Sarnak, Peter (1991), "A proof of Siegel's weight formula.", International Mathematics Research Notices, 1991 (5): 65–69, doi:10.1155/S1073792891000090, MR 1131433
• King, Oliver (2003), "A mass formula for unimodular lattices with no roots", Mathematics of Computation, 72 (242): 839–863, arXiv:math.NT/0012231, Bibcode:2003MaCom..72..839K, doi:10.1090/S0025-5718-02-01455-2.
• Kitaoka, Yoshiyuki (1999), Arithmetic of Quadratic Forms, Cambridge Tracts in Mathematics, Cambridge: Cambridge Univ. Press, ISBN 978-0-521-64996-4
• Minkowski, Hermann (1885), "Untersuchungen über quadratische Formen I. Bestimmung der Anzahl verschiedener Formen, welche ein gegebenes Genus enthält", Acta Mathematica, 7 (1): 201–258, doi:10.1007/BF02402203
• Siegel, Carl Ludwig (1935), "Uber Die Analytische Theorie Der Quadratischen Formen", Annals of Mathematics, Second Series, 36 (3): 527–606, doi:10.2307/1968644, JSTOR 1968644
• Smith, H. J. Stephen (1867), "On the Orders and Genera of Quadratic Forms Containing More than Three Indeterminates", Proceedings of the Royal Society of London, 16: 197–208, doi:10.1098/rspl.1867.0036, JSTOR 112491
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Smith–Volterra–Cantor set
In mathematics, the Smith–Volterra–Cantor set (SVC), fat Cantor set, or ε-Cantor set[1] is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–Volterra–Cantor set is named after the mathematicians Henry Smith, Vito Volterra and Georg Cantor. In an 1875 paper, Smith discussed a nowhere-dense set of positive measure on the real line,[2] and Volterra introduced a similar example in 1881.[3] The Cantor set as we know it today followed in 1883. The Smith–Volterra–Cantor set is topologically equivalent to the middle-thirds Cantor set.
Construction
Similar to the construction of the Cantor set, the Smith–Volterra–Cantor set is constructed by removing certain intervals from the unit interval $[0,1].$
The process begins by removing the middle 1/4 from the interval $[0,1]$ (the same as removing 1/8 on either side of the middle point at 1/2) so the remaining set is
$\left[0,{\tfrac {3}{8}}\right]\cup \left[{\tfrac {5}{8}},1\right].$
The following steps consist of removing subintervals of width $1/4^{n}$ from the middle of each of the $2^{n-1}$ remaining intervals. So for the second step the intervals $(5/32,7/32)$ and $(25/32,27/32)$ are removed, leaving
$\left[0,{\tfrac {5}{32}}\right]\cup \left[{\tfrac {7}{32}},{\tfrac {3}{8}}\right]\cup \left[{\tfrac {5}{8}},{\tfrac {25}{32}}\right]\cup \left[{\tfrac {27}{32}},1\right].$
Continuing indefinitely with this removal, the Smith–Volterra–Cantor set is then the set of points that are never removed. The image below shows the initial set and five iterations of this process.
Each subsequent iterate in the Smith–Volterra–Cantor set's construction removes proportionally less from the remaining intervals. This stands in contrast to the Cantor set, where the proportion removed from each interval remains constant. Thus, the former has positive measure while the latter has zero measure.
Properties
By construction, the Smith–Volterra–Cantor set contains no intervals and therefore has empty interior. It is also the intersection of a sequence of closed sets, which means that it is closed. During the process, intervals of total length
$\sum _{n=0}^{\infty }{\tfrac {2^{n}}{2^{2n+2}}}={\tfrac {1}{4}}+{\tfrac {1}{8}}+{\tfrac {1}{16}}+\cdots ={\tfrac {1}{2}}\,$
are removed from $[0,1],$ showing that the set of the remaining points has a positive measure of 1/2. This makes the Smith–Volterra–Cantor set an example of a closed set whose boundary has positive Lebesgue measure.
Other fat Cantor sets
In general, one can remove $r_{n}$ from each remaining subinterval at the $n$th step of the algorithm, and end up with a Cantor-like set. The resulting set will have positive measure if and only if the sum of the sequence is less than the measure of the initial interval. For instance, suppose the middle intervals of length $a^{n}$ are removed from $[0,1]$ for each $n$th iteration, for some $0\leq a\leq {\dfrac {1}{3}}.$ Then, the resulting set has Lebesgue measure
${\begin{aligned}1-\sum _{n=0}^{\infty }2^{n}a^{n+1}&=1-a\sum _{n=0}^{\infty }(2a)^{n}\\&=1-a{\dfrac {1}{1-2a}}\\&={\dfrac {1-3a}{1-2a}}\end{aligned}}$
which goes from $0$ to $1$ as $a$ goes from $1/3$ to $0.$ ($a>1/3$ is impossible in this construction.)
Cartesian products of Smith–Volterra–Cantor sets can be used to find totally disconnected sets in higher dimensions with nonzero measure. By applying the Denjoy–Riesz theorem to a two-dimensional set of this type, it is possible to find an Osgood curve, a Jordan curve such that the points on the curve have positive area.[4]
See also
• The Smith–Volterra–Cantor set is used in the construction of Volterra's function (see external link).
• The Smith–Volterra–Cantor set is an example of a compact set that is not Jordan measurable, see Jordan measure#Extension to more complicated sets.
• The indicator function of the Smith–Volterra–Cantor set is an example of a bounded function that is not Riemann integrable on (0,1) and moreover, is not equal almost everywhere to a Riemann integrable function, see Riemann integral#Examples.
• List of topologies – List of concrete topologies and topological spaces
References
1. Aliprantis and Burkinshaw (1981), Principles of Real Analysis
2. Smith, Henry J.S. (1874). "On the integration of discontinuous functions". Proceedings of the London Mathematical Society. First series. 6: 140–153
3. Ponce Campuzano, Juan; Maldonado, Miguel (2015). "Vito Volterra's construction of a nonconstant function with a bounded, non Riemann integrable derivative". BSHM Bulletin Journal of the British Society for the History of Mathematics. 30 (2): 143–152. doi:10.1080/17498430.2015.1010771. S2CID 34546093.
4. Balcerzak, M.; Kharazishvili, A. (1999), "On uncountable unions and intersections of measurable sets", Georgian Mathematical Journal, 6 (3): 201–212, doi:10.1023/A:1022102312024, MR 1679442.
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Smooth algebra
In algebra, a commutative k-algebra A is said to be 0-smooth if it satisfies the following lifting property: given a k-algebra C, an ideal N of C whose square is zero and a k-algebra map $u:A\to C/N$, there exists a k-algebra map $v:A\to C$ such that u is v followed by the canonical map. If there exists at most one such lifting v, then A is said to be 0-unramified (or 0-neat). A is said to be 0-étale if it is 0-smooth and 0-unramified. The notion of 0-smoothness is also called formal smoothness.
A finitely generated k-algebra A is 0-smooth over k if and only if Spec A is a smooth scheme over k.
A separable algebraic field extension L of k is 0-étale over k.[1] The formal power series ring $k[\![t_{1},\ldots ,t_{n}]\!]$ is 0-smooth only when $\operatorname {char} k=p>0$ and $[k:k^{p}]<\infty $ (i.e., k has a finite p-basis.)[2]
I-smooth
Let B be an A-algebra and suppose B is given the I-adic topology, I an ideal of B. We say B is I-smooth over A if it satisfies the lifting property: given an A-algebra C, an ideal N of C whose square is zero and an A-algebra map $u:B\to C/N$ that is continuous when $C/N$ is given the discrete topology, there exists an A-algebra map $v:B\to C$ such that u is v followed by the canonical map. As before, if there exists at most one such lift v, then B is said to be I-unramified over A (or I-neat). B is said to be I-étale if it is I-smooth and I-unramified. If I is the zero ideal and A is a field, these notions coincide with 0-smooth etc. as defined above.
A standard example is this: let A be a ring, $B=A[\![t_{1},\ldots ,t_{n}]\!]$ and $I=(t_{1},\ldots ,t_{n}).$ Then B is I-smooth over A.
Let A be a noetherian local k-algebra with maximal ideal ${\mathfrak {m}}$. Then A is ${\mathfrak {m}}$-smooth over $k$ if and only if $A\otimes _{k}k'$ is a regular ring for any finite extension field $k'$ of $k$.[3]
See also
• étale morphism
• formally smooth morphism
• Popescu's theorem
References
1. Matsumura 1989, Theorem 25.3
2. Matsumura 1989, pg. 215
3. Matsumura 1989, Theorem 28.7
• Matsumura, H. (1989). Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. Translated by Reid, M. Cambridge University Press. ISBN 978-0-521-36764-6.
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Smooth coarea formula
In Riemannian geometry, the smooth coarea formulas relate integrals over the domain of certain mappings with integrals over their codomains.
Let $\scriptstyle M,\,N$ be smooth Riemannian manifolds of respective dimensions $\scriptstyle m\,\geq \,n$. Let $\scriptstyle F:M\,\longrightarrow \,N$ be a smooth surjection such that the pushforward (differential) of $\scriptstyle F$ is surjective almost everywhere. Let $\scriptstyle \varphi :M\,\longrightarrow \,[0,\infty )$ a measurable function. Then, the following two equalities hold:
$\int _{x\in M}\varphi (x)\,dM=\int _{y\in N}\int _{x\in F^{-1}(y)}\varphi (x){\frac {1}{N\!J\;F(x)}}\,dF^{-1}(y)\,dN$
$\int _{x\in M}\varphi (x)N\!J\;F(x)\,dM=\int _{y\in N}\int _{x\in F^{-1}(y)}\varphi (x)\,dF^{-1}(y)\,dN$
where $\scriptstyle N\!J\;F(x)$ is the normal Jacobian of $\scriptstyle F$, i.e. the determinant of the derivative restricted to the orthogonal complement of its kernel.
Note that from Sard's lemma almost every point $\scriptstyle y\,\in \,N$ is a regular point of $\scriptstyle F$ and hence the set $\scriptstyle F^{-1}(y)$ is a Riemannian submanifold of $\scriptstyle M$, so the integrals in the right-hand side of the formulas above make sense.
References
• Chavel, Isaac (2006) Riemannian Geometry. A Modern Introduction. Second Edition.
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Smooth completion
In algebraic geometry, the smooth completion (or smooth compactification) of a smooth affine algebraic curve X is a complete smooth algebraic curve which contains X as an open subset.[1] Smooth completions exist and are unique over a perfect field.
Examples
An affine form of a hyperelliptic curve may be presented as $y^{2}=P(x)$ where $(x,y)\in \mathbb {C} ^{2}$ and P(x) has distinct roots and has degree at least 5. The Zariski closure of the affine curve in $\mathbb {C} \mathbb {P} ^{2}$ is singular at the unique infinite point added. Nonetheless, the affine curve can be embedded in a unique compact Riemann surface called its smooth completion. The projection of the Riemann surface to $\mathbb {C} \mathbb {P} ^{1}$ is 2-to-1 over the singular point at infinity if $P(x)$ has even degree, and 1-to-1 (but ramified) otherwise.
This smooth completion can also be obtained as follows. Project the affine curve to the affine line using the x-coordinate. Embed the affine line into the projective line, then take the normalization of the projective line in the function field of the affine curve.
Applications
A smooth connected curve over an algebraically closed field is called hyperbolic if $2g-2+r>0$ where g is the genus of the smooth completion and r is the number of added points.
Over an algebraically closed field of characteristic 0, the fundamental group of X is free with $2g+r-1$ generators if r>0.
(Analogue of Dirichlet's unit theorem) Let X be a smooth connected curve over a finite field. Then the units of the ring of regular functions O(X) on X is a finitely generated abelian group of rank r -1.
Construction
Suppose the base field is perfect. Any affine curve X is isomorphic to an open subset of an integral projective (hence complete) curve. Taking the normalization (or blowing up the singularities) of the projective curve then gives a smooth completion of X. Their points correspond to the discrete valuations of the function field that are trivial on the base field.
By construction, the smooth completion is a projective curve which contains the given curve as an everywhere dense open subset, and the added new points are smooth. Such a (projective) completion always exists and is unique.
If the base field is not perfect, a smooth completion of a smooth affine curve doesn't always exist. But the above process always produces a regular completion if we start with a regular affine curve (smooth varieties are regular, and the converse is true over perfect fields). A regular completion is unique and, by the valuative criterion of properness, any morphism from the affine curve to a complete algebraic variety extends uniquely to the regular completion.
Generalization
If X is a separated algebraic variety, a theorem of Nagata[2] says that X can be embedded as an open subset of a complete algebraic variety. If X is moreover smooth and the base field has characteristic 0, then by Hironaka's theorem X can even be embedded as an open subset of a complete smooth algebraic variety, with boundary a normal crossing divisor. If X is quasi-projective, the smooth completion can be chosen to be projective.
However, contrary to the one-dimensional case, there is no uniqueness of the smooth completion, nor is it canonical.
See also
• Hyperelliptic curve
• Bolza surface
References
1. Griffiths, 1972, p. 286.
2. Conrad, Brian (2007). "Deligne's notes on Nagata compactifications" (PDF). Journal of the Ramanujan Mathematical Society. 22 (3): 205–257. MR 2356346.
Bibliography
• Griffiths, Phillip A. (1972). "Function theory of finite order on algebraic varieties. I(A)". Journal of Differential Geometry. 6 (3): 285–306. MR 0325999. Zbl 0269.14003.
• Hartshorne, Robin (1977). Algebraic geometry. Graduate Texts in Mathematics. Vol. 52. New York, Heidelberg: Springer-Verlag. ISBN 0387902449. (see chapter 4).
Topics in algebraic curves
Rational curves
• Five points determine a conic
• Projective line
• Rational normal curve
• Riemann sphere
• Twisted cubic
Elliptic curves
Analytic theory
• Elliptic function
• Elliptic integral
• Fundamental pair of periods
• Modular form
Arithmetic theory
• Counting points on elliptic curves
• Division polynomials
• Hasse's theorem on elliptic curves
• Mazur's torsion theorem
• Modular elliptic curve
• Modularity theorem
• Mordell–Weil theorem
• Nagell–Lutz theorem
• Supersingular elliptic curve
• Schoof's algorithm
• Schoof–Elkies–Atkin algorithm
Applications
• Elliptic curve cryptography
• Elliptic curve primality
Higher genus
• De Franchis theorem
• Faltings's theorem
• Hurwitz's automorphisms theorem
• Hurwitz surface
• Hyperelliptic curve
Plane curves
• AF+BG theorem
• Bézout's theorem
• Bitangent
• Cayley–Bacharach theorem
• Conic section
• Cramer's paradox
• Cubic plane curve
• Fermat curve
• Genus–degree formula
• Hilbert's sixteenth problem
• Nagata's conjecture on curves
• Plücker formula
• Quartic plane curve
• Real plane curve
Riemann surfaces
• Belyi's theorem
• Bring's curve
• Bolza surface
• Compact Riemann surface
• Dessin d'enfant
• Differential of the first kind
• Klein quartic
• Riemann's existence theorem
• Riemann–Roch theorem
• Teichmüller space
• Torelli theorem
Constructions
• Dual curve
• Polar curve
• Smooth completion
Structure of curves
Divisors on curves
• Abel–Jacobi map
• Brill–Noether theory
• Clifford's theorem on special divisors
• Gonality of an algebraic curve
• Jacobian variety
• Riemann–Roch theorem
• Weierstrass point
• Weil reciprocity law
Moduli
• ELSV formula
• Gromov–Witten invariant
• Hodge bundle
• Moduli of algebraic curves
• Stable curve
Morphisms
• Hasse–Witt matrix
• Riemann–Hurwitz formula
• Prym variety
• Weber's theorem (Algebraic curves)
Singularities
• Acnode
• Crunode
• Cusp
• Delta invariant
• Tacnode
Vector bundles
• Birkhoff–Grothendieck theorem
• Stable vector bundle
• Vector bundles on algebraic curves
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Smooth functor
In differential topology, a branch of mathematics, a smooth functor is a type of functor defined on finite-dimensional real vector spaces. Intuitively, a smooth functor is smooth in the sense that it sends smoothly parameterized families of vector spaces to smoothly parameterized families of vector spaces. Smooth functors may therefore be uniquely extended to functors defined on vector bundles.
Let Vect be the category of finite-dimensional real vector spaces whose morphisms consist of all linear mappings, and let F be a covariant functor that maps Vect to itself. For vector spaces T, U ∈ Vect, the functor F induces a mapping
$F:\mathrm {Hom} _{\mathbf {Vect} }(T,U)\rightarrow \mathrm {Hom} _{\mathbf {Vect} }(F(T),F(U)),$
where Hom is notation for Hom functor. If this map is smooth as a map of infinitely differentiable manifolds then F is said to be a smooth functor.[1]
Common smooth functors include, for some vector space W:[2]
F(W) = ⊗nW, the nth iterated tensor product;
F(W) = Λn(W), the nth exterior power; and
F(W) = Symn(W), the nth symmetric power.
Smooth functors are significant because any smooth functor can be applied fiberwise to a differentiable vector bundle on a manifold. Smoothness of the functor is the condition required to ensure that the patching data for the bundle are smooth as mappings of manifolds.[2] For instance, because the nth exterior power of a vector space defines a smooth functor, the nth exterior power of a smooth vector bundle is also a smooth vector bundle.
Although there are established methods for proving smoothness of standard constructions on finite-dimensional vector bundles, smooth functors can be generalized to categories of topological vector spaces and vector bundles on infinite-dimensional Fréchet manifolds.[3]
See also
• Smooth infinitesimal analysis
• Synthetic differential geometry
Notes
1. Antonelli 2003, p. 1420; Kriegl & Michor 1997, p. 290. Lee 2002, pp.122–23 defines smooth functors over a different category, whose morphisms are linear isomorphisms rather than all linear mappings.
2. Kriegl & Michor 1997, p. 290
3. Kriegl & Michor 1997 have developed an infinite-dimensional theory for so-called "convenient vector spaces" – a class of locally convex spaces that includes Fréchet spaces.
References
• Antonelli, P. L. (2003), Handbook of Finsler geometry, Springer, p. 1420, ISBN 1-4020-1556-9.
• Kriegl, Andreas; Michor, Peter W. (1997), The convenient setting of global analysis, AMS Bookstore, p. 290, ISBN 0-8218-0780-3.
• Lee, John M. (2002), Introduction to smooth manifolds, Springer, pp. 122–23, ISBN 0-387-95448-1.
Functor types
• Additive
• Adjoint
• Conservative
• Derived
• Diagonal
• Enriched
• Essentially surjective
• Exact
• Forgetful
• Full and faithful
• Logical
• Monoidal
• Representable
• Smooth
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Smooth infinitesimal analysis
Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of being expressed in terms of discrete entities. As a theory, it is a subset of synthetic differential geometry.
The nilsquare or nilpotent infinitesimals are numbers ε where ε² = 0 is true, but ε = 0 need not be true at the same time.
Overview
This approach departs from the classical logic used in conventional mathematics by denying the law of the excluded middle, e.g., NOT (a ≠ b) does not imply a = b. In particular, in a theory of smooth infinitesimal analysis one can prove for all infinitesimals ε, NOT (ε ≠ 0); yet it is provably false that all infinitesimals are equal to zero.[1] One can see that the law of excluded middle cannot hold from the following basic theorem (again, understood in the context of a theory of smooth infinitesimal analysis):
Every function whose domain is R, the real numbers, is continuous and infinitely differentiable.
Despite this fact, one could attempt to define a discontinuous function f(x) by specifying that f(x) = 1 for x = 0, and f(x) = 0 for x ≠ 0. If the law of the excluded middle held, then this would be a fully defined, discontinuous function. However, there are plenty of x, namely the infinitesimals, such that neither x = 0 nor x ≠ 0 holds, so the function is not defined on the real numbers.
In typical models of smooth infinitesimal analysis, the infinitesimals are not invertible, and therefore the theory does not contain infinite numbers. However, there are also models that include invertible infinitesimals.
Other mathematical systems exist which include infinitesimals, including nonstandard analysis and the surreal numbers. Smooth infinitesimal analysis is like nonstandard analysis in that (1) it is meant to serve as a foundation for analysis, and (2) the infinitesimal quantities do not have concrete sizes (as opposed to the surreals, in which a typical infinitesimal is 1/ω, where ω is a von Neumann ordinal). However, smooth infinitesimal analysis differs from nonstandard analysis in its use of nonclassical logic, and in lacking the transfer principle. Some theorems of standard and nonstandard analysis are false in smooth infinitesimal analysis, including the intermediate value theorem and the Banach–Tarski paradox. Statements in nonstandard analysis can be translated into statements about limits, but the same is not always true in smooth infinitesimal analysis.
Intuitively, smooth infinitesimal analysis can be interpreted as describing a world in which lines are made out of infinitesimally small segments, not out of points. These segments can be thought of as being long enough to have a definite direction, but not long enough to be curved. The construction of discontinuous functions fails because a function is identified with a curve, and the curve cannot be constructed pointwise. We can imagine the intermediate value theorem's failure as resulting from the ability of an infinitesimal segment to straddle a line. Similarly, the Banach–Tarski paradox fails because a volume cannot be taken apart into points.
See also
• Category theory
• Non-standard analysis
• Synthetic differential geometry
• Dual number
References
1. Bell, John L. (2008). A Primer of Infinitesimal Analysis, 2nd Edition. Cambridge University Press. ISBN 9780521887182.
Further reading
• John Lane Bell, Invitation to Smooth Infinitesimal Analysis (PDF file)
• Ieke Moerdijk and Reyes, G.E., Models for Smooth Infinitesimal Analysis, Springer-Verlag, 1991.
External links
• Michael O'Connor, An Introduction to Smooth Infinitesimal Analysis
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Wikipedia
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Smooth maximum
In mathematics, a smooth maximum of an indexed family x1, ..., xn of numbers is a smooth approximation to the maximum function $\max(x_{1},\ldots ,x_{n}),$ meaning a parametric family of functions $m_{\alpha }(x_{1},\ldots ,x_{n})$ such that for every α, the function $m_{\alpha }$ is smooth, and the family converges to the maximum function $m_{\alpha }\to \max $ as $\alpha \to \infty $. The concept of smooth minimum is similarly defined. In many cases, a single family approximates both: maximum as the parameter goes to positive infinity, minimum as the parameter goes to negative infinity; in symbols, $m_{\alpha }\to \max $ as $\alpha \to \infty $ and $m_{\alpha }\to \min $ as $\alpha \to -\infty $. The term can also be used loosely for a specific smooth function that behaves similarly to a maximum, without necessarily being part of a parametrized family.
Examples
Boltzmann operator
For large positive values of the parameter $\alpha >0$, the following formulation is a smooth, differentiable approximation of the maximum function. For negative values of the parameter that are large in absolute value, it approximates the minimum.
${\mathcal {S}}_{\alpha }(x_{1},\ldots ,x_{n})={\frac {\sum _{i=1}^{n}x_{i}e^{\alpha x_{i}}}{\sum _{i=1}^{n}e^{\alpha x_{i}}}}$
${\mathcal {S}}_{\alpha }$ has the following properties:
1. ${\mathcal {S}}_{\alpha }\to \max $ as $\alpha \to \infty $
2. ${\mathcal {S}}_{0}$ is the arithmetic mean of its inputs
3. ${\mathcal {S}}_{\alpha }\to \min $ as $\alpha \to -\infty $
The gradient of ${\mathcal {S}}_{\alpha }$ is closely related to softmax and is given by
$\nabla _{x_{i}}{\mathcal {S}}_{\alpha }(x_{1},\ldots ,x_{n})={\frac {e^{\alpha x_{i}}}{\sum _{j=1}^{n}e^{\alpha x_{j}}}}[1+\alpha (x_{i}-{\mathcal {S}}_{\alpha }(x_{1},\ldots ,x_{n}))].$
This makes the softmax function useful for optimization techniques that use gradient descent.
This operator is sometimes called the Boltzmann operator,[1] after the Boltzmann distribution.
LogSumExp
Main article: LogSumExp
Another smooth maximum is LogSumExp:
$\mathrm {LSE} _{\alpha }(x_{1},\ldots ,x_{n})={\frac {1}{\alpha }}\log \sum _{i=1}^{n}\exp \alpha x_{i}$
This can also be normalized if the $x_{i}$ are all non-negative, yielding a function with domain $[0,\infty )^{n}$ and range $[0,\infty )$:
$g(x_{1},\ldots ,x_{n})=\log \left(\sum _{i=1}^{n}\exp x_{i}-(n-1)\right)$
The $(n-1)$ term corrects for the fact that $\exp(0)=1$ by canceling out all but one zero exponential, and $\log 1=0$ if all $x_{i}$ are zero.
Mellowmax
The mellowmax operator[1] is defined as follows:
$\mathrm {mm} _{\alpha }(x)={\frac {1}{\alpha }}\log {\frac {1}{n}}\sum _{i=1}^{n}\exp \alpha x_{i}$
It is a non-expansive operator. As $\alpha \to \infty $, it acts like a maximum. As $\alpha \to 0$, it acts like an arithmetic mean. As $\alpha \to -\infty $, it acts like a minimum. This operator can be viewed as a particular instantiation of the quasi-arithmetic mean. It can also be derived from information theoretical principles as a way of regularizing policies with a cost function defined by KL divergence. The operator has previously been utilized in other areas, such as power engineering.[2]
p-Norm
Main article: P-norm
Another smooth maximum is the p-norm:
$\|(x_{1},\ldots ,x_{n})\|_{p}=\left(\sum _{i=1}^{n}|x_{i}|^{p}\right)^{\frac {1}{p}}$
which converges to $\|(x_{1},\ldots ,x_{n})\|_{\infty }=\max _{1\leq i\leq n}|x_{i}|$ as $p\to \infty $.
An advantage of the p-norm is that it is a norm. As such it is scale invariant (homogeneous): $\|(\lambda x_{1},\ldots ,\lambda x_{n})\|_{p}=|\lambda |\cdot \|(x_{1},\ldots ,x_{n})\|_{p}$, and it satisfies the triangle inequality.
Smooth maximum unit
The following binary operator is called the Smooth Maximum Unit (SMU):[3]
${\begin{aligned}\textstyle \max _{\varepsilon }(a,b)&={\frac {a+b+|a-b|_{\varepsilon }}{2}}\\&={\frac {a+b+{\sqrt {(a-b)^{2}+\varepsilon }}}{2}}\end{aligned}}$
where $\varepsilon \geq 0$ is a parameter. As $\varepsilon \to 0$, $|\cdot |_{\varepsilon }\to |\cdot |$ and thus $\textstyle \max _{\varepsilon }\to \max $.
See also
• LogSumExp
• Softmax function
• Generalized mean
References
1. Asadi, Kavosh; Littman, Michael L. (2017). "An Alternative Softmax Operator for Reinforcement Learning". PMLR. 70: 243–252. arXiv:1612.05628. Retrieved January 6, 2023.
2. Safak, Aysel (February 1993). "Statistical analysis of the power sum of multiple correlated log-normal components". IEEE Transactions on Vehicular Technology. 42 (1): {58–61. doi:10.1109/25.192387. Retrieved January 6, 2023.
3. Biswas, Koushik; Kumar, Sandeep; Banerjee, Shilpak; Ashish Kumar Pandey (2021). "SMU: Smooth activation function for deep networks using smoothing maximum technique". arXiv:2111.04682 [cs.LG].
https://www.johndcook.com/soft_maximum.pdf
M. Lange, D. Zühlke, O. Holz, and T. Villmann, "Applications of lp-norms and their smooth approximations for gradient based learning vector quantization," in Proc. ESANN, Apr. 2014, pp. 271-276. (https://www.elen.ucl.ac.be/Proceedings/esann/esannpdf/es2014-153.pdf)
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Wikipedia
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Smooth number
In number theory, an n-smooth (or n-friable) number is an integer whose prime factors are all less than or equal to n.[1][2] For example, a 7-smooth number is a number whose every prime factor is at most 7, so 49 = 72 and 15750 = 2 × 32 × 53 × 7 are both 7-smooth, while 11 and 702 = 2 × 33 × 13 are not 7-smooth. The term seems to have been coined by Leonard Adleman.[3] Smooth numbers are especially important in cryptography, which relies on factorization of integers. The 2-smooth numbers are just the powers of 2, while 5-smooth numbers are known as regular numbers.
Definition
A positive integer is called B-smooth if none of its prime factors are greater than B. For example, 1,620 has prime factorization 22 × 34 × 5; therefore 1,620 is 5-smooth because none of its prime factors are greater than 5. This definition includes numbers that lack some of the smaller prime factors; for example, both 10 and 12 are 5-smooth, even though they miss out the prime factors 3 and 5, respectively. All 5-smooth numbers are of the form 2a × 3b × 5c, where a, b and c are non-negative integers.
The 3-smooth numbers have also been called "harmonic numbers", although that name has other more widely used meanings.[4] 5-smooth numbers are also called regular numbers or Hamming numbers;[5] 7-smooth numbers are also called humble numbers,[6] and sometimes called highly composite,[7] although this conflicts with another meaning of highly composite numbers.
Here, note that B itself is not required to appear among the factors of a B-smooth number. If the largest prime factor of a number is p then the number is B-smooth for any B ≥ p. In many scenarios B is prime, but composite numbers are permitted as well. A number is B-smooth if and only if it is p-smooth, where p is the largest prime less than or equal to B.
Applications
An important practical application of smooth numbers is the fast Fourier transform (FFT) algorithms (such as the Cooley–Tukey FFT algorithm), which operates by recursively breaking down a problem of a given size n into problems the size of its factors. By using B-smooth numbers, one ensures that the base cases of this recursion are small primes, for which efficient algorithms exist. (Large prime sizes require less-efficient algorithms such as Bluestein's FFT algorithm.)
5-smooth or regular numbers play a special role in Babylonian mathematics.[8] They are also important in music theory (see Limit (music)),[9] and the problem of generating these numbers efficiently has been used as a test problem for functional programming.[10]
Smooth numbers have a number of applications to cryptography.[11] While most applications center around cryptanalysis (e.g. the fastest known integer factorization algorithms, for example: General number field sieve algorithm), the VSH hash function is another example of a constructive use of smoothness to obtain a provably secure design.
Distribution
Let $\Psi (x,y)$ denote the number of y-smooth integers less than or equal to x (the de Bruijn function).
If the smoothness bound B is fixed and small, there is a good estimate for $\Psi (x,B)$:
$\Psi (x,B)\sim {\frac {1}{\pi (B)!}}\prod _{p\leq B}{\frac {\log x}{\log p}}.$
where $\pi (B)$ denotes the number of primes less than or equal to $B$.
Otherwise, define the parameter u as u = log x / log y: that is, x = yu. Then,
$\Psi (x,y)=x\cdot \rho (u)+O\left({\frac {x}{\log y}}\right)$
where $\rho (u)$ is the Dickman function.
The average size of the smooth part of a number of given size is known as $\zeta (u)$, and it is known to decay much more slowly than $\rho (u)$.[12]
For any k, almost all natural numbers will not be k-smooth.
Powersmooth numbers
Further, m is called B-powersmooth (or B-ultrafriable) if all prime powers $p^{\nu }$ dividing m satisfy:
$p^{\nu }\leq B.\,$
For example, 720 (24 × 32 × 51) is 5-smooth but not 5-powersmooth (because there are several prime powers greater than 5, e.g. $3^{2}=9\nleq 5$ and $2^{4}=16\nleq 5$). It is 16-powersmooth since its greatest prime factor power is 24 = 16. The number is also 17-powersmooth, 18-powersmooth, etc.
B-smooth and B-powersmooth numbers have applications in number theory, such as in Pollard's p − 1 algorithm and ECM. Such applications are often said to work with "smooth numbers," with no B specified; this means the numbers involved must be B-powersmooth, for some unspecified small number B. As B increases, the performance of the algorithm or method in question degrades rapidly. For example, the Pohlig–Hellman algorithm for computing discrete logarithms has a running time of O(B1/2)—for groups of B-smooth order.
Smooth over a set A
Moreover, m is said to be smooth over a set A if there exists a factorization of m where the factors are powers of elements in A. For example, since 12 = 4 × 3, 12 is smooth over the sets A1 = {4, 3}, A2 = {2, 3}, and $\mathbb {Z} $, however it would not be smooth over the set A3 = {3, 5}, as 12 contains the factor 4 = 22, which is not in A3.
Note the set A does not have to be a set of prime factors, but it is typically a proper subset of the primes as seen in the factor base of Dixon's factorization method and the quadratic sieve. Likewise, it is what the general number field sieve uses to build its notion of smoothness, under the homomorphism $\phi :\mathbb {Z} [\theta ]\to \mathbb {Z} /n\mathbb {Z} $ :\mathbb {Z} [\theta ]\to \mathbb {Z} /n\mathbb {Z} } .[13]
See also
• Highly composite number
• Rough number
• Round number
• Størmer's theorem
• Unusual number
Notes and references
1. "P-Smooth Numbers or P-friable Number". GeeksforGeeks. 2018-02-12. Retrieved 2019-12-12.
2. Weisstein, Eric W. "Smooth Number". mathworld.wolfram.com. Retrieved 2019-12-12.
3. Hellman, M. E.; Reyneri, J. M. (1983). "Fast Computation of Discrete Logarithms in GF (q)". Advances in Cryptology – Proceedings of Crypto 82. pp. 3–13. doi:10.1007/978-1-4757-0602-4_1. ISBN 978-1-4757-0604-8.
4. Sloane, N. J. A. (ed.). "Sequence A003586 (3-smooth numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
5. "Python: Get the Hamming numbers upto a given numbers also check whether a given number is an Hamming number". w3resource. Retrieved 2019-12-12.
6. "Problem H: Humble Numbers". www.eecs.qmul.ac.uk. Retrieved 2019-12-12.
7. Sloane, N. J. A. (ed.). "Sequence A002473 (7-smooth numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
8. Aaboe, Asger (1965), "Some Seleucid mathematical tables (extended reciprocals and squares of regular numbers)", Journal of Cuneiform Studies, 19 (3): 79–86, doi:10.2307/1359089, JSTOR 1359089, MR 0191779, S2CID 164195082.
9. Longuet-Higgins, H. C. (1962), "Letter to a musical friend", Music Review (August): 244–248.
10. Dijkstra, Edsger W. (1981), Hamming's exercise in SASL (PDF), Report EWD792. Originally a privately circulated handwritten note.
11. Naccache, David; Shparlinski, Igor (17 October 2008). "Divisibility, Smoothness and Cryptographic Applications" (PDF). eprint.iacr.org. arXiv:0810.2067. Retrieved 26 July 2017.f
12. Tanaka, Keisuke; Suga, Yuji (20 August 2015). Advances in Information and Computer Security: 10th International Workshop on Security, IWSEC 2015, Nara, Japan, August 26-28, 2015, Proceedings. Springer. pp. 49–51. ISBN 9783319224251.
13. Briggs, Matthew E. (17 April 1998). "An Introduction to the General Number Field Sieve" (PDF). math.vt.edu. Blacksburg, Virginia: Virginia Polytechnic Institute and State University. Retrieved 26 July 2017.
Bibliography
• G. Tenenbaum, Introduction to analytic and probabilistic number theory, (AMS, 2015) ISBN 978-0821898543
• A. Granville, Smooth numbers: Computational number theory and beyond, Proc. of MSRI workshop, 2008
External links
• Weisstein, Eric W. "Smooth Number". MathWorld.
The On-Line Encyclopedia of Integer Sequences (OEIS) lists B-smooth numbers for small Bs:
• 2-smooth numbers: A000079 (2i)
• 3-smooth numbers: A003586 (2i3j)
• 5-smooth numbers: A051037 (2i3j5k)
• 7-smooth numbers: A002473 (2i3j5k7l)
• 11-smooth numbers: A051038 (etc...)
• 13-smooth numbers: A080197
• 17-smooth numbers: A080681
• 19-smooth numbers: A080682
• 23-smooth numbers: A080683
Divisibility-based sets of integers
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Wikipedia
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Smooth scheme
In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smooth variety over a field. Smooth schemes play the role in algebraic geometry of manifolds in topology.
Definition
First, let X be an affine scheme of finite type over a field k. Equivalently, X has a closed immersion into affine space An over k for some natural number n. Then X is the closed subscheme defined by some equations g1 = 0, ..., gr = 0, where each gi is in the polynomial ring k[x1,..., xn]. The affine scheme X is smooth of dimension m over k if X has dimension at least m in a neighborhood of each point, and the matrix of derivatives (∂gi/∂xj) has rank at least n−m everywhere on X.[1] (It follows that X has dimension equal to m in a neighborhood of each point.) Smoothness is independent of the choice of immersion of X into affine space.
The condition on the matrix of derivatives is understood to mean that the closed subset of X where all (n−m) × (n − m) minors of the matrix of derivatives are zero is the empty set. Equivalently, the ideal in the polynomial ring generated by all gi and all those minors is the whole polynomial ring.
In geometric terms, the matrix of derivatives (∂gi/∂xj) at a point p in X gives a linear map Fn → Fr, where F is the residue field of p. The kernel of this map is called the Zariski tangent space of X at p. Smoothness of X means that the dimension of the Zariski tangent space is equal to the dimension of X near each point; at a singular point, the Zariski tangent space would be bigger.
More generally, a scheme X over a field k is smooth over k if each point of X has an open neighborhood which is a smooth affine scheme of some dimension over k. In particular, a smooth scheme over k is locally of finite type.
There is a more general notion of a smooth morphism of schemes, which is roughly a morphism with smooth fibers. In particular, a scheme X is smooth over a field k if and only if the morphism X → Spec k is smooth.
Properties
A smooth scheme over a field is regular and hence normal. In particular, a smooth scheme over a field is reduced.
Define a variety over a field k to be an integral separated scheme of finite type over k. Then any smooth separated scheme of finite type over k is a finite disjoint union of smooth varieties over k.
For a smooth variety X over the complex numbers, the space X(C) of complex points of X is a complex manifold, using the classical (Euclidean) topology. Likewise, for a smooth variety X over the real numbers, the space X(R) of real points is a real manifold, possibly empty.
For any scheme X that is locally of finite type over a field k, there is a coherent sheaf Ω1 of differentials on X. The scheme X is smooth over k if and only if Ω1 is a vector bundle of rank equal to the dimension of X near each point.[2] In that case, Ω1 is called the cotangent bundle of X. The tangent bundle of a smooth scheme over k can be defined as the dual bundle, TX = (Ω1)*.
Smoothness is a geometric property, meaning that for any field extension E of k, a scheme X is smooth over k if and only if the scheme XE := X ×Spec k Spec E is smooth over E. For a perfect field k, a scheme X is smooth over k if and only if X is locally of finite type over k and X is regular.
Generic smoothness
A scheme X is said to be generically smooth of dimension n over k if X contains an open dense subset that is smooth of dimension n over k. Every variety over a perfect field (in particular an algebraically closed field) is generically smooth.[3]
Examples
• Affine space and projective space are smooth schemes over a field k.
• An example of a smooth hypersurface in projective space Pn over k is the Fermat hypersurface x0d + ... + xnd = 0, for any positive integer d that is invertible in k.
• An example of a singular (non-smooth) scheme over a field k is the closed subscheme x2 = 0 in the affine line A1 over k.
• An example of a singular (non-smooth) variety over k is the cuspidal cubic curve x2 = y3 in the affine plane A2, which is smooth outside the origin (x,y) = (0,0).
• A 0-dimensional variety X over a field k is of the form X = Spec E, where E is a finite extension field of k. The variety X is smooth over k if and only if E is a separable extension of k. Thus, if E is not separable over k, then X is a regular scheme but is not smooth over k. For example, let k be the field of rational functions Fp(t) for a prime number p, and let E = Fp(t1/p); then Spec E is a variety of dimension 0 over k which is a regular scheme, but not smooth over k.
• Schubert varieties are in general not smooth.
Notes
1. The definition of smoothness used in this article is equivalent to Grothendieck's definition of smoothness by Theorems 30.2 and Theorem 30.3 in: Matsumura, Commutative Ring Theory (1989).
2. Theorem 30.3, Matsumura, Commutative Ring Theory (1989).
3. Lemma 1 in section 28 and Corollary to Theorem 30.5, Matsumura, Commutative Ring Theory (1989).
References
• D. Gaitsgory's notes on flatness and smoothness at http://www.math.harvard.edu/~gaitsgde/Schemes_2009/BR/SmoothMaps.pdf
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
• Matsumura, Hideyuki (1989), Commutative Ring Theory, Cambridge Studies in Advanced Mathematics (2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6, MR 1011461
See also
• Étale morphism
• Dimension of an algebraic variety
• Glossary of scheme theory
• Smooth completion
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Wikipedia
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Smooth projective plane
In geometry, smooth projective planes are special projective planes. The most prominent example of a smooth projective plane is the real projective plane ${\mathcal {E}}$. Its geometric operations of joining two distinct points by a line and of intersecting two lines in a point are not only continuous but even smooth (infinitely differentiable $=C^{\infty }$). Similarly, the classical planes over the complex numbers, the quaternions, and the octonions are smooth planes. However, these are not the only such planes.
Definition and basic properties
A smooth projective plane ${\mathcal {P}}=(P,{\mathfrak {L}})$ consists of a point space $P$ and a line space ${\mathfrak {L}}$ that are smooth manifolds and where both geometric operations of joining and intersecting are smooth.
The geometric operations of smooth planes are continuous; hence, each smooth plane is a compact topological plane.[1] Smooth planes exist only with point spaces of dimension 2m where $1\leq m\leq 4$, because this is true for compact connected projective topological planes.[2][3] These four cases will be treated separately below.
Theorem. The point manifold of a smooth projective plane is homeomorphic to its classical counterpart, and so is the line manifold.[4]
Automorphisms
Automorphisms play a crucial role in the study of smooth planes. A bijection of the point set of a projective plane is called a collineation, if it maps lines onto lines. The continuous collineations of a compact projective plane ${\mathcal {P}}$ form the group $\operatorname {Aut} {\mathcal {P}}$. This group is taken with the topology of uniform convergence. We have:[5]
Theorem. If ${\mathcal {P}}=(P,{\mathfrak {L}})$ is a smooth plane, then each continuous collineation of ${\mathcal {P}}$ is smooth; in other words, the group of automorphisms of a smooth plane ${\mathcal {P}}$ coincides with $\operatorname {Aut} {\mathcal {P}}$. Moreover, $\operatorname {Aut} {\mathcal {P}}$ is a smooth Lie transformation group of $P$ and of ${\mathfrak {L}}$.
The automorphism groups of the four classical planes are simple Lie groups of dimension 8, 16, 35, or 78, respectively. All other smooth planes have much smaller groups. See below.
Translation planes
A projective plane is called a translation plane if its automorphism group has a subgroup that fixes each point on some line $W$ and acts sharply transitively on the set of points not on $W$.
Theorem. Every smooth projective translation plane ${\mathcal {P}}$ is isomorphic to one of the four classical planes.[6]
This shows that there are many compact connected topological projective planes that are not smooth. On the other hand, the following construction yields real analytic non-Desarguesian planes of dimension 2, 4, and 8, with a compact group of automorphisms of dimension 1, 4, and 13, respectively:[7] represent points and lines in the usual way by homogeneous coordinates over the real or complex numbers or the quaternions, say, by vectors of length $1$. Then the incidence of the point $(x,y,z)$ and the line $(a,b,c)$ is defined by $ax+by+cz=t|c|^{2}|z|^{2}cz$, where $t$ is a fixed real parameter such that $|t|<1/9$. These planes are self-dual.
2-dimensional planes
Compact 2-dimensional projective planes can be described in the following way: the point space is a compact surface $S$, each line is a Jordan curve in $S$ (a closed subset homeomorphic to the circle), and any two distinct points are joined by a unique line. Then $S$ is homeomorphic to the point space of the real plane ${\mathcal {E}}$, any two distinct lines intersect in a unique point, and the geometric operations are continuous (apply Salzmann et al. 1995, §31 to the complement of a line). A familiar family of examples was given by Moulton in 1902.[8][9] These planes are characterized by the fact that they have a 4-dimensional automorphism group. They are not isomorphic to a smooth plane.[10] More generally, all non-classical compact 2-dimensional planes ${\mathcal {P}}$ such that $\dim \operatorname {Aut} {\mathcal {P}}\geq 3$ are known explicitly; none of these is smooth:
Theorem. If ${\mathcal {P}}$ is a smooth 2-dimensional plane and if $\dim \operatorname {Aut} {\mathcal {P}}\geq 3$, then ${\mathcal {P}}$ is the classical real plane ${\mathcal {E}}$.[11]
4-dimensional planes
All compact planes ${\mathcal {P}}$ with a 4-dimensional point space and $\operatorname {Aut} {\mathcal {P}}\geq 7$ have been classified.[12] Up to duality, they are either translation planes or they are isomorphic to a unique so-called shift plane.[13] According to Bödi (1996, Chap. 10), this shift plane is not smooth. Hence, the result on translation planes implies:
Theorem. A smooth 4-dimensional plane is isomorphic to the classical complex plane, or $\dim \operatorname {Aut} {\mathcal {P}}\leq 6$.[14]
8-dimensional planes
Compact 8-dimensional topological planes ${\mathcal {P}}$ have been discussed in Salzmann et al. (1995, Chapter 8) and, more recently, in Salzmann (2014). Put $\Sigma =\operatorname {Aut} {\mathcal {P}}$. Either ${\mathcal {P}}$ is the classical quaternion plane or $\dim \Sigma \leq 18$. If $\dim \Sigma \geq 17$, then ${\mathcal {P}}$ is a translation plane, or a dual translation plane, or a Hughes plane.[15] The latter can be characterized as follows: $\Sigma $ leaves some classical complex subplane ${\mathcal {C}}$ invariant and induces on ${\mathcal {C}}$ the connected component of its full automorphism group.[16][17] The Hughes planes are not smooth.[18][19] This yields a result similar to the case of 4-dimensional planes:
Theorem. If ${\mathcal {P}}$ is a smooth 8-dimensional plane, then ${\mathcal {P}}$ is the classical quaternion plane or $\dim \Sigma \leq 16$.
16-dimensional planes
Let $\Sigma $ denote the automorphism group of a compact 16-dimensional topological projective plane ${\mathcal {P}}$. Either ${\mathcal {P}}$ is the smooth classical octonion plane or $\dim \Sigma \leq 40$. If $\dim \Sigma =40$, then $\Sigma $ fixes a line $W$ and a point $v\in W$, and the affine plane ${\mathcal {P}}\smallsetminus W$ and its dual are translation planes.[20] If $\dim \Sigma =39$, then $\Sigma $ also fixes an incident point-line pair, but neither ${\mathcal {P}}$ nor $\Sigma $ are known explicitly. Nevertheless, none of these planes can be smooth:[21][22][23]
Theorem. If ${\mathcal {P}}$ is a 16-dimensional smooth projective plane, then ${\mathcal {P}}$ is the classical octonion plane or $\dim \Sigma \leq 38$.
Main theorem
The last four results combine to give the following theorem:
If $c_{m}$ is the largest value of $\dim \operatorname {Aut} {\mathcal {P}}$, where ${\mathcal {P}}$ is a non-classical compact 2m-dimensional topological projective plane, then $\dim \operatorname {Aut} {\mathcal {P}}\leq c_{m}-2$ whenever ${\mathcal {P}}$ is even smooth.
Complex analytic planes
The condition, that the geometric operations of a projective plane are complex analytic, is very restrictive. In fact, it is satisfied only in the classical complex plane.[24][25]
Theorem. Every complex analytic projective plane is isomorphic as an analytic plane to the complex plane with its standard analytic structure.
Notes
1. Salzmann et al. 1995, 42.4
2. Löwen, R. (1983), "Topology and dimension of stable planes: On a conjecture of H. Freudenthal", J. Reine Angew. Math., 343: 108–122
3. Salzmann et al. 1995, 54.11
4. Kramer, L. (1994), "The topology of smooth projective planes", Arch. Math., 63: 85–91, doi:10.1007/bf01196303, S2CID 15480568
5. Bödi, R. (1998), "Collineations of smooth stable planes", Forum Math., 10 (6): 751–773, doi:10.1515/form.10.6.751, hdl:11475/3260, S2CID 54504153
6. Otte, J. (1995), "Smooth Projective Translation Planes", Geom. Dedicata, 58 (2): 203–212, doi:10.1007/bf01265639, S2CID 120238728
7. Immervoll, S. (2003), "Real analytic projective planes with large automorphism groups", Adv. Geom., 3 (2): 163–176, doi:10.1515/advg.2003.011
8. Moulton, F. R. (1902), "A simple non-desarguesian plane geometry", Trans. Amer. Math. Soc., 3 (2): 192–195, doi:10.1090/s0002-9947-1902-1500595-3
9. Salzmann et al. 1995, §34
10. Betten, D. (1971), "2-dimensionale differenzierbare projektive Ebenen", Arch. Math., 22: 304–309, doi:10.1007/bf01222580, S2CID 119885473
11. Bödi 1996, (9.1)
12. Salzmann et al. 1995, 74.27
13. Salzmann et al. 1995, §74
14. Bödi 1996, (10.11)
15. Salzmann 2014, 1.10
16. Salzmann et al. 1995, §86
17. Salzmann, H. (2003), "Baer subplanes", Illinois J. Math., 47 (1–2): 485–513, doi:10.1215/ijm/1258488168 3.19
18. Bödi, R. (1999), "Smooth Hughes planes are classical", Arch. Math., 73: 73–80, doi:10.1007/s000130050022, hdl:11475/3229, S2CID 120222293
19. Salzmann 2014, 9.17
20. Salzmann et al. 1995, 87.7
21. Bödi 1996, Chap. 12
22. Bödi, R. (1998), "16-dimensional smooth projective planes with large collineation groups", Geom. Dedicata, 72 (3): 283–298, doi:10.1023/A:1005020223604, hdl:11475/3238, S2CID 56094550
23. Salzmann 2014, 9.18 for a sketch of the proof
24. Breitsprecher, S. (1967), "Einzigkeit der reellen und der komplexen projektiven Ebene", Math. Z., 99 (5): 429–432, doi:10.1007/bf01111021, S2CID 120984088
25. Salzmann et al. 1995, 75.1
References
• Bödi, R. (1996), "Smooth stable and projective planes", Thesis, Tübingen
• Salzmann, H.; Betten, D.; Grundhöfer, T.; Hähl, H.; Löwen, R.; Stroppel, M. (1995), Compact Projective Planes, W. de Gruyter
• Salzmann, H. (2014), Compact planes, mostly 8-dimensional. A retrospect, arXiv:1402.0304, Bibcode:2014arXiv1402.0304S
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Wikipedia
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Admissible representation
In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups. They were introduced by Harish-Chandra.
Real or complex reductive Lie groups
Let G be a connected reductive (real or complex) Lie group. Let K be a maximal compact subgroup. A continuous representation (π, V) of G on a complex Hilbert space V[1] is called admissible if π restricted to K is unitary and each irreducible unitary representation of K occurs in it with finite multiplicity. The prototypical example is that of an irreducible unitary representation of G.
An admissible representation π induces a $({\mathfrak {g}},K)$-module which is easier to deal with as it is an algebraic object. Two admissible representations are said to be infinitesimally equivalent if their associated $({\mathfrak {g}},K)$-modules are isomorphic. Though for general admissible representations, this notion is different than the usual equivalence, it is an important result that the two notions of equivalence agree for unitary (admissible) representations. Additionally, there is a notion of unitarity of $({\mathfrak {g}},K)$-modules. This reduces the study of the equivalence classes of irreducible unitary representations of G to the study of infinitesimal equivalence classes of admissible representations and the determination of which of these classes are infinitesimally unitary. The problem of parameterizing the infinitesimal equivalence classes of admissible representations was fully solved by Robert Langlands and is called the Langlands classification.
Totally disconnected groups
Let G be a locally compact totally disconnected group (such as a reductive algebraic group over a nonarchimedean local field or over the finite adeles of a global field). A representation (π, V) of G on a complex vector space V is called smooth if the subgroup of G fixing any vector of V is open. If, in addition, the space of vectors fixed by any compact open subgroup is finite dimensional then π is called admissible. Admissible representations of p-adic groups admit more algebraic description through the action of the Hecke algebra of locally constant functions on G.
Deep studies of admissible representations of p-adic reductive groups were undertaken by Casselman and by Bernstein and Zelevinsky in the 1970s. Progress was made more recently by Howe, Moy, Gopal Prasad and Bushnell and Kutzko, who developed a theory of types and classified the admissible dual (i.e. the set of equivalence classes of irreducible admissible representations) in many cases.
Notes
1. I.e. a homomorphism π : G → GL(V) (where GL(V) is the group of bounded linear operators on V whose inverse is also bounded and linear) such that the associated map G × V → V is continuous.
References
• Bushnell, Colin J.; Henniart, Guy (2006), The local Langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 335, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-31511-X, ISBN 978-3-540-31486-8, MR 2234120
• Bushnell, Colin J.; Philip C. Kutzko (1993). The admissible dual of GL(N) via compact open subgroups. Annals of Mathematics Studies 129. Princeton University Press. ISBN 0-691-02114-7.
• Chapter VIII of Knapp, Anthony W. (2001). Representation Theory of Semisimple Groups: An Overview Based on Examples. Princeton University Press. ISBN 0-691-09089-0.
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Wikipedia
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Morphism of algebraic stacks
In algebraic geometry, given algebraic stacks $p:X\to C,\,q:Y\to C$ over a base category C, a morphism $f:X\to Y$ of algebraic stacks is a functor such that $q\circ f=p$.
More generally, one can also consider a morphism between prestacks; (a stackification would be an example.)
Types
One particular important example is a presentation of a stack, which is widely used in the study of stacks.
An algebraic stack X is said to be smooth of dimension n - j if there is a smooth presentation $U\to X$ of relative dimension j for some smooth scheme U of dimension n. For example, if $\operatorname {Vect} _{n}$ denotes the moduli stack of rank-n vector bundles, then there is a presentation $\operatorname {Spec} (k)\to \operatorname {Vect} _{n}$ given by the trivial bundle $\mathbb {A} _{k}^{n}$ over $\operatorname {Spec} (k)$.
A quasi-affine morphism between algebraic stacks is a morphism that factorizes as a quasi-compact open immersion followed by an affine morphism.[1]
Notes
1. § 8.6 of F. Meyer, Notes on algebraic stacks
References
• Stacks Project, Ch, 83, Morphisms of algebraic stacks
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Wikipedia
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Smooth structure
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold.[1]
Definition
A smooth structure on a manifold $M$ is a collection of smoothly equivalent smooth atlases. Here, a smooth atlas for a topological manifold $M$ is an atlas for $M$ such that each transition function is a smooth map, and two smooth atlases for $M$ are smoothly equivalent provided their union is again a smooth atlas for $M.$ This gives a natural equivalence relation on the set of smooth atlases.
A smooth manifold is a topological manifold $M$ together with a smooth structure on $M.$
Maximal smooth atlases
By taking the union of all atlases belonging to a smooth structure, we obtain a maximal smooth atlas. This atlas contains every chart that is compatible with the smooth structure. There is a natural one-to-one correspondence between smooth structures and maximal smooth atlases. Thus, we may regard a smooth structure as a maximal smooth atlas and vice versa.
In general, computations with the maximal atlas of a manifold are rather unwieldy. For most applications, it suffices to choose a smaller atlas. For example, if the manifold is compact, then one can find an atlas with only finitely many charts.
Equivalence of smooth structures
Let $\mu $ and $\nu $ be two maximal atlases on $M.$ The two smooth structures associated to $\mu $ and $\nu $ are said to be equivalent if there is a diffeomorphism $f:M\to M$ such that $\mu \circ f=\nu .$
Exotic spheres
John Milnor showed in 1956 that the 7-dimensional sphere admits a smooth structure that is not equivalent to the standard smooth structure. A sphere equipped with a nonstandard smooth structure is called an exotic sphere.
E8 manifold
The E8 manifold is an example of a topological manifold that does not admit a smooth structure. This essentially demonstrates that Rokhlin's theorem holds only for smooth structures, and not topological manifolds in general.
Related structures
The smoothness requirements on the transition functions can be weakened, so that we only require the transition maps to be $k$-times continuously differentiable; or strengthened, so that we require the transition maps to be real-analytic. Accordingly, this gives a $C^{k}$ or (real-)analytic structure on the manifold rather than a smooth one. Similarly, we can define a complex structure by requiring the transition maps to be holomorphic.
See also
• Smooth frame – Generalization of an ordered basis of a vector spacePages displaying short descriptions of redirect targets
• Atlas (topology) – Set of charts that describes a manifold
References
1. Callahan, James J. (1974). "Singularities and plane maps". Amer. Math. Monthly. 81: 211–240. doi:10.2307/2319521.
• Hirsch, Morris (1976). Differential Topology. Springer-Verlag. ISBN 3-540-90148-5.
• Lee, John M. (2006). Introduction to Smooth Manifolds. Springer-Verlag. ISBN 978-0-387-95448-6.
• Sepanski, Mark R. (2007). Compact Lie Groups. Springer-Verlag. ISBN 978-0-387-30263-8.
Manifolds (Glossary)
Basic concepts
• Topological manifold
• Atlas
• Differentiable/Smooth manifold
• Differential structure
• Smooth atlas
• Submanifold
• Riemannian manifold
• Smooth map
• Submersion
• Pushforward
• Tangent space
• Differential form
• Vector field
Main results (list)
• Atiyah–Singer index
• Darboux's
• De Rham's
• Frobenius
• Generalized Stokes
• Hopf–Rinow
• Noether's
• Sard's
• Whitney embedding
Maps
• Curve
• Diffeomorphism
• Local
• Geodesic
• Exponential map
• in Lie theory
• Foliation
• Immersion
• Integral curve
• Lie derivative
• Section
• Submersion
Types of
manifolds
• Closed
• (Almost) Complex
• (Almost) Contact
• Fibered
• Finsler
• Flat
• G-structure
• Hadamard
• Hermitian
• Hyperbolic
• Kähler
• Kenmotsu
• Lie group
• Lie algebra
• Manifold with boundary
• Oriented
• Parallelizable
• Poisson
• Prime
• Quaternionic
• Hypercomplex
• (Pseudo−, Sub−) Riemannian
• Rizza
• (Almost) Symplectic
• Tame
Tensors
Vectors
• Distribution
• Lie bracket
• Pushforward
• Tangent space
• bundle
• Torsion
• Vector field
• Vector flow
Covectors
• Closed/Exact
• Covariant derivative
• Cotangent space
• bundle
• De Rham cohomology
• Differential form
• Vector-valued
• Exterior derivative
• Interior product
• Pullback
• Ricci curvature
• flow
• Riemann curvature tensor
• Tensor field
• density
• Volume form
• Wedge product
Bundles
• Adjoint
• Affine
• Associated
• Cotangent
• Dual
• Fiber
• (Co) Fibration
• Jet
• Lie algebra
• (Stable) Normal
• Principal
• Spinor
• Subbundle
• Tangent
• Tensor
• Vector
Connections
• Affine
• Cartan
• Ehresmann
• Form
• Generalized
• Koszul
• Levi-Civita
• Principal
• Vector
• Parallel transport
Related
• Classification of manifolds
• Gauge theory
• History
• Morse theory
• Moving frame
• Singularity theory
Generalizations
• Banach manifold
• Diffeology
• Diffiety
• Fréchet manifold
• K-theory
• Orbifold
• Secondary calculus
• over commutative algebras
• Sheaf
• Stratifold
• Supermanifold
• Stratified space
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Wikipedia
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Smooth topology
In algebraic geometry, the smooth topology is a certain Grothendieck topology, which is finer than étale topology. Its main use is to define the cohomology of an algebraic stack with coefficients in, say, the étale sheaf $\mathbb {Q} _{l}$.
To understand the problem that motivates the notion, consider the classifying stack $B\mathbb {G} _{m}$ over $\operatorname {Spec} \mathbf {F} _{q}$. Then $B\mathbb {G} _{m}=\operatorname {Spec} \mathbf {F} _{q}$ in the étale topology;[1] i.e., just a point. However, we expect the "correct" cohomology ring of $B\mathbb {G} _{m}$ to be more like that of $\mathbb {C} P^{\infty }$ as the ring should classify line bundles. Thus, the cohomology of $B\mathbb {G} _{m}$ should be defined using smooth topology for formulae like Behrend's fixed point formula to hold.
Notes
1. Behrend 2003, Proposition 5.2.9; in particular, the proof.
References
• Behrend, K. (2003). "Derived l-adic categories for algebraic stacks" (PDF). Memoirs of the American Mathematical Society. 163.
• Laumon, Gérard; Moret-Bailly, Laurent (2000), Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 39, Berlin, New York: Springer-Verlag, ISBN 978-3-540-65761-3, MR 1771927 Unfortunately this book uses the incorrect assertion that morphisms of algebraic stacks induce morphisms of lisse-étale topoi. Some of these errors were fixed by Olsson (2007) harvtxt error: no target: CITEREFOlsson2007 (help).
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Wikipedia
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Smoothed analysis
In theoretical computer science, smoothed analysis is a way of measuring the complexity of an algorithm. Since its introduction in 2001, smoothed analysis has been used as a basis for considerable research, for problems ranging from mathematical programming, numerical analysis, machine learning, and data mining.[1] It can give a more realistic analysis of the practical performance (e.g., running time, success rate, approximation quality) of the algorithm compared to analysis that uses worst-case or average-case scenarios.
Smoothed analysis is a hybrid of worst-case and average-case analyses that inherits advantages of both. It measures the expected performance of algorithms under slight random perturbations of worst-case inputs. If the smoothed complexity of an algorithm is low, then it is unlikely that the algorithm will take a long time to solve practical instances whose data are subject to slight noises and imprecisions. Smoothed complexity results are strong probabilistic results, roughly stating that, in every large enough neighbourhood of the space of inputs, most inputs are easily solvable. Thus, a low smoothed complexity means that the hardness of inputs is a "brittle" property.
Although worst-case analysis has been widely successful in explaining the practical performance of many algorithms, this style of analysis gives misleading results for a number of problems. Worst-case complexity measures the time it takes to solve any input, although hard-to-solve inputs might never come up in practice. In such cases, the worst-case running time can be much worse than the observed running time in practice. For example, the worst-case complexity of solving a linear program using the simplex algorithm is exponential,[2] although the observed number of steps in practice is roughly linear.[3][4] The simplex algorithm is in fact much faster than the ellipsoid method in practice, although the latter has polynomial-time worst-case complexity.
Average-case analysis was first introduced to overcome the limitations of worst-case analysis. However, the resulting average-case complexity depends heavily on the probability distribution that is chosen over the input. The actual inputs and distribution of inputs may be different in practice from the assumptions made during the analysis: a random input may be very unlike a typical input. Because of this choice of data model, a theoretical average-case result might say little about practical performance of the algorithm.
Smoothed analysis generalizes both worst-case and average-case analysis and inherits strengths of both. It is intended to be much more general than average-case complexity, while still allowing low complexity bounds to be proven.
History
ACM and the European Association for Theoretical Computer Science awarded the 2008 Gödel Prize to Daniel Spielman and Shanghua Teng for developing smoothed analysis. The name Smoothed Analysis was coined by Alan Edelman.[1] In 2010 Spielman received the Nevanlinna Prize for developing smoothed analysis. Spielman and Teng's JACM paper "Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time" was also one of the three winners of the 2009 Fulkerson Prize sponsored jointly by the Mathematical Programming Society (MPS) and the American Mathematical Society (AMS).
Examples
Simplex algorithm for linear programming
The simplex algorithm is a very efficient algorithm in practice, and it is one of the dominant algorithms for linear programming in practice. On practical problems, the number of steps taken by the algorithm is linear in the number of variables and constraints.[3][4] Yet in the theoretical worst case it takes exponentially many steps for most successfully analyzed pivot rules. This was one of the main motivations for developing smoothed analysis.[5]
For the perturbation model, we assume that the input data is perturbed by noise from a Gaussian distribution. For normalization purposes, we assume the unperturbed data ${\bar {\mathbf {A} }}\in \mathbb {R} ^{n\times d},{\bar {\mathbf {b} }}\in \mathbb {R} ^{n},\mathbf {c} \in \mathbb {R} ^{d}$ satisfies $\|({\bar {\mathbf {a} }}_{i},{\bar {b}}_{i})\|_{2}\leq 1$ for all rows $({\bar {\mathbf {a} }}_{i},{\bar {b}}_{i})$ of the matrix $({\bar {\mathbf {A} }},{\bar {\mathbf {b} }}).$ The noise $({\hat {\mathbf {A} }},{\hat {\mathbf {b} }})$ has independent entries sampled from a Gaussian distribution with mean $0$ and standard deviation $\sigma $. We set $\mathbf {A} ={\bar {\mathbf {A} }}+{\hat {\mathbf {A} }},\mathbf {b} ={\bar {\mathbf {b} }}+{\hat {\mathbf {b} }}$. The smoothed input data consists of the linear program
maximize
$\mathbf {c^{T}} \cdot \mathbf {x} $
subject to
$\mathbf {A} \mathbf {x} \leq \mathbf {b} $.
If the running time of our algorithm on data $\mathbf {A} ,\mathbf {b} ,\mathbf {c} $ is given by $T(\mathbf {A} ,\mathbf {b} ,\mathbf {c} )$ then the smoothed complexity of the simplex method is[6]
$C_{s}(n,d,\sigma ):=\max _{{\bar {\mathbf {A} }},{\bar {\mathbf {b} }},\mathbf {c} }~\mathbb {E} _{{\hat {\mathbf {A} }},{\hat {\mathbf {b} }}}[T({\bar {\mathbf {A} }}+{\hat {\mathbf {A} }},{\bar {\mathbf {b} }}+{\hat {\mathbf {b} }},\mathbf {c} )]={\rm {poly}}(d,\log n,\sigma ^{-1}).$
This bound holds for a specific pivot rule called the shadow vertex rule. The shadow vertex rule is slower than more commonly used pivot rules such as Dantzig's rule or the steepest edge rule[7] but it has properties that make it very well-suited to probabilistic analysis.[8]
Local search for combinatorial optimization
A number of local search algorithms have bad worst-case running times but perform well in practice.[9]
One example is the 2-opt heuristic for the traveling salesman problem. It can take exponentially many iterations until it finds a locally optimal solution, although in practice the running time is subquadratic in the number of vertices.[10] The approximation ratio, which is the ratio between the length of the output of the algorithm and the length of the optimal solution, tends to be good in practice but can also be bad in the theoretical worst case.
One class of problem instances can be given by $n$ points in the box $[0,1]^{d}$, where their pairwise distances come from a norm. Already in two dimensions, the 2-opt heuristic might take exponentially many iterations until finding a local optimum. In this setting, one can analyze the perturbation model where the vertices $v_{1},\dots ,v_{n}$ are independently sampled according to probability distributions with probability density function $f_{1},\dots ,f_{n}:[0,1]^{d}\rightarrow [0,\theta ]$. For $\theta =1$, the points are uniformly distributed. When $\theta >1$ is big, the adversary has more ability to increase the likelihood of hard problem instances. In this perturbation model, the expected number of iterations of the 2-opt heuristic, as well as the approximation ratios of resulting output, are bounded by polynomial functions of $n$ and $\theta $.[10]
Another local search algorithm for which smoothed analysis was successful is the k-means method. Given $n$ points in $[0,1]^{d}$, it is NP-hard to find a good partition into clusters with small pairwise distances between points in the same cluster. Lloyd's algorithm is widely used and very fast in practice, although it can take $e^{\Omega (n)}$ iterations in the worst case to find a locally optimal solution. However, assuming that the points have independent Gaussian distributions, each with expectation in $[0,1]^{d}$ and standard deviation $\sigma $, the expected number of iterations of the algorithm is bounded by a polynomial in $n$, $d$ and $\sigma $. [11]
See also
• Average-case complexity
• Pseudo-polynomial time
• Worst-case complexity
References
1. Spielman, Daniel; Teng, Shang-Hua (2009), "Smoothed analysis: an attempt to explain the behavior of algorithms in practice" (PDF), Communications of the ACM, ACM, 52 (10): 76–84, doi:10.1145/1562764.1562785, S2CID 7904807
2. Amenta, Nina; Ziegler, Günter (1999), "Deformed products and maximal shadows of polytopes", Contemporary Mathematics, American Mathematical Society, 223: 10–19, CiteSeerX 10.1.1.80.3241, doi:10.1090/conm/223, ISBN 9780821806746, MR 1661377
3. Shamir, Ron (1987), "The Efficiency of the Simplex Method: A Survey", Management Science, 33 (3): 301–334, doi:10.1287/mnsc.33.3.301
4. Andrei, Neculai (2004), "Andrei, Neculai. "On the complexity of MINOS package for linear programming", Studies in Informatics and Control, 13 (1): 35–46
5. Spielman, Daniel; Teng, Shang-Hua (2001), "Smoothed analysis of algorithms", Proceedings of the thirty-third annual ACM symposium on Theory of computing, ACM, pp. 296–305, arXiv:cs/0111050, Bibcode:2001cs.......11050S, doi:10.1145/380752.380813, ISBN 978-1-58113-349-3, S2CID 1471{{citation}}: CS1 maint: date and year (link)
6. Dadush, Daniel; Huiberts, Sophie (2018), "A friendly smoothed analysis of the simplex method", Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pp. 390–403, arXiv:1711.05667, doi:10.1145/3188745.3188826, ISBN 9781450355599, S2CID 11868079{{citation}}: CS1 maint: date and year (link)
7. Borgwardt, Karl-Heinz; Damm, Renate; Donig, Rudolf; Joas, Gabriele (1993), "Empirical studies on the average efficiency of simplex variants under rotation symmetry", ORSA Journal on Computing, Operations Research Society of America, 5 (3): 249–260, doi:10.1287/ijoc.5.3.249
8. Borgwardt, Karl-Heinz (1987), The Simplex Method: A Probabilistic Analysis, Algorithms and Combinatorics, vol. 1, Springer-Verlag, doi:10.1007/978-3-642-61578-8, ISBN 978-3-540-17096-9
9. Manthey, Bodo (2021), Roughgarden, Tim (ed.), "Smoothed Analysis of Local Search", Beyond the Worst-Case Analysis of Algorithms, Cambridge: Cambridge University Press, pp. 285–308, doi:10.1017/9781108637435.018, ISBN 978-1-108-49431-1, S2CID 221680879, retrieved 2022-06-15
10. Englert, Matthias; Röglin, Heiko; Vöcking, Berthold (2007), "Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP", Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 68: 190–264, arXiv:2302.06889, doi:10.1007/s00453-013-9801-4
11. Arthur, David; Manthey, Bodo; Röglin, Heiko (2011), "Smoothed Analysis of the k-Means Method" (PDF), Journal of the ACM, 58 (5): 1–31, doi:10.1145/2027216.2027217, S2CID 5253105
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Wikipedia
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Smoothed finite element method
Smoothed finite element methods (S-FEM)[1] are a particular class of numerical simulation algorithms for the simulation of physical phenomena. It was developed by combining meshfree methods[2] with the finite element method. S-FEM are applicable to solid mechanics as well as fluid dynamics problems, although so far they have mainly been applied to the former.
Description
The essential idea in the S-FEM is to use a finite element mesh (in particular triangular mesh) to construct numerical models of good performance. This is achieved by modifying the compatible strain field, or construct a strain field using only the displacements, hoping a Galerkin model using the modified/constructed strain field can deliver some good properties. Such a modification/construction can be performed within elements but more often beyond the elements (meshfree concepts): bring in the information from the neighboring elements. Naturally, the strain field has to satisfy certain conditions, and the standard Galerkin weak form needs to be modified accordingly to ensure the stability and convergence. A comprehensive review of S-FEM covering both methodology and applications can be found in[3] ("Smoothed Finite Element Methods (S-FEM): An Overview and Recent Developments").
History
The development of S-FEM started from the works on meshfree methods, where the so-called weakened weak (W2) formulation based on the G space theory[4] were developed. The W2 formulation offers possibilities for formulate various (uniformly) "soft" models that works well with triangular meshes. Because triangular mesh can be generated automatically, it becomes much easier in re-meshing and hence automation in modeling and simulation. In addition, W2 models can be made soft enough (in uniform fashion) to produce upper bound solutions (for force-driving problems). Together with stiff models (such as the fully compatible FEM models), one can conveniently bound the solution from both sides. This allows easy error estimation for generally complicated problems, as long as a triangular mesh can be generated. Typical W2 models are the Smoothed Point Interpolation Methods (or S-PIM).[5] The S-PIM can be node-based (known as NS-PIM or LC-PIM),[6] edge-based (ES-PIM),[7] and cell-based (CS-PIM).[8] The NS-PIM was developed using the so-called SCNI technique.[9] It was then discovered that NS-PIM is capable of producing upper bound solution and volumetric locking free.[10] The ES-PIM is found superior in accuracy, and CS-PIM behaves in between the NS-PIM and ES-PIM. Moreover, W2 formulations allow the use of polynomial and radial basis functions in the creation of shape functions (it accommodates the discontinuous displacement functions, as long as it is in G1 space), which opens further rooms for future developments.
The S-FEM is largely the linear version of S-PIM, but with most of the properties of the S-PIM and much simpler. It has also variations of NS-FEM, ES-FEM and CS-FEM. The major property of S-PIM can be found also in S-FEM.[11]
List of S-FEM models
• Node-based Smoothed FEM (NS-FEM)[12]
• Edge-based Smoothed FEM (ES-FEM)[13]
• Face-based Smoothed FEM (FS-FEM)[14]
• Cell-based Smoothed FEM (CS-FEM)[15][16][17]
• Node/Edge-based Smoothed FEM (NS/ES-FEM)[18][19]
• Alpha FEM method (Alpha FEM)[20][21]
• Beta FEM method (Beta FEM)[22][23]
Applications
S-FEM has been applied to solve the following physical problems:
1. Mechanics for solid structures and piezoelectrics;[24][25]
2. Fracture mechanics and crack propagation;[26][27][28][29]
3. Nonlinear and contact problems;[30][31]
4. Stochastic analysis;[32]
5. Heat transfer;[33][34]
6. Structural acoustics;[35][36][37]
7. Adaptive analysis;[38][18]
8. Limited analysis;[39]
9. Crystal plasticity modeling.[40]
See also
• Finite element method
• Meshfree methods
• Weakened weak form
• Loubignac iteration
References
1. Liu, G.R., 2010 Smoothed Finite Element Methods, CRC Press, ISBN 978-1-4398-2027-8.
2. Liu, G.R. 2nd edn: 2009 Mesh Free Methods, CRC Press. 978-1-4200-8209-9
3. W. Zeng, G.R. Liu. Smoothed finite element methods (S-FEM): An overview and recent developments. Archives of Computational Methods in Engineering, 2016, doi: 10.1007/s11831-016-9202-3
4. G.R. Liu. A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory and Part II applications to solid mechanics problems. International Journal for Numerical Methods in Engineering, 81: 1093-1126, 2010
5. Liu, G.R. 2nd edn: 2009 Mesh Free Methods, CRC Press. 978-1-4200-8209-9
6. Liu GR, Zhang GY, Dai KY, Wang YY, Zhong ZH, Li GY and Han X, A linearly conforming point interpolation method (LC-PIM) for 2D solid mechanics problems, International Journal of Computational Methods, 2(4): 645-665, 2005.
7. G.R. Liu, G.R. Zhang. Edge-based Smoothed Point Interpolation Methods. International Journal of Computational Methods, 5(4): 621-646, 2008
8. G.R. Liu, G.R. Zhang. A normed G space and weakened weak (W2) formulation of a cell-based Smoothed Point Interpolation Method. International Journal of Computational Methods, 6(1): 147-179, 2009
9. Chen, J. S., Wu, C. T., Yoon, S. and You, Y. (2001). A stabilized conforming nodal integration for Galerkin mesh-free methods. Int. J. Numer. Meth. Eng. 50: 435–466.
10. G. R. Liu and G. Y. Zhang. Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC-PIM). International Journal for Numerical Methods in Engineering, 74: 1128-1161, 2008.
11. Zhang ZQ, Liu GR, Upper and lower bounds for natural frequencies: A property of the smoothed finite element methods, International Journal for Numerical Methods in Engineering Vol. 84 Issue: 2, 149-178, 2010
12. Liu GR, Nguyen-Thoi T, Nguyen-Xuan H, Lam KY (2009) A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems. Computers and Structures; 87: 14-26.
13. Liu GR, Nguyen-Thoi T, Lam KY (2009) An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses in solids. Journal of Sound and Vibration; 320: 1100-1130.
14. Nguyen-Thoi T, Liu GR, Lam KY, GY Zhang (2009) A Face-based Smoothed Finite Element Method (FS-FEM) for 3D linear and nonlinear solid mechanics problems using 4-node tetrahedral elements. International Journal for Numerical Methods in Engineering; 78: 324-353
15. Liu GR, Dai KY, Nguyen-Thoi T (2007) A smoothed finite element method for mechanics problems. Computational Mechanics; 39: 859-877
16. Dai KY, Liu GR (2007) Free and forced vibration analysis using the smoothed finite element method (SFEM). Journal of Sound and Vibration; 301: 803-820.
17. Dai KY, Liu GR, Nguyen-Thoi T (2007) An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics. Finite Elements in Analysis and Design; 43: 847－860.
18. Li Y, Liu GR, Zhang GY, An adaptive NS/ES-FEM approach for 2D contact problems using triangular elements, Finite Elements in Analysis and Design Vol.47 Issue: 3, 256-275, 2011
19. Jiang C, Zhang ZQ, Liu GR, Han X, Zeng W, An edge-based/node-based selective smoothed finite element method using tetrahedrons for cardiovascular tissues, Engineering Analysis with Boundary Elements Vol.59, 62-77, 2015
20. Liu GR, Nguyen-Thoi T, Lam KY (2009) A novel FEM by scaling the gradient of strains with factor α (αFEM). Computational Mechanics; 43: 369-391
21. Liu GR, Nguyen-Xuan H, Nguyen-Thoi T, Xu X (2009) A novel weak form and a superconvergent alpha finite element method (SαFEM) for mechanics problems using triangular meshes. Journal of Computational Physics; 228: 4055-4087
22. Zeng W, Liu GR, Li D, Dong XW (2016) A smoothing technique based beta finite element method (βFEM) for crystal plasticity modeling. Computers and Structures; 162: 48-67
23. Zeng W, Liu GR, Jiang C, Nguyen-Thoi T, Jiang Y (2016) A generalized beta finite element method with coupled smoothing techniques for solid mechanics. Engineering Analysis with Boundary Elements; 73: 103-119
24. Cui XY, Liu GR, Li GY, et al. A thin plate formulation without rotation DOFs based on the radial point interpolation method and triangular cells, International Journal for Numerical Methods in Engineering Vol.85 Issue: 8 , 958-986, 2011
25. Liu GR, Nguyen-Xuan H, Nguyen-Thoi T, A theoretical study on the smoothed FEM (S-FEM) models: Properties, accuracy and convergence rates, International Journal for Numerical Methods in Engineering Vol. 84 Issue: 10, 1222-1256, 2010
26. Liu GR, Nourbakhshnia N, Zhang YW, A novel singular ES-FEM method for simulating singular stress fields near the crack tips for linear fracture problems, Engineering Fracture Mechanics Vol.78 Issue: 6 Pages: 863-876, 2011
27. Liu GR, Chen L, Nguyen-Thoi T, et al. A novel singular node-based smoothed finite element method (NS-FEM) for upper bound solutions of fracture problems, International Journal for Numerical Methods in Engineering Vol.83 Issue: 11, 1466-1497, 2010
28. Zeng W, Liu GR, Kitamura Y, Nguyen-Xuan H. "A three-dimensional ES-FEM for fracture mechanics problems in elastic solids", Engineering Fracture Mechanics Vol. 114, 127-150, 2013
29. Zeng W, Liu GR, Jiang C, Dong XW, Chen HD, Bao Y, Jiang Y. "An effective fracture analysis method based on the virtual crack closure-integral technique implemented in CS-FEM", Applied Mathematical Modelling Vol. 40, Issue: 5-6, 3783-3800, 2016
30. Zhang ZQ, Liu GR, An edge-based smoothed finite element method (ES-FEM) using 3-node triangular elements for 3D non-linear analysis of spatial membrane structures, International Journal for Numerical Methods in Engineering, Vol. 86 Issue: 2 135-154, 2011
31. Jiang C, Liu GR, Han X, Zhang ZQ, Zeng W, A smoothed finite element method for analysis of anisotropic large deformation of passive rabbit ventricles in diastole, International Journal for Numerical Methods in Biomedical Engineering, Vol. 31 Issue: 1,1-25, 2015
32. Liu GR, Zeng W, Nguyen-Xuan H. Generalized stochastic cell-based smoothed finite element method (GS_CS-FEM) for solid mechanics, Finite Elements in Analysis and Design Vol.63, 51-61, 2013
33. Zhang ZB, Wu SC, Liu GR, et al. Nonlinear Transient Heat Transfer Problems using the Meshfree ES-PIM, International Journal of Nonlinear Sciences and Numerical Simulation Vol.11 Issue: 12, 1077-1091, 2010
34. Wu SC, Liu GR, Cui XY, et al. An edge-based smoothed point interpolation method (ES-PIM) for heat transfer analysis of rapid manufacturing system, International Journal of Heat and Mass Transfer Vol.53 Issue: 9-10, 1938-1950, 2010
35. He ZC, Cheng AG, Zhang GY, et al. Dispersion error reduction for acoustic problems using the edge-based smoothed finite element method (ES-FEM), International Journal for Numerical Methods in Engineering Vol. 86 Issue: 11 Pages: 1322-1338, 2011
36. He ZC, Liu GR, Zhong ZH, et al. A coupled ES-FEM/BEM method for fluid-structure interaction problems, Engineering Analysis with Boundary Elements Vol. 35 Issue: 1, 140-147, 2011
37. Zhang ZQ, Liu GR, Upper and lower bounds for natural frequencies: A property of the smoothed finite element methods, International Journal for Numerical Methods in Engineering Vol.84 Issue: 2,149-178, 2010
38. Nguyen-Thoi T, Liu GR, Nguyen-Xuan H, et al. Adaptive analysis using the node-based smoothed finite element method (NS-FEM), International Journal for Numerical Methods in Biomedical Engineering Vol. 27 Issue: 2, 198-218, 2011
39. Tran TN, Liu GR, Nguyen-Xuan H, et al. An edge-based smoothed finite element method for primal-dual shakedown analysis of structures, International Journal for Numerical Methods in Engineering Vol.82 Issue: 7, 917-938, 2010
40. Zeng W, Larsen JM, Liu GR. Smoothing technique based crystal plasticity finite element modeling of crystalline materials, International Journal of Plasticity Vol.65, 250-268, 2015
External links
Numerical methods for partial differential equations
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Wikipedia
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Smoothed octagon
The smoothed octagon is a region in the plane found by Karl Reinhardt in 1934 and conjectured by him to have the lowest maximum packing density of the plane of all centrally symmetric convex shapes.[1] It was also independently discovered by Kurt Mahler in 1947.[2] It is constructed by replacing the corners of a regular octagon with a section of a hyperbola that is tangent to the two sides adjacent to the corner and asymptotic to the sides adjacent to these.
Construction
The shape of the smoothed octagon can be derived from its packings, which place octagons at the points of a triangular lattice. The requirement that these packings have the same density no matter how the lattice and smoothed octagon are rotated relative to each other, with shapes that remain in contact with each neighboring shape, can be used to determine the shape of the corners. One of the figures shows three octagons that rotate while the area of the triangle formed by their centres remains constant, keeping them packed together as closely as possible. For regular octagons, the red and blue shapes would overlap, so to enable the rotation to proceed the corners are clipped to a point that lies halfway between their centres, generating the required curve, which turns out to be a hyperbola.
The hyperbola is constructed tangent to two sides of the octagon, and asymptotic to the two adjacent to these. The following details apply to a regular octagon of circumradius ${\sqrt {2}}$ with its centre at the point $(2+{\sqrt {2}},0)$ and one vertex at the point $(2,0)$. For two constants $\ell ={\sqrt {2}}-1$ and $m=(1/2)^{1/4}$, the hyperbola is given by the equation
$\ell ^{2}x^{2}-y^{2}=m^{2}$
or the equivalent parameterization (for the right-hand branch only)
${\begin{aligned}x&={\frac {m}{\ell }}\cosh {t}\\y&=m\sinh {t}\\\end{aligned}}$
for the portion of the hyperbola that forms the corner, given by the range of parameter values
$-{\frac {\ln {2}}{4}}<t<{\frac {\ln {2}}{4}}.$
The lines of the octagon tangent to the hyperbola are $y=\pm \left({\sqrt {2}}+1\right)\left(x-2\right)$, and the lines asymptotic to the hyperbola are simply $y=\pm \ell x$.
Packing
The smoothed octagon has a maximum packing density given by[2]
${\frac {8-4{\sqrt {2}}-\ln {2}}{2{\sqrt {2}}-1}}\approx 0.902414\,.$
This is lower than the maximum packing density of circles, which is
${\frac {\pi }{\sqrt {12}}}\approx 0.906899.$
The maximum known packing density of the ordinary regular octagon is
${\frac {4+4{\sqrt {2}}}{5+4{\sqrt {2}}}}\approx 0.906163,$
also slightly less than the maximum packing density of circles, but higher than that of the smoothed octagon.[3]
Unsolved problem in mathematics:
Is the smoothed octagon the centrally symmetric shape with the lowest maximum packing density?
(more unsolved problems in mathematics)
The smoothed octagon achieves its maximum packing density, not just for a single packing, but for a 1-parameter family. All of these are lattice packings. Reinhardt's conjecture that the smoothed octagon has the lowest maximum packing density of all centrally symmetric convex shapes in the plane remains unsolved. If central symmetry is not required, the regular heptagon has even lower packing density, but its optimality is also unproven. In three dimensions, Ulam's packing conjecture states that no convex shape has a lower maximum packing density than the ball.[4]
References
1. Reinhardt, K. (1934). "Über die dichteste gitterförmige Lagerung kongruenter Bereiche in der Ebene und eine besondere Art konvexer Kurven". Abh. Math. Sem. Univ. Hamburg. 10: 216–230. doi:10.1007/BF02940676. S2CID 120336230.
2. Mahler, Kurt (1947). "On the minimum determinant and the circumscribed hexagons of a convex domain" (PDF). Proc. Kon. Ned. Akad. Wet. 50: 692–703.
3. Atkinson, Steven; Jiao, Yang; Torquato, Salvatore (2012-09-10). "Maximally dense packings of two-dimensional convex and concave noncircular particles" (PDF). Physical Review E. 86 (3): 031302. arXiv:1405.0245. Bibcode:2012PhRvE..86c1302A. doi:10.1103/physreve.86.031302. PMID 23030907. S2CID 9806947. Archived from the original (PDF) on 2014-08-24.
4. Kallus, Yoav; Kusner, Wöden (2016). "The local optimality of the double lattice packing". Discrete & Computational Geometry. 56 (2): 449–471. arXiv:1509.02241. doi:10.1007/s00454-016-9792-4. MR 3530975. S2CID 254036374.
External links
• The thinnest densest two-dimensional packing?. Peter Scholl, 2001.
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Wikipedia
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Smoothing problem (stochastic processes)
The smoothing problem (not to be confused with smoothing in statistics, image processing and other contexts) is the problem of estimating an unknown probability density function recursively over time using incremental incoming measurements. It is one of the main problems defined by Norbert Wiener.[1][2] A smoother is an algorithm that implements a solution to this problem, typically based on recursive Bayesian estimation. The smoothing problem is closely related to the filtering problem, both of which are studied in Bayesian smoothing theory.
A smoother is often a two-pass process, composed of forward and backward passes. Consider doing estimation (prediction/retrodiction) about an ongoing process (e.g. tracking a missile) based on incoming observations. When new observations arrive, estimations about past needs to be updated to have a smoother (more accurate) estimation of the whole estimated path until now (taking into account the newer observations). Without a backward pass (for retrodiction), the sequence of predictions in an online filtering algorithm does not look smooth. In other words, retrospectively, it is as if we are using future observations for improving estimation of a point in past, when those observations about future points become available. Note that time of estimation (which determines which observations are available) can be different to the time of the point that the prediction is about (that is subject to prediction/retrodiction). The observations about later times can be used to update and improved the estimations about earlier times. Doing so leads to smoother-looking estimations (retrodiction) about the whole path.
Examples of smoothers
Some variants include:[3]
• Rauch–Tung–Striebel (RTS) smoother
• Gaussian smoothers (e.g., extended Kalman smoother or sigma-point smoothers) for non-linear state-space models.
• Particle smoothers
The confusion in terms and the relation between Filtering and Smoothing problems
The terms Smoothing and Filtering are used for four concepts that may initially be confusing: Smoothing (in two senses: estimation and convolution), and Filtering (again in two senses: estimation and convolution).
Smoothing (estimation) and smoothing (convolution) despite being labelled with the same name in English language, can mean totally different mathematical procedures. The requirements of problems they solve are different. These concepts are distinguished by the context (signal processing versus estimation of stochastic processes).
The historical reason for this confusion is that initially, the Wiener's suggested a "smoothing" filter that was just a convolution. Later on his proposed solutions for obtaining a smoother estimation separate developments as two distinct concepts. One was about attaining a smoother estimation by taking into account past observations, and the other one was smoothing using filter design (design of a convolution filter).
Both the smoothing problem (in sense of estimation) and the filtering problem (in sense of estimation) are often confused with smoothing and filtering in other contexts (especially non-stochastic signal processing, often a name of various types of convolution). These names are used in the context of World War 2 with problems framed by people like Norbert Wiener.[1][2] One source of confusion is the Wiener Filter is in form of a simple convolution. However, in Wiener's filter, two time-series are given. When the filter is defined, a straightforward convolution is the answer. However, in later developments such as Kalman filtering, the nature of filtering is different to convolution and it deserves a different name.
The distinction is described in the following two senses:
1. Convolution: The smoothing in the sense of convolution is simpler. For example, moving average, low-pass filtering, convolution with a kernel, or blurring using Laplace filters in image processing. It is often a filter design problem. Especially non-stochastic and non-Bayesian signal processing, without any hidden variables.
2. Estimation: The smoothing problem (or Smoothing in the sense of estimation) uses Bayesian and state-space models to estimate the hidden state variables. This is used in the context of World War 2 defined by people like Norbert Wiener, in (stochastic) control theory, radar, signal detection, tracking, etc. The most common use is the Kalman Smoother used with Kalman Filter, which is actually developed by Rauch. The procedure is called Kalman-Rauch recursion. It is one of the main problems solved by Norbert Wiener.[1][2] Most importantly, in the Filtering problem (sense 2) the information from observation up to the time of the current sample is used. In smoothing (also sense 2) all observation samples (from future) are used. Filtering is causal but smoothing is batch processing of the same problem, namely, estimation of a time-series process based on serial incremental observations.
But the usual and more common smoothing and filtering (in the sense of 1.) do not have such distinction because there is no distinction between hidden and observable.
The distinction between Smoothing (estimation) and Filtering (estimation): In smoothing all observation samples are used (from future). Filtering is causal, whereas smoothing is batch processing of the given data. Filtering is the estimation of a (hidden) time-series process based on serial incremental observations.
See also
• Filtering problem
• Filter (signal processing)
• Kalman filter, a well-known filtering algorithm related both to the filtering problem and the smoothing problem
• Generalized filtering
• Smoothing
References
1. 1942, Extrapolation, Interpolation and Smoothing of Stationary Time Series. A war-time classified report nicknamed "the yellow peril" because of the color of the cover and the difficulty of the subject. Published postwar 1949 MIT Press. http://www.isss.org/lumwiener.htm Archived 2015-08-16 at the Wayback Machine
2. Wiener, Norbert (1949). Extrapolation, Interpolation, and Smoothing of Stationary Time Series. New York: Wiley. ISBN 0-262-73005-7.
3. Simo Särkkä. Bayesian Filtering and Smoothing. Publisher: Cambridge University Press (5 Sept. 2013) Language: English ISBN 1107619289 ISBN 978-1107619289
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Wikipedia
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Woodbury matrix identity
In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max A. Woodbury,[1][2] says that the inverse of a rank-k correction of some matrix can be computed by doing a rank-k correction to the inverse of the original matrix. Alternative names for this formula are the matrix inversion lemma, Sherman–Morrison–Woodbury formula or just Woodbury formula. However, the identity appeared in several papers before the Woodbury report.[3][4]
The Woodbury matrix identity is[5]
$\left(A+UCV\right)^{-1}=A^{-1}-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1},$
where A, U, C and V are conformable matrices: A is n×n, C is k×k, U is n×k, and V is k×n. This can be derived using blockwise matrix inversion.
While the identity is primarily used on matrices, it holds in a general ring or in an Ab-category.
The Woodbury matrix identity allows cheap computation of inverses and solutions to linear equations. However, little is known about the numerical stability of the formula. There are no published results concerning its error bounds. Anecdotal evidence [6] suggests that it may diverge even for seemingly benign examples (when both the original and modified matrices are well-conditioned).
Discussion
To prove this result, we will start by proving a simpler one. Replacing A and C with the identity matrix I, we obtain another identity which is a bit simpler:
$\left(I+UV\right)^{-1}=I-U\left(I+VU\right)^{-1}V.$
To recover the original equation from this reduced identity, set $U=A^{-1}X$ and $V=CY$.
This identity itself can be viewed as the combination of two simpler identities. We obtain the first identity from
$I=(I+P)^{-1}(I+P)=(I+P)^{-1}+(I+P)^{-1}P$,
thus,
$(I+P)^{-1}=I-(I+P)^{-1}P$,
and similarly
$(I+P)^{-1}=I-P(I+P)^{-1}.$
The second identity is the so-called push-through identity[7]
$(I+UV)^{-1}U=U(I+VU)^{-1}$
that we obtain from
$U(I+VU)=(I+UV)U$
after multiplying by $(I+VU)^{-1}$ on the right and by $(I+UV)^{-1}$ on the left.
Putting all together,
$\left(I+UV\right)^{-1}=I-UV\left(I+UV\right)^{-1}=I-U\left(I+VU\right)^{-1}V.$
where the first and second equality come from the first and second identity, respectively.
Special cases
When $V,U$ are vectors, the identity reduces to the Sherman–Morrison formula.
In the scalar case, the reduced version is simply
${\frac {1}{1+uv}}=1-{\frac {uv}{1+uv}}.$
Inverse of a sum
If n = k and U = V = In is the identity matrix, then
${\begin{aligned}\left({A}+{B}\right)^{-1}&=A^{-1}-A^{-1}(B^{-1}+A^{-1})^{-1}A^{-1}\\&={A}^{-1}-{A}^{-1}\left({A}{B}^{-1}+{I}\right)^{-1}.\end{aligned}}$
Continuing with the merging of the terms of the far right-hand side of the above equation results in Hua's identity
$\left({A}+{B}\right)^{-1}={A}^{-1}-\left({A}+{A}{B}^{-1}{A}\right)^{-1}.$
Another useful form of the same identity is
$\left({A}-{B}\right)^{-1}={A}^{-1}+{A}^{-1}{B}\left({A}-{B}\right)^{-1},$
which, unlike those above, is valid even if $B$ is singular, and has a recursive structure that yields
$\left({A}-{B}\right)^{-1}=\sum _{k=0}^{\infty }\left({A}^{-1}{B}\right)^{k}{A}^{-1}$
if the spectral radius of $A^{-1}B$ is less than one. That is, if the above sum converges then it is equal to $(A-B)^{-1}$.
This form can be used in perturbative expansions where B is a perturbation of A.
Binomial inverse theorem
If A, B, U, V are matrices of sizes n×n, k×k, n×k, k×n, respectively, then
$\left(A+UBV\right)^{-1}=A^{-1}-A^{-1}UB\left(B+BVA^{-1}UB\right)^{-1}BVA^{-1}$
provided A and B + BVA−1UB are nonsingular. Nonsingularity of the latter requires that B−1 exist since it equals B(I + VA−1UB) and the rank of the latter cannot exceed the rank of B.[7]
Since B is invertible, the two B terms flanking the parenthetical quantity inverse in the right-hand side can be replaced with (B−1)−1, which results in the original Woodbury identity.
A variation for when B is singular and possibly even non-square:[7]
$(A+UBV)^{-1}=A^{-1}-A^{-1}U(I+BVA^{-1}U)^{-1}BVA^{-1}.$
Formulas also exist for certain cases in which A is singular.[8]
Pseudoinverse with positive semidefinite matrices
In general Woodbury's identity is not valid if one or more inverses are replaced by (Moore–Penrose) pseudoinverses. However, if $A$ and $C$ are positive semidefinite, and $V=U^{\mathrm {H} }$ (implying that $A+UCV$ is itself positive semidefinite), then the following formula provides a generalization:[9][10]
${\begin{aligned}(XX^{\mathrm {H} }+YY^{\mathrm {H} })^{+}&=(ZZ^{\mathrm {H} })^{+}+(I-YZ^{+})^{\mathrm {H} }X^{+\mathrm {H} }EX^{+}(I-YZ^{+}),\\Z&=(I-XX^{+})Y,\\E&=I-X^{+}Y(I-Z^{+}Z)F^{-1}(X^{+}Y)^{\mathrm {H} },\\F&=I+(I-Z^{+}Z)Y^{\mathrm {H} }(XX^{\mathrm {H} })^{+}Y(I-Z^{+}Z),\end{aligned}}$
where $A+UCU^{\mathrm {H} }$ can be written as $XX^{\mathrm {H} }+YY^{\mathrm {H} }$ because any positive semidefinite matrix is equal to $MM^{\mathrm {H} }$ for some $M$.
Derivations
Direct proof
The formula can be proven by checking that $(A+UCV)$ times its alleged inverse on the right side of the Woodbury identity gives the identity matrix:
${\begin{aligned}&\left(A+UCV\right)\left[A^{-1}-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}\right]\\={}&\left\{I-U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}\right\}+\left\{UCVA^{-1}-UCVA^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}\right\}\\={}&\left\{I+UCVA^{-1}\right\}-\left\{U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}+UCVA^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}\right\}\\={}&I+UCVA^{-1}-\left(U+UCVA^{-1}U\right)\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}\\={}&I+UCVA^{-1}-UC\left(C^{-1}+VA^{-1}U\right)\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}\\={}&I+UCVA^{-1}-UCVA^{-1}\\={}&I.\end{aligned}}$
Alternative proofs
Algebraic proof
First consider these useful identities,
${\begin{aligned}U+UCVA^{-1}U&=UC\left(C^{-1}+VA^{-1}U\right)=\left(A+UCV\right)A^{-1}U\\\left(A+UCV\right)^{-1}UC&=A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}\end{aligned}}$
Now,
${\begin{aligned}A^{-1}&=\left(A+UCV\right)^{-1}\left(A+UCV\right)A^{-1}\\&=\left(A+UCV\right)^{-1}\left(I+UCVA^{-1}\right)\\&=\left(A+UCV\right)^{-1}+\left(A+UCV\right)^{-1}UCVA^{-1}\\&=\left(A+UCV\right)^{-1}+A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}.\end{aligned}}$
Derivation via blockwise elimination
Deriving the Woodbury matrix identity is easily done by solving the following block matrix inversion problem
${\begin{bmatrix}A&U\\V&-C^{-1}\end{bmatrix}}{\begin{bmatrix}X\\Y\end{bmatrix}}={\begin{bmatrix}I\\0\end{bmatrix}}.$
Expanding, we can see that the above reduces to
${\begin{cases}AX+UY=I\\VX-C^{-1}Y=0\end{cases}}$
which is equivalent to $(A+UCV)X=I$. Eliminating the first equation, we find that $X=A^{-1}(I-UY)$, which can be substituted into the second to find $VA^{-1}(I-UY)=C^{-1}Y$. Expanding and rearranging, we have $VA^{-1}=\left(C^{-1}+VA^{-1}U\right)Y$, or $\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}=Y$. Finally, we substitute into our $AX+UY=I$, and we have $AX+U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}=I$. Thus,
$(A+UCV)^{-1}=X=A^{-1}-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}.$
We have derived the Woodbury matrix identity.
Derivation from LDU decomposition
We start by the matrix
${\begin{bmatrix}A&U\\V&C\end{bmatrix}}$
By eliminating the entry under the A (given that A is invertible) we get
${\begin{bmatrix}I&0\\-VA^{-1}&I\end{bmatrix}}{\begin{bmatrix}A&U\\V&C\end{bmatrix}}={\begin{bmatrix}A&U\\0&C-VA^{-1}U\end{bmatrix}}$
Likewise, eliminating the entry above C gives
${\begin{bmatrix}A&U\\V&C\end{bmatrix}}{\begin{bmatrix}I&-A^{-1}U\\0&I\end{bmatrix}}={\begin{bmatrix}A&0\\V&C-VA^{-1}U\end{bmatrix}}$
Now combining the above two, we get
${\begin{bmatrix}I&0\\-VA^{-1}&I\end{bmatrix}}{\begin{bmatrix}A&U\\V&C\end{bmatrix}}{\begin{bmatrix}I&-A^{-1}U\\0&I\end{bmatrix}}={\begin{bmatrix}A&0\\0&C-VA^{-1}U\end{bmatrix}}$
Moving to the right side gives
${\begin{bmatrix}A&U\\V&C\end{bmatrix}}={\begin{bmatrix}I&0\\VA^{-1}&I\end{bmatrix}}{\begin{bmatrix}A&0\\0&C-VA^{-1}U\end{bmatrix}}{\begin{bmatrix}I&A^{-1}U\\0&I\end{bmatrix}}$
which is the LDU decomposition of the block matrix into an upper triangular, diagonal, and lower triangular matrices.
Now inverting both sides gives
${\begin{aligned}{\begin{bmatrix}A&U\\V&C\end{bmatrix}}^{-1}&={\begin{bmatrix}I&A^{-1}U\\0&I\end{bmatrix}}^{-1}{\begin{bmatrix}A&0\\0&C-VA^{-1}U\end{bmatrix}}^{-1}{\begin{bmatrix}I&0\\VA^{-1}&I\end{bmatrix}}^{-1}\\[8pt]&={\begin{bmatrix}I&-A^{-1}U\\0&I\end{bmatrix}}{\begin{bmatrix}A^{-1}&0\\0&\left(C-VA^{-1}U\right)^{-1}\end{bmatrix}}{\begin{bmatrix}I&0\\-VA^{-1}&I\end{bmatrix}}\\[8pt]&={\begin{bmatrix}A^{-1}+A^{-1}U\left(C-VA^{-1}U\right)^{-1}VA^{-1}&-A^{-1}U\left(C-VA^{-1}U\right)^{-1}\\-\left(C-VA^{-1}U\right)^{-1}VA^{-1}&\left(C-VA^{-1}U\right)^{-1}\end{bmatrix}}\qquad \mathrm {(1)} \end{aligned}}$
We could equally well have done it the other way (provided that C is invertible) i.e.
${\begin{bmatrix}A&U\\V&C\end{bmatrix}}={\begin{bmatrix}I&UC^{-1}\\0&I\end{bmatrix}}{\begin{bmatrix}A-UC^{-1}V&0\\0&C\end{bmatrix}}{\begin{bmatrix}I&0\\C^{-1}V&I\end{bmatrix}}$
Now again inverting both sides,
${\begin{aligned}{\begin{bmatrix}A&U\\V&C\end{bmatrix}}^{-1}&={\begin{bmatrix}I&0\\C^{-1}V&I\end{bmatrix}}^{-1}{\begin{bmatrix}A-UC^{-1}V&0\\0&C\end{bmatrix}}^{-1}{\begin{bmatrix}I&UC^{-1}\\0&I\end{bmatrix}}^{-1}\\[8pt]&={\begin{bmatrix}I&0\\-C^{-1}V&I\end{bmatrix}}{\begin{bmatrix}\left(A-UC^{-1}V\right)^{-1}&0\\0&C^{-1}\end{bmatrix}}{\begin{bmatrix}I&-UC^{-1}\\0&I\end{bmatrix}}\\[8pt]&={\begin{bmatrix}\left(A-UC^{-1}V\right)^{-1}&-\left(A-UC^{-1}V\right)^{-1}UC^{-1}\\-C^{-1}V\left(A-UC^{-1}V\right)^{-1}&C^{-1}+C^{-1}V\left(A-UC^{-1}V\right)^{-1}UC^{-1}\end{bmatrix}}\qquad \mathrm {(2)} \end{aligned}}$
Now comparing elements (1, 1) of the RHS of (1) and (2) above gives the Woodbury formula
$\left(A-UC^{-1}V\right)^{-1}=A^{-1}+A^{-1}U\left(C-VA^{-1}U\right)^{-1}VA^{-1}.$
Applications
This identity is useful in certain numerical computations where A−1 has already been computed and it is desired to compute (A + UCV)−1. With the inverse of A available, it is only necessary to find the inverse of C−1 + VA−1U in order to obtain the result using the right-hand side of the identity. If C has a much smaller dimension than A, this is more efficient than inverting A + UCV directly. A common case is finding the inverse of a low-rank update A + UCV of A (where U only has a few columns and V only a few rows), or finding an approximation of the inverse of the matrix A + B where the matrix B can be approximated by a low-rank matrix UCV, for example using the singular value decomposition.
This is applied, e.g., in the Kalman filter and recursive least squares methods, to replace the parametric solution, requiring inversion of a state vector sized matrix, with a condition equations based solution. In case of the Kalman filter this matrix has the dimensions of the vector of observations, i.e., as small as 1 in case only one new observation is processed at a time. This significantly speeds up the often real time calculations of the filter.
In the case when C is the identity matrix I, the matrix $I+VA^{-1}U$ is known in numerical linear algebra and numerical partial differential equations as the capacitance matrix.[4]
See also
• Sherman–Morrison formula
• Schur complement
• Matrix determinant lemma, formula for a rank-k update to a determinant
• Invertible matrix
• Moore–Penrose pseudoinverse#Updating the pseudoinverse
Notes
1. Max A. Woodbury, Inverting modified matrices, Memorandum Rept. 42, Statistical Research Group, Princeton University, Princeton, NJ, 1950, 4pp MR38136
2. Max A. Woodbury, The Stability of Out-Input Matrices. Chicago, Ill., 1949. 5 pp. MR32564
3. Guttmann, Louis (1946). "Enlargement methods for computing the inverse matrix". Ann. Math. Statist. 17 (3): 336–343. doi:10.1214/aoms/1177730946.
4. Hager, William W. (1989). "Updating the inverse of a matrix". SIAM Review. 31 (2): 221–239. doi:10.1137/1031049. JSTOR 2030425. MR 0997457.
5. Higham, Nicholas (2002). Accuracy and Stability of Numerical Algorithms (2nd ed.). SIAM. p. 258. ISBN 978-0-89871-521-7. MR 1927606.
6. "MathOverflow discussion". MathOverflow.
7. Henderson, H. V.; Searle, S. R. (1981). "On deriving the inverse of a sum of matrices" (PDF). SIAM Review. 23 (1): 53–60. doi:10.1137/1023004. hdl:1813/32749. JSTOR 2029838.
8. Kurt S. Riedel, "A Sherman–Morrison–Woodbury Identity for Rank Augmenting Matrices with Application to Centering", SIAM Journal on Matrix Analysis and Applications, 13 (1992)659-662, doi:10.1137/0613040 preprint MR1152773
9. Bernstein, Dennis S. (2018). Scalar, Vector, and Matrix Mathematics: Theory, Facts, and Formulas (Revised and expanded ed.). Princeton: Princeton University Press. p. 638. ISBN 9780691151205.
10. Schott, James R. (2017). Matrix analysis for statistics (Third ed.). Hoboken, New Jersey: John Wiley & Sons, Inc. p. 219. ISBN 9781119092483.
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 2.7.3. Woodbury Formula", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
External links
• Some matrix identities
• Weisstein, Eric W. "Woodbury formula". MathWorld.
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Wikipedia
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Snaith's theorem
In algebraic topology, a branch of mathematics, Snaith's theorem, introduced by Victor Snaith, identifies the complex K-theory spectrum with the localization of the suspension spectrum of $\mathbb {C} P^{\infty }$ away from the Bott element.
References
For a proof, see http://people.fas.harvard.edu/~amathew/snaith.pdf
• Victor Snaith, Algebraic Cobordism and K-theory, Mem. Amer. Math. Soc. no 221 (1979)
External links
• Snaith theorem at the nLab
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Wikipedia
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Snake-in-the-box
The snake-in-the-box problem in graph theory and computer science deals with finding a certain kind of path along the edges of a hypercube. This path starts at one corner and travels along the edges to as many corners as it can reach. After it gets to a new corner, the previous corner and all of its neighbors must be marked as unusable. The path should never travel to a corner which has been marked unusable.
In other words, a snake is a connected open path in the hypercube where each node connected with path, with the exception of the head (start) and the tail (finish), it has exactly two neighbors that are also in the snake. The head and the tail each have only one neighbor in the snake. The rule for generating a snake is that a node in the hypercube may be visited if it is connected to the current node and it is not a neighbor of any previously visited node in the snake, other than the current node.
In graph theory terminology, this is called finding the longest possible induced path in a hypercube; it can be viewed as a special case of the induced subgraph isomorphism problem. There is a similar problem of finding long induced cycles in hypercubes, called the coil-in-the-box problem.
The snake-in-the-box problem was first described by Kautz (1958), motivated by the theory of error-correcting codes. The vertices of a solution to the snake or coil in the box problems can be used as a Gray code that can detect single-bit errors. Such codes have applications in electrical engineering, coding theory, and computer network topologies. In these applications, it is important to devise as long a code as is possible for a given dimension of hypercube. The longer the code, the more effective are its capabilities.
Finding the longest snake or coil becomes notoriously difficult as the dimension number increases and the search space suffers a serious combinatorial explosion. Some techniques for determining the upper and lower bounds for the snake-in-the-box problem include proofs using discrete mathematics and graph theory, exhaustive search of the search space, and heuristic search utilizing evolutionary techniques.
Known lengths and bounds
The maximum length for the snake-in-the-box problem is known for dimensions one through eight; it is
1, 2, 4, 7, 13, 26, 50, 98 (sequence A099155 in the OEIS).
Beyond that length, the exact length of the longest snake is not known; the best lengths found so far for dimensions nine through thirteen are
190, 370, 712, 1373, 2687.
For cycles (the coil-in-the-box problem), a cycle cannot exist in a hypercube of dimension less than two. The maximum lengths of the longest possible cycles are
0, 4, 6, 8, 14, 26, 48, 96 (sequence A000937 in the OEIS).
Beyond that length, the exact length of the longest cycle is not known; the best lengths found so far for dimensions nine through thirteen are
188, 366, 692, 1344, 2594.
Doubled coils are a special case: cycles whose second half repeats the structure of their first half, also known as symmetric coils. For dimensions two through seven the lengths of the longest possible doubled coils are
4, 6, 8, 14, 26, 46.
Beyond that, the best lengths found so far for dimensions eight through thirteen are
94, 186, 362, 662, 1222, 2354.
For both the snake and the coil in the box problems, it is known that the maximum length is proportional to 2n for an n-dimensional box, asymptotically as n grows large, and bounded above by 2n − 1. However the constant of proportionality is not known, but is observed to be in the range 0.3 - 0.4 for current best known values.[1]
Notes
1. For asymptotic lower bounds, see Evdokimov (1969), Wojciechowski (1989), and Abbot & Katchalski (1991). For upper bounds, see Douglas (1969), Deimer (1985), Solov'eva (1987), Abbot & Katchalski (1988), Snevily (1994), and Zémor (1997).
References
• Abbot, H. L.; Katchalski, M. (1988), "On the snake in the box problem", Journal of Combinatorial Theory, Series B, 45: 13–24, doi:10.1016/0095-8956(88)90051-2
• Abbot, H. L.; Katchalski, M. (1991), "On the construction of snake-in-the-box codes", Utilitas Mathematica, 40: 97–116
• Allison, David; Paulusma, Daniel (2016), New Bounds for the Snake-in-the-Box Problem, arXiv:1603.05119, Bibcode:2016arXiv160305119A
• Bitterman, D. S. (2004), New Lower Bounds for the Snake-In-The-Box Problem: A Prolog Genetic Algorithm and Heuristic Search Approach (PDF) (M.S. thesis), Department of Computer Science, University of Georgia
• Blaum, Mario; Etzion, Tuvi (2002), Use of snake-in-the-box codes for reliable identification of tracks in servo fields of a disk drive, U.S. Patent 6,496,312
• Casella, D. A.; Potter, W. D. (2005), "Using Evolutionary Techniques to Hunt for Snakes and Coils", 2005 IEEE Congress on Evolutionary Computation (CEC2005), vol. 3, pp. 2499–2504
• Casella, D. A. (2005), New Lower Bounds for the Snake-in-the-Box and the Coil-in-the-Box Problems (PDF) (M.S. thesis), Department of Computer Science, University of Georgia
• Danzer, L.; Klee, V. (1967), "Length of snakes in boxes", Journal of Combinatorial Theory, 2 (3): 258–265, doi:10.1016/S0021-9800(67)80026-7
• Davies, D. W. (1965), "Longest 'separated' paths and loops in an N-cube", IEEE Transactions on Electronic Computers, EC-14 (2): 261, doi:10.1109/PGEC.1965.264259
• Deimer, Knut (1985), "A new upper bound for the length of snakes", Combinatorica, 5 (2): 109–120, doi:10.1007/BF02579373
• Diaz-Gomez, P. A.; Hougen, D. F. (2006), "The snake in the box problem: mathematical conjecture and a genetic algorithm approach", Proceedings of the 8th Conference on Genetic and Evolutionary Computation, Seattle, Washington, USA, pp. 1409–1410, doi:10.1145/1143997.1144219{{citation}}: CS1 maint: location missing publisher (link)
• Douglas, Robert J. (1969), "Upper bounds on the length of circuits of even spread in the d-cube", Journal of Combinatorial Theory, 7 (3): 206–214, doi:10.1016/S0021-9800(69)80013-X
• Evdokimov, A. A. (1969), "Maximal length of a chain in a unit n-dimensional cube", Matematicheskie Zametki, 6: 309–319
• Kautz, William H. (June 1958), "Unit-distance error-checking codes", IRE Transactions on Electronic Computers, EC-7 (2): 179–180, doi:10.1109/TEC.1958.5222529, S2CID 26649532
• Kim, S.; Neuhoff, D. L. (2000), "Snake-in-the-box codes as robust quantizer index assignments", Proceedings of the IEEE International Symposium on Information Theory, p. 402, doi:10.1109/ISIT.2000.866700
• Kinny, D. (2012), "A New Approach to the Snake-In-The-Box Problem", Proceedings of the 20th European Conference on Artificial Intelligence, ECAI-2012, pp. 462–467
• Kinny, D. (2012), "Monte-Carlo Search for Snakes and Coils", Proceedings of the 6th International WS on Multi-disciplinary Trends in Artificial Intelligence, MIWAI-2012, pp. 271–283
• Klee, V. (1970), "What is the maximum length of a d-dimensional snake?", American Mathematical Monthly, 77 (1): 63–65, doi:10.2307/2316860, JSTOR 2316860
• Kochut, K. J. (1996), "Snake-in-the-box codes for dimension 7", Journal of Combinatorial Mathematics and Combinatorial Computing, 20: 175–185
• Lukito, A.; van Zanten, A. J. (2001), "A new non-asymptotic upper bound for snake-in-the-box codes", Journal of Combinatorial Mathematics and Combinatorial Computing, 39: 147–156
• Paterson, Kenneth G.; Tuliani, Jonathan (1998), "Some new circuit codes", IEEE Transactions on Information Theory, 44 (3): 1305–1309, doi:10.1109/18.669420
• Potter, W. D.; Robinson, R. W.; Miller, J. A.; Kochut, K. J.; Redys, D. Z. (1994), "Using the genetic algorithm to find snake in the box codes", Proceedings of the Seventh International Conference on Industrial & Engineering Applications of Artificial Intelligence and Expert Systems, Austin, Texas, USA, pp. 421–426{{citation}}: CS1 maint: location missing publisher (link)
• Snevily, H. S. (1994), "The snake-in-the-box problem: a new upper bound", Discrete Mathematics, 133 (1–3): 307–314, doi:10.1016/0012-365X(94)90039-6
• Solov'eva, F. I. (1987), "An upper bound on the length of a cycle in an n-dimensional unit cube", Metody Diskretnogo Analiza (in Russian), 45: 71–76, 96–97
• Tuohy, D. R.; Potter, W. D.; Casella, D. A. (2007), "Searching for Snake-in-the-Box Codes with Evolved Pruning Models", Proceedings of the 2007 Int. Conf. on Genetic and Evolutionary Methods (GEM'2007), Las Vegas, Nevada, USA, pp. 3–9{{citation}}: CS1 maint: location missing publisher (link)
• Wojciechowski, J. (1989), "A new lower bound for snake-in-the-box codes", Combinatorica, 9 (1): 91–99, doi:10.1007/BF02122688
• Yang, Yuan Sheng; Sun, Fang; Han, Song (2000), "A backward search algorithm for the snake in the box problem", Journal of the Dalian University of Technology, 40 (5): 509–511
• Zémor, Gilles (1997), "An upper bound on the size of the snake-in-the-box", Combinatorica, 17 (2): 287–298, doi:10.1007/BF01200911
• Zinovik, I.; Kroening, D.; Chebiryak, Y. (2008), "Computing binary combinatorial gray codes via exhaustive search with SAT solvers", IEEE Transactions on Information Theory, 54 (4): 1819–1823, doi:10.1109/TIT.2008.917695, hdl:20.500.11850/11304
External links
• Kinny, David (2012), Snake-in-the-Box Research Page (Kyoto University)
• Potter, W. D. (2011), A list of current records for the Snake-in-the-Box Problem (UGA)
• Weisstein, Eric W., "Snake", MathWorld
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Wikipedia
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Snake lemma
The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology. Homomorphisms constructed with its help are generally called connecting homomorphisms.
Statement
In an abelian category (such as the category of abelian groups or the category of vector spaces over a given field), consider a commutative diagram:
where the rows are exact sequences and 0 is the zero object.
Then there is an exact sequence relating the kernels and cokernels of a, b, and c:
$\ker a~{\color {Gray}\longrightarrow }~\ker b~{\color {Gray}\longrightarrow }~\ker c~{\overset {d}{\longrightarrow }}~\operatorname {coker} a~{\color {Gray}\longrightarrow }~\operatorname {coker} b~{\color {Gray}\longrightarrow }~\operatorname {coker} c$
where d is a homomorphism, known as the connecting homomorphism.
Furthermore, if the morphism f is a monomorphism, then so is the morphism $\ker a~{\color {Gray}\longrightarrow }~\ker b$, and if g' is an epimorphism, then so is $\operatorname {coker} b~{\color {Gray}\longrightarrow }~\operatorname {coker} c$.
The cokernels here are: $\operatorname {coker} a=A'/\operatorname {im} a$, $\operatorname {coker} b=B'/\operatorname {im} b$, $\operatorname {coker} c=C'/\operatorname {im} c$.
Explanation of the name
To see where the snake lemma gets its name, expand the diagram above as follows:
and then the exact sequence that is the conclusion of the lemma can be drawn on this expanded diagram in the reversed "S" shape of a slithering snake.
Construction of the maps
The maps between the kernels and the maps between the cokernels are induced in a natural manner by the given (horizontal) maps because of the diagram's commutativity. The exactness of the two induced sequences follows in a straightforward way from the exactness of the rows of the original diagram. The important statement of the lemma is that a connecting homomorphism d exists which completes the exact sequence.
In the case of abelian groups or modules over some ring, the map d can be constructed as follows:
Pick an element x in ker c and view it as an element of C; since g is surjective, there exists y in B with g(y) = x. Because of the commutativity of the diagram, we have g'(b(y)) = c(g(y)) = c(x) = 0 (since x is in the kernel of c), and therefore b(y) is in the kernel of g' . Since the bottom row is exact, we find an element z in A' with f '(z) = b(y). z is unique by injectivity of f '. We then define d(x) = z + im(a). Now one has to check that d is well-defined (i.e., d(x) only depends on x and not on the choice of y), that it is a homomorphism, and that the resulting long sequence is indeed exact. One may routinely verify the exactness by diagram chasing (see the proof of Lemma 9.1 in [1]).
Once that is done, the theorem is proven for abelian groups or modules over a ring. For the general case, the argument may be rephrased in terms of properties of arrows and cancellation instead of elements. Alternatively, one may invoke Mitchell's embedding theorem.
Naturality
In the applications, one often needs to show that long exact sequences are "natural" (in the sense of natural transformations). This follows from the naturality of the sequence produced by the snake lemma.
If
is a commutative diagram with exact rows, then the snake lemma can be applied twice, to the "front" and to the "back", yielding two long exact sequences; these are related by a commutative diagram of the form
Example
Let $k$ be field, $V$ be a $k$-vector space. $V$ is $k[t]$-module by $t:V\to V$ being a $k$-linear transformation, so we can tensor $V$ and $k$ over $k[t]$.
$V\otimes _{k[t]}k=V\otimes _{k[t]}(k[t]/(t))=V/tV=\operatorname {coker} (t).$
Given a short exact sequence of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): k -vector spaces $0\to M\to N\to P\to 0$, we can induce an exact sequence $M\otimes _{k[t]}k\to N\otimes _{k[t]}k\to P\otimes _{k[t]}k\to 0$ by right exactness of tensor product. But the sequence $0\to M\otimes _{k[t]}k\to N\otimes _{k[t]}k\to P\otimes _{k[t]}k\to 0$ is not exact in general. Hence, a natural question arises. Why is this sequence not exact?
According to the diagram above, we can induce an exact sequence $\ker(t_{M})\to \ker(t_{N})\to \ker(t_{P})\to M\otimes _{k[t]}k\to N\otimes _{k[t]}k\to P\otimes _{k[t]}k\to 0$ by applying the snake lemma. Thus, the snake lemma reflects the tensor product's failure to be exact.
In the category of groups
While many results of homological algebra, such as the five lemma or the nine lemma, hold for abelian categories as well as in the category of groups, the snake lemma does not. Indeed, arbitrary cokernels do not exist. However, one can replace cokernels by (left) cosets $A'/\operatorname {im} a$, $B'/\operatorname {im} b$, and $C'/\operatorname {im} c$. Then the connecting homomorphism can still be defined, and one can write down a sequence as in the statement of the snake lemma. This will always be a chain complex, but it may fail to be exact. Exactness can be asserted, however, when the vertical sequences in the diagram are exact, that is, when the images of a, b, and c are normal subgroups.
Counterexample
Consider the alternating group $A_{5}$: this contains a subgroup isomorphic to the symmetric group $S_{3}$, which in turn can be written as a semidirect product of cyclic groups: $S_{3}\simeq C_{3}\rtimes C_{2}$.[2] This gives rise to the following diagram with exact rows:
${\begin{matrix}&1&\to &C_{3}&\to &C_{3}&\to 1\\&\downarrow &&\downarrow &&\downarrow \\1\to &1&\to &S_{3}&\to &A_{5}\end{matrix}}$
Note that the middle column is not exact: $C_{2}$ is not a normal subgroup in the semidirect product.
Since $A_{5}$ is simple, the right vertical arrow has trivial cokernel. Meanwhile the quotient group $S_{3}/C_{3}$ is isomorphic to $C_{2}$. The sequence in the statement of the snake lemma is therefore
$1\longrightarrow 1\longrightarrow 1\longrightarrow 1\longrightarrow C_{2}\longrightarrow 1$,
which indeed fails to be exact.
In popular culture
The proof of the snake lemma is taught by Jill Clayburgh's character at the very beginning of the 1980 film It's My Turn.[3]
See also
• Zig-zag lemma
References
1. Lang 2002, p. 159
2. "Extensions of C2 by C3". GroupNames. Retrieved 2021-11-06.
3. Schochet, C. L. (1999). "The Topological Snake Lemma and Corona Algebras" (PDF). New York Journal of Mathematics. 5: 131–7. CiteSeerX 10.1.1.73.1568. Archived (PDF) from the original on 2022-10-09.
• Lang, Serge (2002). "III §9 The Snake Lemma". Algebra (3rd ed.). Springer. pp. 157–9. ISBN 978-0-387-95385-4.
• Atiyah, M.F.; Macdonald, I. G. (1969). Introduction to Commutative Algebra. Addison–Wesley. ISBN 0-201-00361-9.
• Hilton, P.; Stammbach, U. (1997). A course in homological algebra. Graduate Texts in Mathematics. Springer. p. 99. ISBN 0-387-94823-6.
External links
• Weisstein, Eric W. "Snake Lemma". MathWorld.
• Snake Lemma at PlanetMath
• Proof of the Snake Lemma in the film It's My Turn
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Wikipedia
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SnapPea
SnapPea is free software designed to help mathematicians, in particular low-dimensional topologists, study hyperbolic 3-manifolds. The primary developer is Jeffrey Weeks, who created the first version[1] as part of his doctoral thesis,[2] supervised by William Thurston. It is not to be confused with the unrelated android malware with the same name.[3][4][5]
The latest version is 3.0d3. Marc Culler, Nathan Dunfield and collaborators have extended the SnapPea kernel and written Python extension modules which allow the kernel to be used in a Python program or in the interpreter. They also provide a graphical user interface written in Python which runs under most operating systems (see external links below).
The following people are credited in SnapPea 2.5.3's list of acknowledgments: Colin Adams, Bill Arveson, Pat Callahan, Joe Christy, Dave Gabai, Charlie Gunn, Martin Hildebrand, Craig Hodgson, Diane Hoffoss, A. C. Manoharan, Al Marden, Dick McGehee, Rob Meyerhoff, Lee Mosher, Walter Neumann, Carlo Petronio, Mark Phillips, Alan Reid, and Makoto Sakuma.
The C source code is extensively commented by Jeffrey Weeks and contains useful descriptions of the mathematics involved with references.
The SnapPeaKernel is released under GNU GPL 2+[6] as is SnapPy.[7]
Algorithms and functions
At the core of SnapPea are two main algorithms. The first attempts to find a minimal ideal triangulation of a given link complement. The second computes the canonical decomposition of a cusped hyperbolic 3-manifold. Almost all the other functions of SnapPea rely in some way on one of these decompositions.
Minimal ideal triangulation
SnapPea inputs data in a variety of formats. Given a link diagram, SnapPea can ideally triangulate the link complement. It then performs a sequence of simplifications to find a locally minimal ideal triangulation.
Once a suitable ideal triangulation is found, SnapPea can try to find a hyperbolic structure. In his Princeton lecture notes, Thurston noted a method for describing the geometric shape of each hyperbolic tetrahedron by a complex number and a set of nonlinear equations of complex variables whose solution would give a complete hyperbolic metric on the 3-manifold. These equations consist of edge equations and cusp (completeness) equations. SnapPea uses an iterative method utilizing Newton's method to search for solutions. If no solution exists, then this is reported to the user.
The local minimality of the triangulation is meant to increase the likelihood that such a solution exists, since heuristically one might expect such a triangulation to be "straightened" without causing degenerations or overlapping of tetrahedra.
From this description of the hyperbolic structure on a link complement, SnapPea can then perform hyperbolic Dehn filling on the cusps to obtain more hyperbolic 3-manifolds. SnapPea does this by taking any given slopes which determine certain Dehn filling equations (also explained in Thurston's notes), and then adjusting the shapes of the ideal tetrahedra to give solutions to these equations and the edge equations. For almost all slopes, this gives an incomplete hyperbolic structure on the link complement, whose completion gives a hyperbolic structure on the Dehn-filled manifold. Its volume is the sum of the volumes of the adjusted tetrahedra.
Canonical decomposition
SnapPea is usually able to compute the canonical decomposition of a cusped hyperbolic 3-manifold from a given ideal triangulation. If not, then it randomly retriangulates and tries again. This has never been known to fail.
The canonical decomposition allows SnapPea to tell two cusped hyperbolic 3-manifolds apart by turning the problem of recognition into a combinatorial question, i.e. checking if the two manifolds have combinatorially equivalent canonical decompositions. SnapPea is also able to check if two closed hyperbolic 3-manifolds are isometric by drilling out short geodesics to create cusped hyperbolic 3-manifolds and then using the canonical decomposition as before.
The recognition algorithm allow SnapPea to tell two hyperbolic knots or links apart. Weeks, et al., were also able to compile different censuses of hyperbolic 3-manifolds by using the algorithm to cull lists of duplicates.
Additionally, from the canonical decomposition, SnapPea is able to:
• Compute the Ford domain
• Compute the symmetry group
Computable invariants
Censuses
SnapPea has several databases of hyperbolic 3-manifolds available for systematic study.
• Cusped census
• Closed census
See also
• Regina incorporates aspects of SnapPea.
• Computational topology
References
1. Weeks, Jeffrey R., SnapPea C source code, (1999)
2. Weeks, Jeffrey R., Convex hulls and isometries of cusped hyperbolic $3$-manifolds. Topology Appl. 52 (1993), no. 2, 127—149.
3. Fox-Brewster, Thomas. "Android 'Gooligan' Hackers Just Scored The Biggest Ever Theft Of Google Accounts". forbes.com. Retrieved 21 May 2017.
4. "Adware or APT – SnapPea Downloader - An Android Malware that implements 12 different exploits". Check Point Blog. 10 July 2015. Retrieved 21 May 2017.
5. "How to Manage Your Android Device from Windows with SnapPea". howtogeek.com. Retrieved 21 May 2017.
6. ReadMe file for the SnapPea kernel, accessed 2013-09-06.
7. "SnapPy — SnapPy 2.1 documentation". Math.uic.edu. Retrieved 2014-03-12.
External links
• SnapPea Jeff Weeks' site
• SnapPy Culler and Dunfield's extension
• Orb Damian Heard's extension, allows :
• hyperbolic manifolds with totally geodesic boundary
• orbifolds where the orbifold locus contains trivalent vertices
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Wikipedia
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Snark (graph theory)
In the mathematical field of graph theory, a snark is an undirected graph with exactly three edges per vertex whose edges cannot be colored with only three colors. In order to avoid trivial cases, snarks are often restricted to have additional requirements on their connectivity and on the length of their cycles. Infinitely many snarks exist.
One of the equivalent forms of the four color theorem is that every snark is a non-planar graph. Research on snarks originated in Peter G. Tait's work on the four color theorem in 1880, but their name is much newer, given to them by Martin Gardner in 1976. Beyond coloring, snarks also have connections to other hard problems in graph theory: writing in the Electronic Journal of Combinatorics, Miroslav Chladný and Martin Škoviera state that
In the study of various important and difficult problems in graph theory (such as the cycle double cover conjecture and the 5-flow conjecture), one encounters an interesting but somewhat mysterious variety of graphs called snarks. In spite of their simple definition...and over a century long investigation, their properties and structure are largely unknown.[1]
As well as the problems they mention, W. T. Tutte's snark conjecture concerns the existence of Petersen graphs as graph minors of snarks; its proof has been long announced but remains unpublished, and would settle a special case of the existence of nowhere zero 4-flows.
History and examples
Snarks were so named by the American mathematician Martin Gardner in 1976, after the mysterious and elusive object of the poem The Hunting of the Snark by Lewis Carroll.[2] However, the study of this class of graphs is significantly older than their name. Peter G. Tait initiated the study of snarks in 1880, when he proved that the four color theorem is equivalent to the statement that no snark is planar.[3] The first graph known to be a snark was the Petersen graph; it was proved to be a snark by Julius Petersen in 1898,[4] although it had already been studied for a different purpose by Alfred Kempe in 1886.[5]
The next four known snarks were
• the Blanuša snarks (two with 18 vertices), discovered by Danilo Blanuša in 1946,[6]
• the Descartes snark (210 vertices), discovered by Bill Tutte in 1948,[7] and
• the Szekeres snark (50 vertices), discovered by George Szekeres in 1973.[8]
In 1975, Rufus Isaacs generalized Blanuša's method to construct two infinite families of snarks: the flower snarks and the Blanuša–Descartes–Szekeres snarks, a family that includes the two Blanuša snarks, the Descartes snark and the Szekeres snark. Isaacs also discovered a 30-vertex snark that does not belong to the Blanuša–Descartes–Szekeres family and that is not a flower snark: the double-star snark.[9] The 50-vertex Watkins snark was discovered in 1989.[10]
Another notable cubic non-three-edge-colorable graph is Tietze's graph, with 12 vertices; as Heinrich Franz Friedrich Tietze discovered in 1910, it forms the boundary of a subdivision of the Möbius strip requiring six colors.[11] However, because it contains a triangle, it is not generally considered a snark. Under strict definitions of snarks, the smallest snarks are the Petersen graph and Blanuša snarks, followed by six different 20-vertex snarks.[12]
A list of all of the snarks up to 36 vertices (according to a strict definition), and up to 34 vertices (under a weaker definition), was generated by Gunnar Brinkmann, Jan Goedgebeur, Jonas Hägglund and Klas Markström in 2012.[12] The number of snarks for a given even number of vertices grows at least exponentially in the number of vertices.[13] (Because they have odd-degree vertices, all snarks must have an even number of vertices by the handshaking lemma.)[14] OEIS sequence A130315 contains the number of non-trivial snarks of $2n$ vertices for small values of $n$.[15]
Definition
The precise definition of snarks varies among authors,[12][9] but generally refers to cubic graphs (having exactly three edges at each vertex) whose edges cannot be colored with only three colors. By Vizing's theorem, the number of colors needed for the edges of a cubic graph is either three ("class one" graphs) or four ("class two" graphs), so snarks are cubic graphs of class two. However, in order to avoid cases where a snark is of class two for trivial reasons, or is constructed in a trivial way from smaller graphs, additional restrictions on connectivity and cycle lengths are often imposed. In particular:
• If a cubic graph has a bridge, an edge whose removal would disconnect it, then it cannot be of class one. By the handshaking lemma, the subgraphs on either side of the bridge have an odd number of vertices each. Whichever of three colors is chosen for the bridge, their odd number of vertices prevents these subgraphs from being covered by cycles that alternate between the other two colors, as would be necessary in a 3-edge-coloring. For this reason, snarks are generally required to be bridgeless.[2][9]
• A loop (an edge connecting a vertex to itself) cannot be colored without causing the same color to appear twice at that vertex, a violation of the usual requirements for graph edge coloring. Additionally, a cycle consisting of two vertices connected by two edges can always be replaced by a single edge connecting their two other neighbors, simplifying the graph without changing its three-edge-colorability. For these reasons, snarks are generally restricted to simple graphs, graphs without loops or multiple adjacencies.[9]
• If a graph contains a triangle, then it can again be simplified without changing its three-edge-colorability, by contracting the three vertices of the triangle into a single vertex. Therefore, many definitions of snarks forbid triangles.[9] However, although this requirement was also stated in Gardner's work giving the name "snark" to these graphs, Gardner lists Tietze's graph, which contains a triangle, as being a snark.[2]
• If a graph contains a four-vertex cycle, it can be simplified in two different ways by removing two opposite edges of the cycle and replacing the resulting paths of degree-two vertices by single edges. It has a three-edge-coloring if and only if at least one of these simplifications does. Therefore, Isaacs requires a "nontrivial" cubic class-two graph to avoid four-vertex cycles,[9] and other authors have followed suit in forbidding these cycles.[12] The requirement that a snark avoid cycles of length four or less can be summarized by stating that the girth of these graphs, the length of their shortest cycles, is at least five.
• More strongly, the definition used by Brinkmann et al. (2012) requires snarks to be cyclically 4-edge-connected. That means there can be no subset of three or fewer edges, the removal of which would disconnect the graph into two subgraphs each of which has at least one cycle. Brinkmann et al. define a snark to be a cubic and cyclically 4-edge-connected graph of girth five or more and class two; they define a "weak snark" to allow girth four.[12]
Although these definitions only consider constraints on the girth up to five, snarks with arbitrarily large girth exist.[16]
Properties
Work by Peter G. Tait established that the four-color theorem is true if and only if every snark is non-planar.[3] This theorem states that every planar graph has a graph coloring of its the vertices with four colors, but Tait showed how to convert 4-vertex-colorings of maximal planar graphs into 3-edge-colorings of their dual graphs, which are cubic and planar, and vice versa. A planar snark would therefore necessarily be dual to a counterexample to the four-color theorem. Thus, the subsequent proof of the four-color theorem[17] also demonstrates that all snarks are non-planar.[18]
All snarks are non-Hamiltonian: when a cubic graph has a Hamiltonian cycle, it is always possible to 3-color its edges, by using two colors in alternation for the cycle, and the third color for the remaining edges. However, many known snarks are close to being Hamiltonian, in the sense that they are hypohamiltonian graphs: the removal of any single vertex leaves a Hamiltonian subgraph. A hypohamiltonian snark must be bicritical: the removal of any two vertices leaves a three-edge-colorable subgraph.[19] The oddness of a cubic graph is defined as the minimum number of odd cycles, in any system of cycles that covers each vertex once (a 2-factor). For the same reason that they have no Hamiltonian cycles, snarks have positive oddness: a completely even 2-factor would lead to a 3-edge-coloring, and vice versa. It is possible to construct infinite families of snarks whose oddness grows linearly with their numbers of vertices.[14]
The cycle double cover conjecture posits that in every bridgeless graph one can find a collection of cycles covering each edge twice, or equivalently that the graph can be embedded onto a surface in such a way that all faces of the embedding are simple cycles. When a cubic graph has a 3-edge-coloring, it has a cycle double cover consisting of the cycles formed by each pair of colors. Therefore, among cubic graphs, the snarks are the only possible counterexamples. More generally, snarks form the difficult case for this conjecture: if it is true for snarks, it is true for all graphs.[20] In this connection, Branko Grünbaum conjectured that no snark could be embedded onto a surface in such a way that all faces are simple cycles and such that every two faces either are disjoint or share only a single edge; if any snark had such an embedding, its faces would form a cycle double cover. However, a counterexample to Grünbaum's conjecture was found by Martin Kochol.[21]
Determining whether a given cyclically 5-connected cubic graph is 3-edge-colorable is NP-complete. Therefore, determining whether a graph is a snark is co-NP-complete.[22]
Snark conjecture
W. T. Tutte conjectured that every snark has the Petersen graph as a minor. That is, he conjectured that the smallest snark, the Petersen graph, may be formed from any other snark by contracting some edges and deleting others. Equivalently (because the Petersen graph has maximum degree three) every snark has a subgraph that can be formed from the Petersen graph by subdividing some of its edges. This conjecture is a strengthened form of the four color theorem, because any graph containing the Petersen graph as a minor must be nonplanar. In 1999, Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas announced a proof of this conjecture.[23] Steps towards this result have been published,[24][25] but as of 2022, the complete proof remains unpublished.[18][24] See the Hadwiger conjecture for other problems and results relating graph coloring to graph minors.
Tutte also conjectured a generalization to arbitrary graphs: every bridgeless graph with no Petersen minor has a nowhere zero 4-flow. That is, the edges of the graph may be assigned a direction, and a number from the set {1, 2, 3}, such that the sum of the incoming numbers minus the sum of the outgoing numbers at each vertex is divisible by four. As Tutte showed, for cubic graphs such an assignment exists if and only if the edges can be colored by three colors, so the conjecture would follow from the snark conjecture in this case. However, proving the snark conjecture would not settle the question of the existence of 4-flows for non-cubic graphs.[26]
References
1. Chladný, Miroslav; Škoviera, Martin (2010), "Factorisation of snarks", Electronic Journal of Combinatorics, 17: R32, doi:10.37236/304, MR 2595492
2. Gardner, Martin (1976), "Snarks, boojums, and other conjectures related to the four-color-map theorem", Mathematical Games, Scientific American, 4 (234): 126–130, Bibcode:1976SciAm.234d.126G, doi:10.1038/scientificamerican0476-126, JSTOR 24950334
3. Tait, Peter Guthrie (1880), "Remarks on the colourings of maps", Proceedings of the Royal Society of Edinburgh, 10: 729, doi:10.1017/S0370164600044643
4. Petersen, Julius (1898), "Sur le théorème de Tait", L'Intermédiaire des Mathématiciens, 5: 225–227
5. Kempe, A. B. (1886), "A memoir on the theory of mathematical form", Philosophical Transactions of the Royal Society of London, 177: 1–70, doi:10.1098/rstl.1886.0002, S2CID 108716533
6. Blanuša, Danilo (1946), "Le problème des quatre couleurs", Glasnik Matematičko-Fizički i Astronomski, Ser. II, 1: 31–42, MR 0026310
7. Descartes, Blanche (1948), "Network-colourings", The Mathematical Gazette, 32 (299): 67–69, doi:10.2307/3610702, JSTOR 3610702, MR 0026309, S2CID 250434686
8. Szekeres, George (1973), "Polyhedral decompositions of cubic graphs", Bulletin of the Australian Mathematical Society, 8 (3): 367–387, doi:10.1017/S0004972700042660
9. Isaacs, Rufus (1975), "Infinite families of non-trivial trivalent graphs which are not Tait-colorable", The American Mathematical Monthly, 82 (3): 221–239, doi:10.2307/2319844, JSTOR 2319844
10. Watkins, John J. (1989), "Snarks", in Capobianco, Michael F.; Guan, Mei Gu; Hsu, D. Frank; Tian, Feng (eds.), Graph theory and its Applications: East and West, Proceedings of the First China–USA International Conference held in Jinan, June 9–20, 1986, Annals of the New York Academy of Sciences, vol. 576, New York: New York Academy of Sciences, pp. 606–622, doi:10.1111/j.1749-6632.1989.tb16441.x, MR 1110857, S2CID 222072657
11. Tietze, Heinrich (1910), "Einige Bemerkungen zum Problem des Kartenfärbens auf einseitigen Flächen" [Some remarks on the problem of map coloring on one-sided surfaces] (PDF), DMV Annual Report, 19: 155–159
12. Brinkmann, Gunnar; Goedgebeur, Jan; Hägglund, Jonas; Markström, Klas (2012), "Generation and properties of snarks", Journal of Combinatorial Theory, Series B, 103 (4): 468–488, arXiv:1206.6690, doi:10.1016/j.jctb.2013.05.001, MR 3071376, S2CID 15284747
13. Skupień, Zdzisław (2007), "Exponentially many hypohamiltonian snarks", 6th Czech-Slovak International Symposium on Combinatorics, Graph Theory, Algorithms and Applications, Electronic Notes in Discrete Mathematics, vol. 28, pp. 417–424, doi:10.1016/j.endm.2007.01.059
14. Lukot'ka, Robert; Máčajová, Edita; Mazák, Ján; Škoviera, Martin (2015), "Small snarks with large oddness", Electronic Journal of Combinatorics, 22 (1), Paper 1.51, arXiv:1212.3641, doi:10.37236/3969, MR 3336565, S2CID 4805178
15. Sloane, N. J. A. (ed.), "Sequence A130315", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
16. Kochol, Martin (1996), "Snarks without small cycles", Journal of Combinatorial Theory, Series B, 67 (1): 34–47, doi:10.1006/jctb.1996.0032, MR 1385382
17. Appel, Kenneth; Haken, Wolfgang (1989), Every Planar Map is Four-Colorable, Contemporary Mathematics, vol. 98, With the collaboration of J. Koch., Providence, RI: American Mathematical Society, doi:10.1090/conm/098, ISBN 0-8218-5103-9, MR 1025335, S2CID 8735627
18. belcastro, sarah-marie (2012), "The continuing saga of snarks", The College Mathematics Journal, 43 (1): 82–87, doi:10.4169/college.math.j.43.1.082, MR 2875562, S2CID 118189042
19. Steffen, E. (1998), "Classification and characterizations of snarks", Discrete Mathematics, 188 (1–3): 183–203, doi:10.1016/S0012-365X(97)00255-0, MR 1630478; Steffen, E. (2001), "On bicritical snarks", Math. Slovaca, 51 (2): 141–150, MR 1841443
20. Jaeger, François (1985), "A survey of the cycle double cover conjecture", in Alspach, B. R.; Godsil, C. D. (eds.), Annals of Discrete Mathematics 27: Cycles in Graphs, North-Holland Mathematics Studies, vol. 27, pp. 1–12, doi:10.1016/S0304-0208(08)72993-1, ISBN 978-0-444-87803-8
21. Kochol, Martin (2009), "Polyhedral embeddings of snarks in orientable surfaces", Proceedings of the American Mathematical Society, vol. 137, pp. 1613–1619
22. Kochol, Martin (2010), "Complexity of 3-edge-coloring in the class of cubic graphs with a polyhedral embedding in an orientable surface", Discrete Applied Mathematics, 158 (16): 1856–1860, doi:10.1016/j.dam.2010.06.019, MR 2679785
23. Thomas, Robin (1999), "Recent excluded minor theorems for graphs" (PDF), Surveys in Combinatorics, 1999, Cambridge University Press, pp. 201–222
24. Edwards, Katherine; Sanders, Daniel P.; Seymour, Paul; Thomas, Robin (2016), "Three-edge-colouring doublecross cubic graphs" (PDF), Journal of Combinatorial Theory, Series B, 119: 66–95, doi:10.1016/j.jctb.2015.12.006, MR 3486338, S2CID 2656843
25. Robertson, Neil; Seymour, Paul; Thomas, Robin (2019), "Excluded minors in cubic graphs", Journal of Combinatorial Theory, Series B, 138: 219–285, arXiv:1403.2118, doi:10.1016/j.jctb.2019.02.002, MR 3979232, S2CID 16237685
26. DeVos, Matthew (March 7, 2007), "4-flow conjecture", Open Problem Garden
External links
Wikimedia Commons has media related to Snarks (graph theory).
• Weisstein, Eric W., "Snark", MathWorld
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Wikipedia
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Snedecor Award
The Snedecor Award, named after George W. Snedecor, is given by the Committee of Presidents of Statistical Societies to a statistician for contribution to biometry.
Winners
• 1977: A. P. Dawid
• 1978: Bruce W. Turnbull
• 1979: Ethel S. Gilbert
• 1981: Barry H. Margolin
• 1982: Byron J. T. Morgan
• 1983: D. S. Robson
• 1984: Stuart H. Hurlbert
• 1985: Mitchell H. Gail
• 1986: Scott L. Zeger
• 1987: George E. Bonney
• 1988: Cyrus R. Mehta
• 1989: Barry I. Graubard
• 1990: Kenneth H. Pollock
• 1993: Kenneth L. Lange
• 1995: Norman E. Breslow
• 1997: Michael A. Newton
• 1999: Daniel Scharfstein
• 2001: Patrick J. Heagerty
• 2003: Paul R. Rosenbaum
• 2005: Nicholas P. Jewell
• 2007: Donald Rubin
• 2009: Marie Davidian
• 2011: Nilanjan Chatterjee
• 2013: Jack Kalbfleisch
• 2015: Danyu Lin
• 2017: Aurore Delaigle
• 2019: Sudipto Banerjee
• 2021: David Dunson
• 2023: Michael Kosorok[1]
See also
• List of mathematics awards
References
1. Doherty, Brennan (2023-04-10). "Public health professor wins biometry award". The Well. Retrieved 2023-05-01.
External links
• Snedecor Award website
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Snell envelope
The Snell envelope, used in stochastics and mathematical finance, is the smallest supermartingale dominating a stochastic process. The Snell envelope is named after James Laurie Snell.
Definition
Given a filtered probability space $(\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\in [0,T]},\mathbb {P} )$ and an absolutely continuous probability measure $\mathbb {Q} \ll \mathbb {P} $ then an adapted process $U=(U_{t})_{t\in [0,T]}$ is the Snell envelope with respect to $\mathbb {Q} $ of the process $X=(X_{t})_{t\in [0,T]}$ if
1. $U$ is a $\mathbb {Q} $-supermartingale
2. $U$ dominates $X$, i.e. $U_{t}\geq X_{t}$ $\mathbb {Q} $-almost surely for all times $t\in [0,T]$
3. If $V=(V_{t})_{t\in [0,T]}$ is a $\mathbb {Q} $-supermartingale which dominates $X$, then $V$ dominates $U$.[1]
Construction
Given a (discrete) filtered probability space $(\Omega ,{\mathcal {F}},({\mathcal {F}}_{n})_{n=0}^{N},\mathbb {P} )$ and an absolutely continuous probability measure $\mathbb {Q} \ll \mathbb {P} $ then the Snell envelope $(U_{n})_{n=0}^{N}$ with respect to $\mathbb {Q} $ of the process $(X_{n})_{n=0}^{N}$ is given by the recursive scheme
$U_{N}:=X_{N},$
$U_{n}:=X_{n}\lor \mathbb {E} ^{\mathbb {Q} }[U_{n+1}\mid {\mathcal {F}}_{n}]$ for $n=N-1,...,0$
where $\lor $ is the join (in this case equal to the maximum of the two random variables).[1]
Application
• If $X$ is a discounted American option payoff with Snell envelope $U$ then $U_{t}$ is the minimal capital requirement to hedge $X$ from time $t$ to the expiration date.[1]
References
1. Föllmer, Hans; Schied, Alexander (2004). Stochastic finance: an introduction in discrete time (2 ed.). Walter de Gruyter. pp. 280–282. ISBN 9783110183467.
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Snellius–Pothenot problem
The Snellius–Pothenot problem is a problem in planar surveying. Given three known points A, B and C, an observer at an unknown point P observes that the segment AC subtends an angle $\alpha $ and the segment CB subtends an angle $\beta $; the problem is to determine the position of the point P. (See figure; the point denoted C is between A and B as seen from P).
Since it involves the observation of known points from an unknown point, the problem is an example of resection. Historically it was first studied by Snellius, who found a solution around 1615.
Formulating the equations
First equation
Denoting the (unknown) angles CAP as x and CBP as y gives:
$x+y=2\pi -\alpha -\beta -C$
by using the sum of the angles formula for the quadrilateral PACB. The variable C represents the (known) internal angle in this quadrilateral at point C. (Note that in the case where the points C and P are on the same side of the line AB, the angle C will be greater than $\pi $).
Second equation
Applying the law of sines in triangles PAC and PBC we can express PC in two different ways:
${\frac {{\rm {AC}}\sin x}{\sin \alpha }}={\rm {PC}}={\frac {{\rm {BC}}\sin y}{\sin \beta }}.$
A useful trick at this point is to define an auxiliary angle $\phi $ such that
$\tan \phi ={\frac {{\rm {BC}}\sin \alpha }{{\rm {AC}}\sin \beta }}.$
(A minor note: one should be concerned about division by zero, but consider that the problem is symmetric, so if one of the two given angles is zero one can, if needed, rename that angle alpha and call the other (non-zero) angle beta, reversing the roles of A and B as well. This will suffice to guarantee that the ratio above is well defined. An alternative approach to the zero angle problem is given in the algorithm below.)
With this substitution the equation becomes
${\frac {\sin x}{\sin y}}=\tan \phi .$
Now two known trigonometric identities can be used, namely
$\tan \left({\frac {\pi }{4}}-\phi \right)={\frac {1-\tan \phi }{\tan \phi +1}}$ and
${\frac {\tan[(x-y)/2]}{\tan[(x+y)/2]}}={\frac {\sin x-\sin y}{\sin x+\sin y}}$
to put this in the form of the second equation;
$\tan {\frac {1}{2}}(x-y)=\tan {\frac {1}{2}}(\alpha +\beta +C)\tan \left({\frac {\pi }{4}}-\phi \right).$
Now these two equations in two unknowns must be solved. Once x and y are known the various triangles can be solved straightforwardly to determine the position of P.[1] The detailed procedure is shown below.
Solution algorithm
Given are two lengths AC and BC, and three angles $\alpha $, $\beta $ and C, the solution proceeds as follows.
• calculate $\phi =\operatorname {atan2} ({\rm {BC}}\sin \alpha ,{\rm {AC}}\sin \beta )$. Where atan2 is a computer function, also called the arctangent of two arguments, that returns the arctangent of the ratio of the two values given. Note that in Microsoft Excel the two arguments are reversed, so the proper syntax would be '$={\rm {atan2(AC^{*}\backslash \sin(beta),BC^{*}\backslash \sin(alpha))}}$'. The atan2 function correctly handles the case where one of the two arguments is zero.
• calculate $K=2\pi -\alpha -\beta -C.$
• calculate $W=2\cdot \operatorname {atan} \left[\tan(\pi /4-\phi )\tan \left({\frac {1}{2}}(\alpha +\beta +C)\right)\right].$
• find $x=(K+W)/2$ and $y=(K-W)/2.$
• if $|\sin \beta |>|\sin \alpha |$ calculate ${\rm {PC}}={\frac {{\rm {BC}}\sin y}{\sin \beta }}$ else use ${\rm {PC}}={\frac {{\rm {AC}}\sin x}{\sin \alpha }}.$
• find ${\rm {PA}}={\sqrt {{\rm {AC}}^{2}+{\rm {PC}}^{2}-2\cdot {\rm {AC}}\cdot {\rm {PC}}\cdot \cos(\pi -\alpha -x)}}.$ (This comes from the law of cosines.)
• find ${\rm {PB}}={\sqrt {{\rm {BC}}^{2}+{\rm {PC}}^{2}-2\cdot {\rm {BC}}\cdot {\rm {PC}}\cdot \cos(\pi -\beta -y)}}.$
If the coordinates of $A:x_{A},y_{A}$ and $C:x_{C},y_{C}$ are known in some appropriate Cartesian coordinate system then the coordinates of $P$ can be found as well.
Geometric (graphical) solution
By the inscribed angle theorem the locus of points from which AC subtends an angle $\alpha $ is a circle having its center on the midline of AC; from the center O of this circle AC subtends an angle $2\alpha $. Similarly the locus of points from which CB subtends an angle $\beta $ is another circle. The desired point P is at the intersection of these two loci.
Therefore, on a map or nautical chart showing the points A, B, C, the following graphical construction can be used:
• Draw the segment AC, the midpoint M and the midline, which crosses AC perpendicularly at M. On this line find the point O such that $MO={\frac {AC}{2\tan \alpha }}$. Draw the circle with center at O passing through A and C.
• Repeat the same construction with points B, C and the angle $\beta $.
• Mark P at the intersection of the two circles (the two circles intersect at two points; one intersection point is C and the other is the desired point P.)
This method of solution is sometimes called Cassini's method.
Rational trigonometry approach
The following solution is based upon a paper by N. J. Wildberger.[2] It has the advantage that it is almost purely algebraic. The only place trigonometry is used is in converting the angles to spreads. There is only one square root required.
• define the following:
• $s(x)=\sin ^{2}(x)$
• $A(x,y,z)=(x+y+z)^{2}-2(x^{2}+y^{2}+z^{2})$
• $r_{1}=s(\beta )$
• $r_{2}=s(\alpha )$
• $r_{3}=s(\alpha +\beta )$
• $Q_{1}=BC^{2}$
• $Q_{2}=AC^{2}$
• $Q_{3}=AB^{2}$
• now let:
• $R_{1}=r_{2}Q_{3}/r_{3}$
• $R_{2}=r_{1}Q_{3}/r_{3}$
• $C_{0}=((Q_{1}+Q_{2}+Q_{3})(r_{1}+r_{2}+r_{3})-2(Q_{1}r_{1}+Q_{2}r_{2}+Q_{3}r_{3}))/(2r_{3})$
• $D_{0}=r_{1}r_{2}A(Q_{1},Q_{2},Q_{3})/r_{3}$
• the following equation gives two possible values for $R_{3}$:
• $(R_{3}-C_{0})^{2}=D_{0}$
• choosing the larger of these values, let:
• $v_{1}=1-(R_{1}+R_{3}-Q_{2})^{2}/(4R_{1}R_{3})$
• $v_{2}=1-(R_{2}+R_{3}-Q_{1})^{2}/(4R_{2}R_{3})$
• finally:
• $AP^{2}=v_{1}R_{1}/r_{2}=v_{1}Q_{3}/r_{3}$
• $BP^{2}=v2R_{2}/r_{1}=v_{2}Q_{3}/r_{3}$
The indeterminate case
When the point P happens to be located on the same circle as A, B and C, the problem has an infinite number of solutions; the reason is that from any other point P' located on the arc APB of this circle the observer sees the same angles alpha and beta as from P (inscribed angle theorem). Thus the solution in this case is not uniquely determined.
The circle through ABC is known as the "danger circle", and observations made on (or very close to) this circle should be avoided. It is helpful to plot this circle on a map before making the observations.
A theorem on cyclic quadrilaterals is helpful in detecting the indeterminate situation. The quadrilateral APBC is cyclic iff a pair of opposite angles (such as the angle at P and the angle at C) are supplementary i.e. iff $\alpha +\beta +C=k\pi ,(k=1,2,\cdots )$. If this condition is observed the computer/spreadsheet calculations should be stopped and an error message ("indeterminate case") returned.
Solved examples
(Adapted form Bowser,[3] exercise 140, page 203). A, B and C are three objects such that AC = 435 (yards), CB = 320, and C = 255.8 degrees. From a station P it is observed that APC = 30 degrees and CPB = 15 degrees. Find the distances of P from A, B and C. (Note that in this case the points C and P are on the same side of the line AB, a different configuration from the one shown in the figure).
Answer: PA = 790, PB = 777, PC = 502.
A slightly more challenging test case for a computer program uses the same data but this time with CPB = 0. The program should return the answers 843, 1157 and 837.
Naming controversy
The British authority on geodesy, George Tyrrell McCaw (1870–1942) wrote that the proper term in English was Snellius problem, while Snellius-Pothenot was the continental European usage.[4]
McCaw thought the name of Laurent Pothenot (1650–1732) did not deserve to be included as he had made no original contribution, but merely restated Snellius 75 years later.
See also
• Solution of triangles
• Triangulation (surveying)
Notes
1. Bowser: A treatise
2. Norman J. Wildberger (2010). "Greek Geometry, Rational Trigonometry, and the Snellius – Pothenot Surveying Problem" (PDF). Chamchuri Journal of Mathematics. 2 (2): 1–14.
3. Bowser: A treatise
4. McCaw, G. T. (1918). "Resection in Survey". The Geographical Journal. 52 (2): 105–126. doi:10.2307/1779558. JSTOR 1779558.
• Gerhard Heindl: Analysing Willerding’s formula for solving the planar three point resection problem, Journal of Applied Geodesy, Band 13, Heft 1, Seiten 27–31, ISSN (Online) 1862-9024, ISSN (Print) 1862-9016, DOI:
References
• Edward A. Bowser: A treatise on plane and spherical trigonometry, Washington D.C., Heath & Co., 1892, page 188 Google books
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Wikipedia
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Snezana Lawrence
Snezana Lawrence FIMA is a Yugoslav and British historian of mathematics and a senior lecturer in mathematics and design engineering at Middlesex University.[1]
Education and career
Lawrence is originally from Yugoslavia, of mixed Serbian and Jewish ancestry.[2] She studied descriptive geometry at the University of Belgrade before moving to England in 1991 during the Breakup of Yugoslavia and ensuing Yugoslav Wars, and later becoming a naturalized British citizen.[3] She earned her PhD from the Open University in 2002. Her dissertation, Geometry of Architecture and Freemasonry in 19th Century England, was supervised by Jeremy Gray.[4]
While working as a secondary school teacher at St Edmund's Catholic School, Dover in 2004–2005, she won a Gatsby Teacher Fellowship in Mathematics, with which she started a popular web site "Maths is Good For You". The site had the aim of providing a resource to bring more work on the history of mathematics into the secondary school curriculum.[5]
Subsequently, Lawrence moved to post-secondary education, including work as a senior lecturer at Bath Spa University,[6] Anglia Ruskin University,[7] and Middlesex University.[1]
Books
Lawrence is the co-editor, with Irish mathematician Mark McCartney, of the book Mathematicians and their Gods: Interactions between mathematics and religious beliefs (Oxford University Press, 2015), on connections between mathematics and religion.[8] She is the author of A New Year’s Present from a Mathematician (Chapman Hall / CRC Press, 2019), on the nature of mathematics and the definition of mathematicians.[9]
Recognition
Lawrence is a Fellow of the Institute of Mathematics and its Applications, for whom she is Diversity Champion and an elected council member.[10]
References
1. "Dr Snezana Lawrence", Academic and research staff, Middlesex University, retrieved 2020-09-09
2. Lawrence, Snezana, Diamond, Hanna (ed.), "Anka's Escape from Belgrade", Fleeing Hitler
3. "About...", The Monge project, archived from the original on 2022-10-02
4. Snezana Lawrence at the Mathematics Genealogy Project
5. Lawrence, Snezana (July 2006), "Maths is good for you: web-based history of mathematics resources for young mathematicians (and their teachers)", BSHM Bulletin: Journal of the British Society for the History of Mathematics, Informa {UK} Limited, 21 (2): 90–96, doi:10.1080/17498430600803375, S2CID 122851275
6. Author profile from Mathematicians and their Gods
7. ORCID profile, retrieved 2020-09-09
8. Reviews of Mathematicians and their Gods:
• Luciano, Erika, zbMATH, Zbl 1365.01002{{citation}}: CS1 maint: untitled periodical (link)
• Bultheel, Adhemar (September 2015), "Review", EMS Reviews, European Mathematical Society
• Sawyer, Megan (November 2015), "Review", MAA Reviews, Mathematical Association of America
• Toller, Owen (June 2016), The Mathematical Gazette, 100 (548): 368–370, doi:10.1017/mag.2016.86, S2CID 184146073{{citation}}: CS1 maint: untitled periodical (link)
• Whiteman, Jamie A. (August 2016), "For your information", The Mathematics Teacher, 110 (1): 79, doi:10.5951/mathteacher.110.1.0078, JSTOR 10.5951/mathteacher.110.1.0078
• Muntersbjorn, Madeline (September 2016), HOPOS: The Journal of the International Society for the History of Philosophy of Science, University of Chicago Press, 6 (2): 333–336, doi:10.1086/687777{{citation}}: CS1 maint: untitled periodical (link)
• Howell, Russell W. (August 2018), Historia Mathematica, 45 (3): 300–302, doi:10.1016/j.hm.2018.06.002{{citation}}: CS1 maint: untitled periodical (link)
9. Review of A New Year’s Present from a Mathematician:
• Caulfield, Michael (August 2020), "Review", MAA Reviews, Mathematical Association of America
10. "Snezana Lawrence", Mathematics Today Editorial Board, Institute of Mathematics and its Applications, retrieved 2020-09-09
External links
• Math is good for you
• The Monge project
Authority control
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• ISNI
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• BnF data
• Germany
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Other
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Snezhana Abarzhi
Snezhana I. Abarzhi (also known as Snejana I. Abarji)[1] is an applied mathematician and mathematical physicist from the former Soviet Union specializing in the dynamics of fluids and plasmas and their applications in nature and technology. Her research has indicated that supernovas explode more slowly and less turbulently than previously thought, changing the understanding of the mechanisms by which heavy atomic nuclei are formed in these explosions.[2] She is Professor and Chair of Applied Mathematics at the University of Western Australia.[1]
Snezhana I. Abarzhi
Education
• Moscow Institute of Physics and Technology
• Russian Academy of Sciences
Known forResearch in fluid instabilities, interfaces, and mixing
Scientific career
Fields
• Applied mathematics
• Mathematical physics
Institutions
• Russian Academy of Sciences
• UNC Chapel Hill
• University of Bayreuth
• Stony Brook University
• Osaka University
• Stanford University
• University of Chicago
• Illinois Inst. Tech.
• Carnegie Mellon University
• University of Western Australia
Education and career
Abarzhi earned bachelor's degrees in physics and applied mathematics and in molecular biology in 1987 from the Moscow Institute of Physics and Technology, and earned a master's degree in physics and applied mathematics there, summa cum laude, in 1990. She completed her doctorate in 1994 through the Landau Institute for Theoretical Physics and Kapitza Institute for Physical Problems of the Russian Academy of Sciences, supervised by Sergei I. Anisimov.[3]
Although she held a position as a researcher for the Russian Academy of Sciences from 1994 to 2004, she came to the US in 1997 as a visiting professor at the University of North Carolina at Chapel Hill, and then in 1998 became an Alexander von Humboldt Fellow at the University of Bayreuth in Germany. In 1999 she took a research position at Stony Brook University. In 2002 she briefly moved to a research professorship at Osaka University before returning to the US as a senior fellow in the Center for Turbulence Research at Stanford University. In 2005 she became a research faculty member at the University of Chicago and in 2006 she added a regular-rank faculty position as an associate professor at the Illinois Institute of Technology.[3] She also worked at Carnegie Mellon University from 2013 to 2016 before moving to the University of Western Australia.[4]
Recognition
In 2020 Abarzhi was named a Fellow of the American Physical Society (APS), after a nomination from the APS Division of Fluid Dynamics, "for deep and abiding work on the Rayleigh-Taylor and related instabilities, and for sustained leadership in that community".[5][6]
Selected publications
• Abarzhi, Snezhana I.; Bhowmick, Aklant K.; Naveh, Annie; Pandian, Arun; Swisher, Nora C.; Stellingwerf, Robert F.; Arnett, W. David (November 2018), "Supernova, nuclear synthesis, fluid instabilities, and interfacial mixing", Proceedings of the National Academy of Sciences, 116 (37): 18184–18192, doi:10.1073/pnas.1714502115, PMC 6744890, PMID 30478062
• Abarzhi, Snezhana I.; Ilyin, Daniil V.; Goddard, William A.; Anisimov, Sergei I. (August 2018), "Interface dynamics: Mechanisms of stabilization and destabilization and structure of flow fields", Proceedings of the National Academy of Sciences, 116 (37): 18218–18226, doi:10.1073/pnas.1714500115, PMC 6744915, PMID 30082395
• Abarzhi, Snezhana I. (April 2010), "Review of theoretical modelling approaches of Rayleigh–Taylor instabilities and turbulent mixing", Philosophical Transactions of the Royal Society A, 368 (1916): 1809–1828, Bibcode:2010RSPTA.368.1809A, doi:10.1098/rsta.2010.0020, PMID 20211884, S2CID 38628393
• Abarzhi, Snezhana I. (July 1998), "Stable steady flows in Rayleigh–Taylor instability", Physical Review Letters, 81 (2): 337–340, Bibcode:1998PhRvL..81..337A, doi:10.1103/physrevlett.81.337
References
1. "Q&A with the ANZIAM 2018 female plenary speakers: Snezhana Abarzhi", ANZIAM 2018, Australian Mathematical Society, retrieved 2020-11-06
2. "New evidence reveals how life was created after the big bang", Scimex, 27 November 2018
3. Curriculum vitae, archived from the original on 2010-06-20
4. Abarzhi, Snezhana I.; Ilyin, Daniil V.; Goddard, William A.; Anisimov, Sergei I. (2019), "Interface dynamics: Mechanisms of stabilization and destabilization and structure of flow fields", CTR tea talk announcement and speaker biography, Center for Turbulence Research, Stanford University, vol. 116, no. 37, p. 18218, Bibcode:2019PNAS..11618218A, doi:10.1073/pnas.1714500115, retrieved 2020-11-06
5. APS Fellows Archive: Fellows nominated by DFD in 2020, retrieved 2020-11-06
6. Top UWA physicist elected to prestigious professional society, University of Western Australia, 8 October 2020, retrieved 2020-11-06
Authority control: Academics
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Snow plow routing problem
The snow plow routing problem is an application of the structure of Arc Routing Problems (ARPs) and Vehicle Routing Problems (VRPs) to snow removal that considers roads as edges of a graph.
The problem is a simple routing problem when the arrival times are not specified.[1] Snow plow problems consider constraints such as the cost of plowing downhill compared to plowing uphill.[2] The Mixed Chinese Postman Problem is applicable to snow routes where directed edges represent one-way streets and undirected edges represent two-way streets.[3]
Background
The routing and scheduling of snow removal vehicles is an important topic for transportation planners and operation researchers[4] This set of problems is part of a larger field of problems referred to as Arc Routing Problems, which is a subset of a larger field named Vehicle Routing Problems. Vehicle routing and scheduling include snow removal, a postman delivering the mail, meter reading to collect money for the city, school bus routing, garbage waste and refuse collection, and street maintenance.[1]
Context
The snow removal problem is to clear the roads to be safe for traffic by vehicles maintained by a public or private body in a minimum amount of time. The problem of snow vehicle routing incorporates higher salaries for vehicle drivers and high fuel costs and high costs of purchasing and maintaining snow vehicles. In the public sector, the objective is less often minimizing cost and more often maximizing safety and convenience, for example by reducing the number of left turns on major roads which are hazardous for vehicles to make.
References
1. Omer, Masoud (2007). "Efficient routing of snow routing of snow removal vehicles vehicles".
2. Dussault, Benjamin; Golden, Bruce; Wasil, Edward (October 2014). "The downhill plow problem with multiple plows". Journal of the Operational Research Society. 65 (10): 1465–1474. doi:10.1057/jors.2013.83. ISSN 0160-5682. S2CID 36977043.
3. Corberán, Ángel (2015). Arc Routing: Problems, Methods, and Applications. ISBN 978-1-61197-366-2.
4. Bodin, Lawrence; Golden, Bruce (Summer 1981). "Classification in vehicle routing and scheduling". Networks. 11 (2): 97–108. doi:10.1002/net.3230110204.
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Wikipedia
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Snub (geometry)
In geometry, a snub is an operation applied to a polyhedron. The term originates from Kepler's names of two Archimedean solids, for the snub cube (cubus simus) and snub dodecahedron (dodecaedron simum).[1] In general, snubs have chiral symmetry with two forms: with clockwise or counterclockwise orientation. By Kepler's names, a snub can be seen as an expansion of a regular polyhedron: moving the faces apart, twisting them about their centers, adding new polygons centered on the original vertices, and adding pairs of triangles fitting between the original edges.
The two snubbed Archimedean solids
Snub cube or
Snub cuboctahedron
Snub dodecahedron or
Snub icosidodecahedron
The terminology was generalized by Coxeter, with a slightly different definition, for a wider set of uniform polytopes.
Conway snubs
John Conway explored generalized polyhedron operators, defining what is now called Conway polyhedron notation, which can be applied to polyhedra and tilings. Conway calls Coxeter's operation a semi-snub.[2]
In this notation, snub is defined by the dual and gyro operators, as s = dg, and it is equivalent to an alternation of a truncation of an ambo operator. Conway's notation itself avoids Coxeter's alternation (half) operation since it only applies for polyhedra with only even-sided faces.
Snubbed regular figures
Forms to snub Polyhedra Euclidean tilings Hyperbolic tilings
Names Tetrahedron Cube or
octahedron
Icosahedron or
dodecahedron
Square tiling Hexagonal tiling or
Triangular tiling
Heptagonal tiling or
Order-7 triangular tiling
Images
Snubbed form Conway
notation
sT sC = sO sI = sD sQ sH = sΔ sΔ7
Image
In 4-dimensions, Conway suggests the snub 24-cell should be called a semi-snub 24-cell because, unlike 3-dimensional snub polyhedra are alternated omnitruncated forms, it is not an alternated omnitruncated 24-cell. It is instead actually an alternated truncated 24-cell.[3]
Coxeter's snubs, regular and quasiregular
Snub cube, derived from cube or cuboctahedron
Seed Rectified
r
Truncated
t
Alternated
h
Name Cube Cuboctahedron
Rectified cube
Truncated cuboctahedron
Cantitruncated cube
Snub cuboctahedron
Snub rectified cube
Conway notation C CO
rC
tCO
trC or trO
htCO = sCO
htrC = srC
Schläfli symbol {4,3} ${\begin{Bmatrix}4\\3\end{Bmatrix}}$ or r{4,3} $t{\begin{Bmatrix}4\\3\end{Bmatrix}}$ or tr{4,3} $ht{\begin{Bmatrix}4\\3\end{Bmatrix}}=s{\begin{Bmatrix}4\\3\end{Bmatrix}}$
htr{4,3} = sr{4,3}
Coxeter diagram or or or
Image
Coxeter's snub terminology is slightly different, meaning an alternated truncation, deriving the snub cube as a snub cuboctahedron, and the snub dodecahedron as a snub icosidodecahedron. This definition is used in the naming of two Johnson solids: the snub disphenoid and the snub square antiprism, and of higher dimensional polytopes, such as the 4-dimensional snub 24-cell, with extended Schläfli symbol s{3,4,3}, and Coxeter diagram .
A regular polyhedron (or tiling), with Schläfli symbol ${\begin{Bmatrix}p,q\end{Bmatrix}}$, and Coxeter diagram , has truncation defined as $t{\begin{Bmatrix}p,q\end{Bmatrix}}$, and , and has snub defined as an alternated truncation $ht{\begin{Bmatrix}p,q\end{Bmatrix}}=s{\begin{Bmatrix}p,q\end{Bmatrix}}$, and . This alternated construction requires q to be even.
A quasiregular polyhedron, with Schläfli symbol ${\begin{Bmatrix}p\\q\end{Bmatrix}}$ or r{p,q}, and Coxeter diagram or , has quasiregular truncation defined as $t{\begin{Bmatrix}p\\q\end{Bmatrix}}$ or tr{p,q}, and or , and has quasiregular snub defined as an alternated truncated rectification $ht{\begin{Bmatrix}p\\q\end{Bmatrix}}=s{\begin{Bmatrix}p\\q\end{Bmatrix}}$ or htr{p,q} = sr{p,q}, and or .
For example, Kepler's snub cube is derived from the quasiregular cuboctahedron, with a vertical Schläfli symbol ${\begin{Bmatrix}4\\3\end{Bmatrix}}$, and Coxeter diagram , and so is more explicitly called a snub cuboctahedron, expressed by a vertical Schläfli symbol $s{\begin{Bmatrix}4\\3\end{Bmatrix}}$, and Coxeter diagram . The snub cuboctahedron is the alternation of the truncated cuboctahedron, $t{\begin{Bmatrix}4\\3\end{Bmatrix}}$, and .
Regular polyhedra with even-order vertices can also be snubbed as alternated truncations, like the snub octahedron, as $s{\begin{Bmatrix}3,4\end{Bmatrix}}$, , is the alternation of the truncated octahedron, $t{\begin{Bmatrix}3,4\end{Bmatrix}}$, and . The snub octahedron represents the pseudoicosahedron, a regular icosahedron with pyritohedral symmetry.
The snub tetratetrahedron, as $s{\begin{Bmatrix}3\\3\end{Bmatrix}}$, and , is the alternation of the truncated tetrahedral symmetry form, $t{\begin{Bmatrix}3\\3\end{Bmatrix}}$, and .
Seed Truncated
t
Alternated
h
Name Octahedron Truncated octahedron Snub octahedron
Conway notation O tO htO or sO
Schläfli symbol {3,4} t{3,4} ht{3,4} = s{3,4}
Coxeter diagram
Image
Coxeter's snub operation also allows n-antiprisms to be defined as $s{\begin{Bmatrix}2\\n\end{Bmatrix}}$ or $s{\begin{Bmatrix}2,2n\end{Bmatrix}}$, based on n-prisms $t{\begin{Bmatrix}2\\n\end{Bmatrix}}$ or $t{\begin{Bmatrix}2,2n\end{Bmatrix}}$, while ${\begin{Bmatrix}2,n\end{Bmatrix}}$ is a regular n-hosohedron, a degenerate polyhedron, but a valid tiling on the sphere with digon or lune-shaped faces.
Snub hosohedra, {2,2p}
Image
Coxeter
diagrams
...
...
Schläfli
symbols
s{2,4} s{2,6} s{2,8} s{2,10} s{2,12} s{2,14} s{2,16}... s{2,∞}
sr{2,2}
$s{\begin{Bmatrix}2\\2\end{Bmatrix}}$
sr{2,3}
$s{\begin{Bmatrix}2\\3\end{Bmatrix}}$
sr{2,4}
$s{\begin{Bmatrix}2\\4\end{Bmatrix}}$
sr{2,5}
$s{\begin{Bmatrix}2\\5\end{Bmatrix}}$
sr{2,6}
$s{\begin{Bmatrix}2\\6\end{Bmatrix}}$
sr{2,7}
$s{\begin{Bmatrix}2\\7\end{Bmatrix}}$
sr{2,8}...
$s{\begin{Bmatrix}2\\8\end{Bmatrix}}$...
sr{2,∞}
$s{\begin{Bmatrix}2\\\infty \end{Bmatrix}}$
Conway
notation
A2 = T A3 = O A4 A5 A6 A7 A8... A∞
The same process applies for snub tilings:
Triangular tiling
Δ
Truncated triangular tiling
tΔ
Snub triangular tiling
htΔ = sΔ
{3,6} t{3,6} ht{3,6} = s{3,6}
Examples
Snubs based on {p,4}
Space Spherical Euclidean Hyperbolic
Image
Coxeter
diagram
...
Schläfli
symbol
s{2,4} s{3,4} s{4,4} s{5,4} s{6,4} s{7,4} s{8,4} ...s{∞,4}
Quasiregular snubs based on r{p,3}
Conway
notation
Spherical Euclidean Hyperbolic
Image
Coxeter
diagram
...
Schläfli
symbol
sr{2,3} sr{3,3} sr{4,3} sr{5,3} sr{6,3} sr{7,3} sr{8,3} ...sr{∞,3}
$s{\begin{Bmatrix}2\\3\end{Bmatrix}}$ $s{\begin{Bmatrix}3\\3\end{Bmatrix}}$ $s{\begin{Bmatrix}4\\3\end{Bmatrix}}$ $s{\begin{Bmatrix}5\\3\end{Bmatrix}}$ $s{\begin{Bmatrix}6\\3\end{Bmatrix}}$ $s{\begin{Bmatrix}7\\3\end{Bmatrix}}$ $s{\begin{Bmatrix}8\\3\end{Bmatrix}}$ $s{\begin{Bmatrix}\infty \\3\end{Bmatrix}}$
Conway
notation
A3 sT sC or sO sD or sI sΗ or sΔ
Quasiregular snubs based on r{p,4}
Space Spherical Euclidean Hyperbolic
Image
Coxeter
diagram
...
Schläfli
symbol
sr{2,4} sr{3,4} sr{4,4} sr{5,4} sr{6,4} sr{7,4} sr{8,4} ...sr{∞,4}
$s{\begin{Bmatrix}2\\4\end{Bmatrix}}$ $s{\begin{Bmatrix}3\\4\end{Bmatrix}}$ $s{\begin{Bmatrix}4\\4\end{Bmatrix}}$ $s{\begin{Bmatrix}5\\4\end{Bmatrix}}$ $s{\begin{Bmatrix}6\\4\end{Bmatrix}}$ $s{\begin{Bmatrix}7\\4\end{Bmatrix}}$ $s{\begin{Bmatrix}8\\4\end{Bmatrix}}$ $s{\begin{Bmatrix}\infty \\4\end{Bmatrix}}$
Conway
notation
A4 sC or sO sQ
Nonuniform snub polyhedra
Nonuniform polyhedra with all even-valance vertices can be snubbed, including some infinite sets; for example:
Snub bipyramids sdt{2,p}
Snub square bipyramid
Snub hexagonal bipyramid
Snub rectified bipyramids srdt{2,p}
Snub antiprisms s{2,2p}
Image ...
Schläfli
symbols
ss{2,4} ss{2,6} ss{2,8} ss{2,10}...
ssr{2,2}
$ss{\begin{Bmatrix}2\\2\end{Bmatrix}}$
ssr{2,3}
$ss{\begin{Bmatrix}2\\3\end{Bmatrix}}$
ssr{2,4}
$ss{\begin{Bmatrix}2\\4\end{Bmatrix}}$
ssr{2,5}...
$ss{\begin{Bmatrix}2\\5\end{Bmatrix}}$
Coxeter's uniform snub star-polyhedra
Snub star-polyhedra are constructed by their Schwarz triangle (p q r), with rational ordered mirror-angles, and all mirrors active and alternated.
Snubbed uniform star-polyhedra
s{3/2,3/2}
s{(3,3,5/2)}
sr{5,5/2}
s{(3,5,5/3)}
sr{5/2,3}
sr{5/3,5}
s{(5/2,5/3,3)}
sr{5/3,3}
s{(3/2,3/2,5/2)}
s{3/2,5/3}
Coxeter's higher-dimensional snubbed polytopes and honeycombs
In general, a regular polychoron with Schläfli symbol ${\begin{Bmatrix}p,q,r\end{Bmatrix}}$, and Coxeter diagram , has a snub with extended Schläfli symbol $s{\begin{Bmatrix}p,q,r\end{Bmatrix}}$, and .
A rectified polychoron ${\begin{Bmatrix}p\\q,r\end{Bmatrix}}$ = r{p,q,r}, and has snub symbol $s{\begin{Bmatrix}p\\q,r\end{Bmatrix}}$ = sr{p,q,r}, and .
Examples
There is only one uniform convex snub in 4-dimensions, the snub 24-cell. The regular 24-cell has Schläfli symbol, ${\begin{Bmatrix}3,4,3\end{Bmatrix}}$, and Coxeter diagram , and the snub 24-cell is represented by $s{\begin{Bmatrix}3,4,3\end{Bmatrix}}$, Coxeter diagram . It also has an index 6 lower symmetry constructions as $s\left\{{\begin{array}{l}3\\3\\3\end{array}}\right\}$ or s{31,1,1} and , and an index 3 subsymmetry as $s{\begin{Bmatrix}3\\3,4\end{Bmatrix}}$ or sr{3,3,4}, and or .
The related snub 24-cell honeycomb can be seen as a $s{\begin{Bmatrix}3,4,3,3\end{Bmatrix}}$ or s{3,4,3,3}, and , and lower symmetry $s{\begin{Bmatrix}3\\3,4,3\end{Bmatrix}}$ or sr{3,3,4,3} and or , and lowest symmetry form as $s\left\{{\begin{array}{l}3\\3\\3\\3\end{array}}\right\}$ or s{31,1,1,1} and .
A Euclidean honeycomb is an alternated hexagonal slab honeycomb, s{2,6,3}, and or sr{2,3,6}, and or sr{2,3[3]}, and .
Another Euclidean (scaliform) honeycomb is an alternated square slab honeycomb, s{2,4,4}, and or sr{2,41,1} and :
The only uniform snub hyperbolic uniform honeycomb is the snub hexagonal tiling honeycomb, as s{3,6,3} and , which can also be constructed as an alternated hexagonal tiling honeycomb, h{6,3,3}, . It is also constructed as s{3[3,3]} and .
Another hyperbolic (scaliform) honeycomb is a snub order-4 octahedral honeycomb, s{3,4,4}, and .
See also
• Snub polyhedron
Polyhedron operators
Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations
t0{p,q}
{p,q}
t01{p,q}
t{p,q}
t1{p,q}
r{p,q}
t12{p,q}
2t{p,q}
t2{p,q}
2r{p,q}
t02{p,q}
rr{p,q}
t012{p,q}
tr{p,q}
ht0{p,q}
h{q,p}
ht12{p,q}
s{q,p}
ht012{p,q}
sr{p,q}
References
1. Kepler, Harmonices Mundi, 1619
2. Conway, (2008) p.287 Coxeter's semi-snub operation
3. Conway, 2008, p.401 Gosset's Semi-snub Polyoctahedron
• Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J. C. P. (1954). "Uniform polyhedra". Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences. The Royal Society. 246 (916): 401–450. Bibcode:1954RSPTA.246..401C. doi:10.1098/rsta.1954.0003. ISSN 0080-4614. JSTOR 91532. MR 0062446. S2CID 202575183.
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (pp. 154–156 8.6 Partial truncation, or alternation)
• 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 , Googlebooks
• (Paper 17) Coxeter, The Evolution of Coxeter–Dynkin diagrams, [Nieuw Archief voor Wiskunde 9 (1991) 233–248]
• (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, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
• Weisstein, Eric W. "Snubification". MathWorld.
• Richard Klitzing, Snubs, alternated facetings, and Stott–Coxeter–Dynkin diagrams, Symmetry: Culture and Science, Vol. 21, No.4, 329–344, (2010)
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Wikipedia
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Snub 24-cell
In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, 432 edges, and 96 vertices. One can build it from the 600-cell by diminishing a select subset of icosahedral pyramids and leaving only their icosahedral bases, thereby removing 480 tetrahedra and replacing them with 24 icosahedra.
Snub 24-cell
Type Uniform 4-polytope
Schläfli symbol[1] s{3,4,3}
sr{3,3,4}
s{31,1,1}
Coxeter-Dynkin
diagrams
or
or
Cells 144 96 3.3.3 (oblique)
24 3.3.3
24 3.3.3.3.3
Faces 480 {3}
Edges 432
Vertices 96
Vertex figure
(Tridiminished icosahedron)
Symmetry groups [3+,4,3], 1/2F4, order 576
[(3,3)+,4], 1/2B4, order 192
[31,1,1]+, 1/2D4, order 96
Dual Dual snub 24-cell
Properties convex
Uniform index 30 31 32
Topologically, under its highest symmetry, [3+,4,3], as an alternation of a truncated 24-cell, it contains 24 pyritohedra (an icosahedron with Th symmetry), 24 regular tetrahedra, and 96 triangular pyramids.
Semiregular polytope
It is one of three semiregular 4-polytopes made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper.[2] He called it a tetricosahedric for being made of tetrahedron and icosahedron cells. (The other two are the rectified 5-cell and rectified 600-cell.)
Alternative names
• Snub icositetrachoron
• Snub demitesseract
• Semi-snub polyoctahedron (John Conway)[3]
• Sadi (Jonathan Bowers) for snub disicositetrachoron
• Tetricosahedric (Thorold Gosset)[2]
Geometry
Coordinates
The vertices of a snub 24-cell centered at the origin of 4-space, with edges of length 2, are obtained by taking even permutations of
(0, ±1, ±φ, ±φ2)
where φ = 1+√5/2 ≈ 1.618 is the golden ratio.
The unit-radius coordinates of the snub 24-cell, with edges of length φ−1 ≈ 0.618, are the even permutations of
(±φ/2, ±1/2, ±φ−1/2, 0)
These 96 vertices can be found by partitioning each of the 96 edges of a 24-cell in the golden ratio in a consistent manner dimensionally analogous to the way the 12 vertices of an icosahedron or "snub octahedron" can be produced by partitioning the 12 edges of an octahedron in the golden ratio. This can be done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector.[4] This is equivalent to the snub truncation construction of the 24-cell described below.
The 96 vertices of the snub 24-cell, together with the 24 vertices of a 24-cell, form the 120 vertices of the 600-cell.
Constructions
The snub 24-cell is derived from the 24-cell by a special form of truncation.
Truncations remove vertices by cutting through the edges incident to the vertex; forms of truncation differ by where on the edge the cut is made. The common truncations of the 24-cell include the recitified 24-cell (which cuts each edge at its midpoint, producing a polytope bounded by 24 cubes and 24 cuboctahedra), and the truncated 24-cell (which cuts each edge one-third of its length from the vertex, producing a polytope bounded by 24 cubes and 24 truncated octahedra). In these truncations a cube is produced in place of the removed vertex, because the vertex figure of the 24-cell is a cube and the cuts are equidistant from the vertex.
The snub truncation of the 24-cell[4] cuts each edge into two golden sections (such that the larger section is in the golden ratio ~1.618 to the smaller section, and the original edge is in the golden ratio to the larger section). The cut must be made in alternate directions on alternate edges incident to each vertex, in order to have a coherent result. The edges incident to a vertex in the 24-cell are the 8 radii of its cubical vertex figure. The only way to choose alternate radii of a cube is to choose the four radii of a tetrahedron (inscribed in the cube) to be cut at the smaller section of their length from the vertex, and the opposite four radii (of the other tetrahedron that can be inscribed in the cube) to be cut at the larger section of their length from the vertex. There are of course two ways to do this; both produce a cluster of five regular tetrahedra in place of the removed vertex, rather than a cube.
This construction has an analogy in 3 dimensions: the construction of the icosahedron (the "snub octahedron") from the octahedron, by the same method.[5] That is how the snub-24 cell's icosahedra are produced from the 24-cell's octahedra during truncation.
The snub 24-cell is related to the truncated 24-cell by an alternation operation. Half the vertices are deleted, the 24 truncated octahedron cells become 24 icosahedron cells, the 24 cubes become 24 tetrahedron cells, and the 96 deleted vertex voids create 96 new tetrahedron cells.
Orthogonal projection, F4 Coxeter plane
Snub 24-cell 600-cell
The snub 24-cell may also be constructed by a particular diminishing of the 600-cell: by removing 24 vertices from the 600-cell corresponding to those of an inscribed 24-cell, and then taking the convex hull of the remaining vertices. This is equivalent to removing 24 icosahedral pyramids from the 600-cell.
Conversely, the 600-cell may be constructed from the snub 24-cell by augmenting it with 24 icosahedral pyramids.
Weyl orbits
Another construction method uses quaternions and the Icosahedral symmetry of Weyl group orbits $O(\Lambda )=W(H_{4})=I$ of order 120.[6] The following describe $T$ and $T'$ 24-cells as quaternion orbit weights of D4 under the Weyl group W(D4):
O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2}
O(1000) : V1
O(0010) : V2
O(0001) : V3
With quaternions $(p,q)$ where ${\bar {p}}$ is the conjugate of $p$ and $[p,q]:r\rightarrow r'=prq$ and $[p,q]^{*}:r\rightarrow r''=p{\bar {r}}q$, then the Coxeter group $W(H_{4})=\lbrace [p,{\bar {p}}]\oplus [p,{\bar {p}}]^{*}\rbrace $ is the symmetry group of the 600-cell and the 120-cell of order 14400.
Given $p\in T$ such that ${\bar {p}}=\pm p^{4},{\bar {p}}^{2}=\pm p^{3},{\bar {p}}^{3}=\pm p^{2},{\bar {p}}^{4}=\pm p$ and $p^{\dagger }$ as an exchange of $-1/\varphi \leftrightarrow \varphi $ within $p$, we can construct the snub 24-cell as $S=\sum _{i=1}^{4}\oplus p^{i}T$
Structure
A net of the snub 24-cell with blue icosahedra, and red and yellow tetrahedra.
The icosahedral cells fit together face-to-face leaving voids between them filled by clusters of five tetrahedral cells.[7]
Each icosahedral cell is joined to 8 other icosahedral cells at 8 triangular faces in the positions corresponding to an inscribing octahedron. The remaining triangular faces are joined to tetrahedral cells, which occur in pairs that share an edge on the icosahedral cell.
The tetrahedral cells may be divided into two groups, of 96 yellow cells and 24 red cells respectively (as colored in the net illustration). Each yellow tetrahedral cell is joined via its triangular faces to 3 blue icosahedral cells and one red tetrahedral cell, while each red tetrahedral cell is joined to 4 yellow tetrahedra. Thus, the tetrahedral cells occur in clusters of five (four yellow cells face-bonded around a red central one, each red/yellow pair lying in a different hyperplane). The red central tetrahedron of the five shares each of its six edges with a different icosahedral cell, and with the pair of yellow tetrahedral cells which shares that edge on the icosahedral cell.
Symmetry
The snub 24-cell has three vertex-transitive colorings based on a Wythoff construction on a Coxeter group from which it is alternated: F4 defines 24 interchangeable icosahedra, while the B4 group defines two groups of icosahedra in an 8:16 counts, and finally the D4 group has 3 groups of icosahedra with 8:8:8 counts.[8]
Symmetry
(order)
Constructive name Coxeter-Dynkin diagram
Extended Schläfli symbol
Vertex figure
(Tridiminished icosahedron)
Cells
(Colored as faces in vertex figures)
1/2F4
[3+,4,3]
(576)
Snub 24-cell
s{3,4,3}
One set of 24 icosahedra (blue)
Two sets of tetrahedra: 96 (yellow) and 24 (cyan)
1/2B4
[(3,3)+,4]
(192)
Snub rectified 16-cell
sr{3,3,4}
Two sets icosahedra: 8, 16 each (red and blue)
Two sets of tetrahedra: 96 (yellow) and 24 (cyan)
1/2D4
[31,1,1]+
(96)
Omnisnub demitesseract
s{31,1,1}
Three sets of 8 icosahedra (red, green, and blue)
Two sets of tetrahedra: 96 (yellow) and 24 (cyan)
Projections
Orthographic projections
orthographic projections
Coxeter plane F4 B4
Graph
Dihedral symmetry [12]+ [8/2]
Coxeter plane D4 / B3 / A2 B2 / A3
Graph
Dihedral symmetry [6]+ [4]
Orthogonal projection
Centered on hyperplane of one icosahedron.
Perspective projections
Perspective projections
Perspective projection centered on an icosahedral cell, with 4D viewpoint placed at a distance of 5 times the vertex-center radius. The nearest icosahedral cell is rendered in solid color, and the other cells are in edge-outline. Cells facing away from the 4D viewpoint are culled, to reduce visual clutter.
The same projection, now with 4 of the 8 icosahedral cells surrounding the central cell shown in green.
The same projection as above, now with the other 4 icosahedral cells surrounding the central cell shown in magenta. The animated version of this image gives a good view on the layout of these cells.
From this particular viewpoint, one of the gaps containing tetrahedral cells can be seen. Each of these gaps are filled by 5 tetrahedral cells, not shown here.
Same projection as above, now with the central tetrahedral cell in the gap filled in. This tetrahedral cell is joined to 4 other tetrahedral cells, two of which fills the two gaps visible in this image. The other two each lies between a green tetrahedral cell, a magenta cell, and the central cell, to the left and right of the yellow tetrahedral cell.
Note that in these images, cells facing away from the 4D viewpoint have been culled; hence there are only a total of 1 + 8 + 6 + 24 = 39 cells accounted for here. The other cells lie on the other side of the snub 24-cell. Part of the edge outline of one of them, an icosahedral cell, can be discerned here, overlying the yellow tetrahedron.
In this image, only the nearest icosahedral cell and the 6 yellow tetrahedral cells from the previous image are shown.
Now the 12 tetrahedral cells joined to the central icosahedral cell and the 6 yellow tetrahedral cells are shown. Each of these cells is surrounded by the central icosahedron and two of the other icosahedral cells shown earlier.
Finally, the other 12 tetrahedral cells joined to the 6 yellow tetrahedral cells are shown here. These cells, together with the 8 icosahedral cells shown earlier, comprise all the cells that share at least 1 vertex with the central cell.
Dual
The Dual snub 24-cell has 144 identical irregular cells. Each cell has faces of two kinds: 3 kites and 6 isosceles triangles. The polytope has a total of 432 faces (144 kites and 288 isosceles triangles) and 480 edges.[9]
Related polytopes
The snub 24-cell can be obtained as a diminishing of the 600-cell at 24 of its vertices, in fact those of a vertex inscribed 24-cell. There is also a further such bi-diminishing, when the vertices of a second vertex inscribed 24-cell would be diminished as well. Accordingly, this one is known as the bi-24-diminished 600-cell.
D4 uniform polychora
{3,31,1}
h{4,3,3}
2r{3,31,1}
h3{4,3,3}
t{3,31,1}
h2{4,3,3}
2t{3,31,1}
h2,3{4,3,3}
r{3,31,1}
{31,1,1}={3,4,3}
rr{3,31,1}
r{31,1,1}=r{3,4,3}
tr{3,31,1}
t{31,1,1}=t{3,4,3}
sr{3,31,1}
s{31,1,1}=s{3,4,3}
The snub 24-cell is also called a semi-snub 24-cell because it is not a true snub (alternation of an omnitruncated 24-cell). The full snub 24-cell can also be constructed although it is not uniform, being composed of irregular tetrahedra on the alternated vertices.
The snub 24-cell is the largest facet of the 4-dimensional honeycomb, the snub 24-cell honeycomb.
The snub 24-cell is a part of the F4 symmetry family of uniform 4-polytopes.
24-cell family polytopes
Name 24-cell truncated 24-cell snub 24-cell rectified 24-cell cantellated 24-cell bitruncated 24-cell cantitruncated 24-cell runcinated 24-cell runcitruncated 24-cell omnitruncated 24-cell
Schläfli
symbol
{3,4,3} t0,1{3,4,3}
t{3,4,3}
s{3,4,3} t1{3,4,3}
r{3,4,3}
t0,2{3,4,3}
rr{3,4,3}
t1,2{3,4,3}
2t{3,4,3}
t0,1,2{3,4,3}
tr{3,4,3}
t0,3{3,4,3} t0,1,3{3,4,3} t0,1,2,3{3,4,3}
Coxeter
diagram
Schlegel
diagram
F4
B4
B3(a)
B3(b)
B2
See also
• Snub 24-cell honeycomb
• Dual snub 24-cell
Citations
1. Klitzing. sfn error: no target: CITEREFKlitzing (help)
2. Gosset 1900.
3. Conway, Burgiel & Goodman-Strass 2008, p. 401, 26. Gosset's semi-snub polyoctahedron.
4. Coxeter 1973, pp. 151–153, §8.4. The snub {3,4,3}.
5. Coxeter 1973, pp. 50–52, §3.7.
6. Koca, Al-Ajmi & Ozdes Koca 2011, pp. 986–988, 6. Dual of the snub 24-cell.
7. Koca, Al-Ajmi & Ozdes Koca 2011, 5. Detailed analysis of cell structure of the snub 24-cell.
8. Koca, Ozdes Koca & Al-Barwani 2012.
9. Koca, Al-Ajmi & Ozdes Koca 2011.
References
• Gosset, Thorold (1900). "On the Regular and Semi-Regular Figures in Space of n Dimensions". Messenger of Mathematics. Macmillan.
• Coxeter, H.S.M. (1973) [1948]. Regular Polytopes (3rd ed.). New York: Dover.
• Conway, John; Burgiel, Heidi; Goodman-Strass, Chaim (2008). The Symmetries of Things. ISBN 978-1-56881-220-5.
• Koca, Mehmet; Ozdes Koca, Nazife; Al-Barwani, Muataz (2012). "Snub 24-Cell Derived from the Coxeter-Weyl Group W(D4)". Int. J. Geom. Methods Mod. Phys. 09 (8). arXiv:1106.3433. doi:10.1142/S0219887812500685. S2CID 119288632.
• Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system". Linear Algebra and Its Applications. 434 (4): 977–989. arXiv:0906.2109. doi:10.1016/j.laa.2010.10.005. ISSN 0024-3795. S2CID 18278359.
• Klitzing, Richard. "4D uniform polytopes (polychora) s3s4o3o - sadi".
External links
• 3. Convex uniform polychora based on the icositetrachoron (24-cell) - Model 31, George Olshevsky.
• Print #11: Snub icositetrachoron net, George Olshevsky.
• Snub icositetrachoron - Data and images
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
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Wikipedia
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Runcinated 5-orthoplexes
In five-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation (runcination) of the regular 5-orthoplex.
5-orthoplex
Runcinated 5-orthoplex
Runcinated 5-cube
Runcitruncated 5-orthoplex
Runcicantellated 5-orthoplex
Runcicantitruncated 5-orthoplex
Runcitruncated 5-cube
Runcicantellated 5-cube
Runcicantitruncated 5-cube
Orthogonal projections in B5 Coxeter plane
There are 8 runcinations of the 5-orthoplex with permutations of truncations, and cantellations. Four are more simply constructed relative to the 5-cube.
Runcinated 5-orthoplex
Runcinated 5-orthoplex
Type Uniform 5-polytope
Schläfli symbol t0,3{3,3,3,4}
Coxeter-Dynkin diagram
4-faces 162
Cells 1200
Faces 2160
Edges 1440
Vertices 320
Vertex figure
Coxeter group B5 [4,3,3,3]
D5 [32,1,1]
Properties convex
Alternate names
• Runcinated pentacross
• Small prismated triacontiditeron (Acronym: spat) (Jonathan Bowers)[1]
Coordinates
The vertices of the can be made in 5-space, as permutations and sign combinations of:
(0,1,1,1,2)
Images
orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]
Runcitruncated 5-orthoplex
Runcitruncated 5-orthoplex
Typeuniform 5-polytope
Schläfli symbolt0,1,3{3,3,3,4}
t0,1,3{3,31,1}
Coxeter-Dynkin diagrams
4-faces162
Cells1440
Faces3680
Edges3360
Vertices960
Vertex figure
Coxeter groupsB5, [3,3,3,4]
D5, [32,1,1]
Propertiesconvex
Alternate names
• Runcitruncated pentacross
• Prismatotruncated triacontiditeron (Acronym: pattit) (Jonathan Bowers)[2]
Coordinates
Cartesian coordinates for the vertices of a runcitruncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of
(±3,±2,±1,±1,0)
Images
orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]
Runcicantellated 5-orthoplex
Runcicantellated 5-orthoplex
Type Uniform 5-polytope
Schläfli symbol t0,2,3{3,3,3,4}
t0,2,3{3,3,31,1}
Coxeter-Dynkin diagram
4-faces162
Cells1200
Faces2960
Edges2880
Vertices960
Vertex figure
Coxeter group B5 [4,3,3,3]
D5 [32,1,1]
Properties convex
Alternate names
• Runcicantellated pentacross
• Prismatorhombated triacontiditeron (Acronym: pirt) (Jonathan Bowers)[3]
Coordinates
The vertices of the runcicantellated 5-orthoplex can be made in 5-space, as permutations and sign combinations of:
(0,1,2,2,3)
Images
orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]
Runcicantitruncated 5-orthoplex
Runcicantitruncated 5-orthoplex
Type Uniform 5-polytope
Schläfli symbol t0,1,2,3{3,3,3,4}
Coxeter-Dynkin
diagram
4-faces162
Cells1440
Faces4160
Edges4800
Vertices1920
Vertex figure
Irregular 5-cell
Coxeter groups B5 [4,3,3,3]
D5 [32,1,1]
Properties convex, isogonal
Alternate names
• Runcicantitruncated pentacross
• Great prismated triacontiditeron (gippit) (Jonathan Bowers)[4]
Coordinates
The Cartesian coordinates of the vertices of a runcicantitruncated 5-orthoplex having an edge length of √2 are given by all permutations of coordinates and sign of:
$\left(0,1,2,3,4\right)$
Images
orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]
Snub 5-demicube
The snub 5-demicube defined as an alternation of the omnitruncated 5-demicube is not uniform, but it can be given Coxeter diagram or and symmetry [32,1,1]+ or [4,(3,3,3)+], and constructed from 10 snub 24-cells, 32 snub 5-cells, 40 snub tetrahedral antiprisms, 80 2-3 duoantiprisms, and 960 irregular 5-cells filling the gaps at the deleted vertices.
Related polytopes
This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.
B5 polytopes
β5
t1β5
t2γ5
t1γ5
γ5
t0,1β5
t0,2β5
t1,2β5
t0,3β5
t1,3γ5
t1,2γ5
t0,4γ5
t0,3γ5
t0,2γ5
t0,1γ5
t0,1,2β5
t0,1,3β5
t0,2,3β5
t1,2,3γ5
t0,1,4β5
t0,2,4γ5
t0,2,3γ5
t0,1,4γ5
t0,1,3γ5
t0,1,2γ5
t0,1,2,3β5
t0,1,2,4β5
t0,1,3,4γ5
t0,1,2,4γ5
t0,1,2,3γ5
t0,1,2,3,4γ5
Notes
1. Klitzing, (x3o3o3x4o - spat)
2. Klitzing, (x3x3o3x4o - pattit)
3. Klitzing, (x3o3x3x4o - pirt)
4. Klitzing, (x3x3x3x4o - gippit)
References
• H.S.M. Coxeter:
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• 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 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]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "5D uniform polytopes (polytera)". x3o3o3x4o - spat, x3x3o3x4o - pattit, x3o3x3x4o - pirt, x3x3x3x4o - gippit
External links
• Glossary for hyperspace, George Olshevsky.
• Polytopes of Various Dimensions, Jonathan Bowers
• Runcinated uniform polytera (spid), Jonathan Bowers
• Multi-dimensional Glossary
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
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Snub apeiroapeirogonal tiling
In geometry, the snub apeiroapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of s{∞,∞}. It has 3 equilateral triangles and 2 apeirogons around every vertex, with vertex figure 3.3.∞.3.∞.
Snub apeiroapeirogonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.3.∞.3.∞
Schläfli symbols{∞,4}
sr{∞,∞} or $s{\begin{Bmatrix}\infty \\\infty \end{Bmatrix}}$
Wythoff symbol| ∞ ∞ 2
Coxeter diagram
or
Symmetry group[∞,∞]+, (∞∞2)
DualInfinitely-infinite-order floret pentagonal tiling
PropertiesVertex-transitive Chiral
Dual tiling
Related polyhedra and tiling
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.∞
The snub tetrapeirogonal tiling is last in an infinite series of snub polyhedra and tilings with vertex figure 3.3.n.3.n.
4n2 symmetry mutations of snub tilings: 3.3.n.3.n
Symmetry
4n2
Spherical Euclidean Compact hyperbolic Paracompact
222 322 442 552 662 772 882 ∞∞2
Snub
figures
Config. 3.3.2.3.2 3.3.3.3.3 3.3.4.3.4 3.3.5.3.5 3.3.6.3.6 3.3.7.3.7 3.3.8.3.8 3.3.∞.3.∞
Gyro
figures
Config. V3.3.2.3.2 V3.3.3.3.3 V3.3.4.3.4 V3.3.5.3.5 V3.3.6.3.6 V3.3.7.3.7 V3.3.8.3.8 V3.3.∞.3.∞
See also
Wikimedia Commons has media related to Uniform tiling 3-3-i-3-i.
• 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
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
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Snub cubic prism
In geometry, a snub cubic prism or snub cuboctahedral prism is a convex uniform polychoron (four-dimensional polytope).
Snub cubic prism
Schlegel diagram
TypePrismatic uniform polychoron
Uniform index56
Schläfli symbolsr{4,3}×{}
Coxeter-Dynkin
Cells40 total:
2 4.3.3.3.3
32 3.4.4
6 4.4.4
Faces136 total:
64 {3}
72 {4}
Edges144
Vertices48
Vertex figure
irr. pentagonal pyramid
Symmetry group[(4,3)+,2], order 48
Propertiesconvex
It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes.
See also
• Snub cubic antiprism s{4,3,2} - A related nonuniform polychoron
Alternative names
• Snub-cuboctahedral dyadic prism (Norman W. Johnson)
• Sniccup (Jonathan Bowers: for snub-cubic prism)
• Snub-cuboctahedral hyperprism
• Snub-cubic hyperprism
External links
• 6. Convex uniform prismatic polychora - Model 56, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora) s3s4s x - sniccup".
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Snub dodecadodecahedron
In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U40. It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices.[1] It is given a Schläfli symbol sr{5⁄2,5}, as a snub great dodecahedron.
Snub dodecadodecahedron
TypeUniform star polyhedron
ElementsF = 84, E = 150
V = 60 (χ = −6)
Faces by sides60{3}+12{5}+12{5/2}
Coxeter diagram
Wythoff symbol| 2 5/2 5
Symmetry groupI, [5,3]+, 532
Index referencesU40, C49, W111
Dual polyhedronMedial pentagonal hexecontahedron
Vertex figure
3.3.5/2.3.5
Bowers acronymSiddid
Cartesian coordinates
Cartesian coordinates for the vertices of a snub dodecadodecahedron are all the even permutations of
(±2α, ±2, ±2β),
(±(α+β/τ+τ), ±(-ατ+β+1/τ), ±(α/τ+βτ-1)),
(±(-α/τ+βτ+1), ±(-α+β/τ-τ), ±(ατ+β-1/τ)),
(±(-α/τ+βτ-1), ±(α-β/τ-τ), ±(ατ+β+1/τ)) and
(±(α+β/τ-τ), ±(ατ-β+1/τ), ±(α/τ+βτ+1)),
with an even number of plus signs, where
β = (α2/τ+τ)/(ατ−1/τ),
where τ = (1+√5)/2 is the golden mean and α is the positive real root of τα4−α3+2α2−α−1/τ, or approximately 0.7964421. Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.
Related polyhedra
Medial pentagonal hexecontahedron
Medial pentagonal hexecontahedron
TypeStar polyhedron
Face
ElementsF = 60, E = 150
V = 84 (χ = −6)
Symmetry groupI, [5,3]+, 532
Index referencesDU40
dual polyhedronSnub dodecadodecahedron
The medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.
See also
• List of uniform polyhedra
• Inverted snub dodecadodecahedron
References
1. Maeder, Roman. "40: snub dodecadodecahedron". MathConsult.{{cite web}}: CS1 maint: url-status (link)
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208
External links
• Weisstein, Eric W. "Medial pentagonal hexecontahedron". MathWorld.
• Weisstein, Eric W. "Snub dodecadodecahedron". MathWorld.
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
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Snub dodecahedral prism
In geometry, a snub dodecahedral prism or snub icosidodecahedral prism is a convex uniform polychoron (four-dimensional polytope).
Snub dodecahedral prism
Schlegel diagram
TypePrismatic uniform polychoron
Uniform index64
Schläfli symbolsr{3,5}×{}
Coxeter-Dynkin
Cells94 total:
2 3.3.3.3.5
80 3.4.4
12 4.4.5
Faces334 total:
160 {3}
150 {4}
24 {5}
Edges360
Vertices120
Vertex figure
irr. pentagonal pyramid
Symmetry group[(5,3)+,2], order 120
Propertiesconvex
It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes, in this case a pair of snub dodecahedra.
Alternative names
• Snub-icosidodecahedral dyadic prism (Norman W. Johnson)
• Sniddip (Jonathan Bowers: for snub-dodecahedral prism)
• Snub-icosidodecahedral hyperprism
• Snub-dodecahedral prism
• Snub-dodecahedral hyperprism
See also
• Snub dodecahedral antiprism ht0,1,2,3{5,3,2}, or - A related nonuniform polychoron
External links
• 6. Convex uniform prismatic polychora - Model 64, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora) x s3s5s - sniddip".
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Snub heptaheptagonal tiling
In geometry, the snub heptaheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{7,7}, constructed from two regular heptagons and three equilateral triangles around every vertex.
Snub heptaheptagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.3.7.3.7
Schläfli symbolsr{7,7} or $s{\begin{Bmatrix}7\\7\end{Bmatrix}}$
Wythoff symbol| 7 7 2
Coxeter diagram
Symmetry group[7,7]+, (772)
[7+,4], (7*2)
DualOrder-7-7 floret pentagonal tiling
PropertiesVertex-transitive
Images
Drawn in chiral pairs, with edges missing between black triangles:
Symmetry
A double symmetry coloring can be constructed from [7,4] symmetry with only one color heptagon.
Related tilings
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
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
4n2 symmetry mutations of snub tilings: 3.3.n.3.n
Symmetry
4n2
Spherical Euclidean Compact hyperbolic Paracompact
222 322 442 552 662 772 882 ∞∞2
Snub
figures
Config. 3.3.2.3.2 3.3.3.3.3 3.3.4.3.4 3.3.5.3.5 3.3.6.3.6 3.3.7.3.7 3.3.8.3.8 3.3.∞.3.∞
Gyro
figures
Config. V3.3.2.3.2 V3.3.3.3.3 V3.3.4.3.4 V3.3.5.3.5 V3.3.6.3.6 V3.3.7.3.7 V3.3.8.3.8 V3.3.∞.3.∞
See also
Wikimedia Commons has media related to Uniform tiling 3-3-7-3-7.
• Square tiling
• 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
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Snub trihexagonal tiling
In geometry, the snub hexagonal tiling (or snub trihexagonal tiling) is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex. It has Schläfli symbol sr{3,6}. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol sr{4,6}.
Snub trihexagonal tiling
TypeSemiregular tiling
Vertex configuration
3.3.3.3.6
Schläfli symbolsr{6,3} or $s{\begin{Bmatrix}6\\3\end{Bmatrix}}$
Wythoff symbol| 6 3 2
Coxeter diagram
Symmetryp6, [6,3]+, (632)
Rotation symmetryp6, [6,3]+, (632)
Bowers acronymSnathat
DualFloret pentagonal tiling
PropertiesVertex-transitive chiral
Conway calls it a snub hextille, constructed as a snub operation applied to a hexagonal tiling (hextille).
There are three regular and eight semiregular tilings in the plane. This is the only one which does not have a reflection as a symmetry.
There is only one uniform coloring of a snub trihexagonal tiling. (Labeling the colors by numbers, "3.3.3.3.6" gives "11213".)
Circle packing
The snub trihexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing (kissing number).[1] The lattice domain (red rhombus) repeats 6 distinct circles. The hexagonal gaps can be filled by exactly one circle, leading to the densest packing from the triangular tiling.
Related polyhedra and tilings
Uniform hexagonal/triangular tilings
Fundamental
domains
Symmetry: [6,3], (*632) [6,3]+, (632)
{6,3} t{6,3} r{6,3} t{3,6} {3,6} rr{6,3} tr{6,3} sr{6,3}
Config. 63 3.12.12 (6.3)2 6.6.6 36 3.4.6.4 4.6.12 3.3.3.3.6
Symmetry mutations
This semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram . These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons.
n32 symmetry mutations of snub tilings: 3.3.3.3.n
Symmetry
n32
Spherical Euclidean Compact hyperbolic Paracomp.
232 332 432 532 632 732 832 ∞32
Snub
figures
Config. 3.3.3.3.2 3.3.3.3.3 3.3.3.3.4 3.3.3.3.5 3.3.3.3.6 3.3.3.3.7 3.3.3.3.8 3.3.3.3.∞
Gyro
figures
Config. V3.3.3.3.2 V3.3.3.3.3 V3.3.3.3.4 V3.3.3.3.5 V3.3.3.3.6 V3.3.3.3.7 V3.3.3.3.8 V3.3.3.3.∞
6-fold pentille tiling
Floret pentagonal tiling
TypeDual semiregular tiling
Facesirregular pentagons
Coxeter diagram
Symmetry groupp6, [6,3]+, (632)
Rotation groupp6, [6,3]+, (632)
Dual polyhedronSnub trihexagonal tiling
Face configurationV3.3.3.3.6
Face figure:
Propertiesface-transitive, chiral
In geometry, the 6-fold pentille or floret pentagonal tiling is a dual semiregular tiling of the Euclidean plane.[2] It is one of the 15 known isohedral pentagon tilings. Its six pentagonal tiles radiate out from a central point, like petals on a flower.[3] Each of its pentagonal faces has four 120° and one 60° angle.
It is the dual of the uniform snub trihexagonal tiling,[4] and has rotational symmetries of orders 6-3-2 symmetry.
Variations
The floret pentagonal tiling has geometric variations with unequal edge lengths and rotational symmetry, which is given as monohedral pentagonal tiling type 5. In one limit, an edge-length goes to zero and it becomes a deltoidal trihexagonal tiling.
General Zero length
degenerate
Special cases
(See animation)
Deltoidal trihexagonal tiling
a=b, d=e
A=60°, D=120°
a=b, d=e, c=0
A=60°, 90°, 90°, D=120°
a=b=2c=2d=2e
A=60°, B=C=D=E=120°
a=b=d=e
A=60°, D=120°, E=150°
2a=2b=c=2d=2e
0°, A=60°, D=120°
a=b=c=d=e
0°, A=60°, D=120°
Related k-uniform and dual k-uniform tilings
There are many k-uniform tilings whose duals mix the 6-fold florets with other tiles; for example, labeling F for V34.6, C for V32.4.3.4, B for V33.42, H for V36:
uniform (snub trihexagonal) 2-uniform 3-uniform
F, p6 (t=3, e=3) FH, p6 (t=5, e=7) FH, p6m (t=3, e=3) FCB, p6m (t=5, e=6) FH2, p6m (t=3, e=4) FH2, p6m (t=5, e=5)
dual uniform (floret pentagonal) dual 2-uniform dual 3-uniform
3-uniform 4-uniform
FH2, p6 (t=7, e=9) F2H, cmm (t=4, e=6) F2H2, p6 (t=6, e=9) F3H, p2 (t=7, e=12) FH3, p6 (t=7, e=10) FH3, p6m (t=7, e=8)
dual 3-uniform dual 4-uniform
Fractalization
Replacing every V36 hexagon by a rhombitrihexagon furnishes a 6-uniform tiling, two vertices of 4.6.12 and two vertices of 3.4.6.4.
Replacing every V36 hexagon by a truncated hexagon furnishes a 8-uniform tiling, five vertices of 32.12, two vertices of 3.4.3.12, and one vertex of 3.4.6.4.
Replacing every V36 hexagon by a truncated trihexagon furnishes a 15-uniform tiling, twelve vertices of 4.6.12, two vertices of 3.42.6, and one vertex of 3.4.6.4.
In each fractal tiling, every vertex in a floret pentagonal domain is in a different orbit since there is no chiral symmetry (the domains have 3:2 side lengths of $1+{\frac {1}{\sqrt {3}}}:2+{\frac {2}{\sqrt {3}}}$ in the rhombitrihexagonal; $1+{\frac {2}{\sqrt {3}}}:2+{\frac {4}{\sqrt {3}}}$ in the truncated hexagonal; and $1+{\sqrt {3}}:2+2{\sqrt {3}}$ in the truncated trihexagonal).
Fractalizing the Snub Trihexagonal Tiling using the Rhombitrihexagonal, Truncated Hexagonal and Truncated Trihexagonal Tilings
Rhombitrihexagonal Truncated Hexagonal Truncated Trihexagonal
Related tilings
Dual uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632) [6,3]+, (632)
V63 V3.122 V(3.6)2 V36 V3.4.6.4 V.4.6.12 V34.6
See also
Wikimedia Commons has media related to Uniform tiling 3-3-3-3-6 (snub trihexagonal tiling).
• Tilings of regular polygons
• List of uniform tilings
References
1. Order in Space: A design source book, Keith Critchlow, p.74-75, pattern E
2. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things, 2008, ISBN 978-1-56881-220-5, "A K Peters, LTD. - The Symmetries of Things". Archived from the original on 2010-09-19. Retrieved 2012-01-20. (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p. 288, table)
3. Five space-filling polyhedra by Guy Inchbald
4. Weisstein, Eric W. "Dual tessellation". MathWorld.
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
• 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. ISBN 0-486-23729-X. p. 39
• Keith Critchlow, Order in Space: A design source book, 1970, p. 69-61, Pattern R, Dual p. 77-76, pattern 5
• Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–56, dual rosette tiling p. 96, p. 114
External links
• Weisstein, Eric W. "Uniform tessellation". MathWorld.
• Weisstein, Eric W. "Semiregular tessellation". MathWorld.
• Klitzing, Richard. "2D Euclidean tilings s3s6s - snathat - O11".
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
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Wikipedia
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Snub cube
In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices.
Snub cube
(Click here for rotating model)
TypeArchimedean solid
Uniform polyhedron
ElementsF = 38, E = 60, V = 24 (χ = 2)
Faces by sides(8+24){3}+6{4}
Conway notationsC
Schläfli symbolssr{4,3} or $s{\begin{Bmatrix}4\\3\end{Bmatrix}}$
ht0,1,2{4,3}
Wythoff symbol| 2 3 4
Coxeter diagram
Symmetry groupO, 1/2B3, [4,3]+, (432), order 24
Rotation groupO, [4,3]+, (432), order 24
Dihedral angle3-3: 153°14′04″ (153.23°)
3-4: 142°59′00″ (142.98°)
ReferencesU12, C24, W17
PropertiesSemiregular convex chiral
Colored faces
3.3.3.3.4
(Vertex figure)
Pentagonal icositetrahedron
(dual polyhedron)
Net
It is a chiral polyhedron; that is, it has two distinct forms, which are mirror images (or "enantiomorphs") of each other. The union of both forms is a compound of two snub cubes, and the convex hull of both sets of vertices is a truncated cuboctahedron.
Kepler first named it in Latin as cubus simus in 1619 in his Harmonices Mundi. H. S. M. Coxeter, noting it could be derived equally from the octahedron as the cube, called it snub cuboctahedron, with a vertical extended Schläfli symbol $s\scriptstyle {\begin{Bmatrix}4\\3\end{Bmatrix}}$, and representing an alternation of a truncated cuboctahedron, which has Schläfli symbol $t\scriptstyle {\begin{Bmatrix}4\\3\end{Bmatrix}}$.
Dimensions
For a snub cube with edge length 1, its surface area and volume are:
${\begin{aligned}A&=6+8{\sqrt {3}}&&\approx 19.856\,406\,460\,6\\V&={\frac {8t+6}{3{\sqrt {2(t^{2}-3)}}}}&&\approx 7.889\,477\,399\,98\end{aligned}}$
where t is the tribonacci constant
$t={\frac {1+{\sqrt[{3}]{19-3{\sqrt {33}}}}+{\sqrt[{3}]{19+3{\sqrt {33}}}}}{3}}\approx 1.839\,286\,755\,21.$
If the original snub cube has edge length 1, its dual pentagonal icositetrahedron has side lengths
${\frac {1}{\sqrt {t+1}}}{\approx 0.593\,465}\quad {\text{and}}\quad {\frac {\sqrt {t+1}}{2}}\approx 0.842\,509.$.
In general, the volume of a snub cube with side length $a$ can be found with this formula, using the t as the tribonacci constant above:[1]
$V=a^{3}\cdot {\frac {3{\sqrt {t-1}}+4{\sqrt {t+1}}}{3{\sqrt {2-t}}}}=a^{3}\cdot {\frac {1}{3}}{\sqrt {188+{\sqrt[{3}]{6448437-45111{\sqrt {33}}}}+{\sqrt[{3}]{6448437+45111{\sqrt {33}}}}}}\approx 7.889\,477\,399\,98a^{3}$.
Cartesian coordinates
Cartesian coordinates for the vertices of a snub cube are all the even permutations of
(±1, ±1/t, ±t)
with an even number of plus signs, along with all the odd permutations with an odd number of plus signs, where t ≈ 1.83929 is the tribonacci constant. Taking the even permutations with an odd number of plus signs, and the odd permutations with an even number of plus signs, gives a different snub cube, the mirror image. Taking all of them together yields the compound of two snub cubes.
This snub cube has edges of length $\alpha ={\sqrt {2+4t-2t^{2}}}$, a number which satisfies the equation
$\alpha ^{6}-4\alpha ^{4}+16\alpha ^{2}-32=0,\,$
and can be written as
${\begin{aligned}\alpha &={\sqrt {{\frac {4}{3}}-{\frac {16}{3\beta }}+{\frac {2\beta }{3}}}}\approx 1.609\,72\\\beta &={\sqrt[{3}]{26+6{\sqrt {33}}}}\end{aligned}}$
To get a snub cube with unit edge length, divide all the coordinates above by the value α given above.
Orthogonal projections
The snub cube has two special orthogonal projections, centered, on two types of faces: triangles, and squares, correspond to the A2 and B2 Coxeter planes.
Orthogonal projections
Centered by Face
Triangle
Face
Square
Edge
Solid
Wireframe
Projective
symmetry
[3] [4]+ [2]
Dual
Spherical tiling
The snub cube can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Great circle arcs (geodesics) on the sphere are projected as circular arcs on the plane.
square-centered
Orthographic projection Stereographic projection
Geometric relations
Cube, rhombicuboctahedron and snub cube (animated expansion and twisting)
The snub cube can be generated by taking the six faces of the cube, pulling them outward so they no longer touch, then giving them each a small rotation on their centers (all clockwise or all counter-clockwise) until the spaces between can be filled with equilateral triangles.
Uniform alternation of a truncated cuboctahedron
The snub cube can also be derived from the truncated cuboctahedron by the process of alternation. 24 vertices of the truncated cuboctahedron form a polyhedron topologically equivalent to the snub cube; the other 24 form its mirror-image. The resulting polyhedron is vertex-transitive but not uniform.
An "improved" snub cube, with a slightly smaller square face and slightly larger triangular faces compared to Archimedes' uniform snub cube, is useful as a spherical design.[2]
Related polyhedra and tilings
The snub cube is one of a family of uniform polyhedra related to the cube and regular octahedron.
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
This semiregular polyhedron is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram . These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n = 6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons.
n32 symmetry mutations of snub tilings: 3.3.3.3.n
Symmetry
n32
Spherical Euclidean Compact hyperbolic Paracomp.
232 332 432 532 632 732 832 ∞32
Snub
figures
Config. 3.3.3.3.2 3.3.3.3.3 3.3.3.3.4 3.3.3.3.5 3.3.3.3.6 3.3.3.3.7 3.3.3.3.8 3.3.3.3.∞
Gyro
figures
Config. V3.3.3.3.2 V3.3.3.3.3 V3.3.3.3.4 V3.3.3.3.5 V3.3.3.3.6 V3.3.3.3.7 V3.3.3.3.8 V3.3.3.3.∞
The snub cube is second in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.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.∞
Snub cubical graph
Snub cubical graph
4-fold symmetry
Vertices24
Edges60
Automorphisms24
PropertiesHamiltonian, regular
Table of graphs and parameters
In the mathematical field of graph theory, a snub cubical graph is the graph of vertices and edges of the snub cube, one of the Archimedean solids. It has 24 vertices and 60 edges, and is an Archimedean graph.[3]
Orthogonal projection
See also
• Compound of two snub cubes
• Snub (geometry)
• Snub dodecahedron
• Snub square tiling
• Truncated cube
References
1. "Snub Cube - Geometry Calculator". rechneronline.de. Retrieved 2020-05-26.
2. "Spherical Designs" by R.H. Hardin and N.J.A. Sloane
3. Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
• Jayatilake, Udaya (March 2005). "Calculations on face and vertex regular polyhedra". Mathematical Gazette. 89 (514): 76–81. doi:10.1017/S0025557200176818. S2CID 125675814.
• 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)
• Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2.
External links
• Eric W. Weisstein, Snub cube (Archimedean solid) at MathWorld.
• Weisstein, Eric W. "Snub cubic graph". MathWorld.
• Klitzing, Richard. "3D convex uniform polyhedra s3s4s - snic".
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
• Editable printable net of a Snub Cube with interactive 3D view
Archimedean solids
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)
Catalan duals
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)
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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Wikipedia
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Snub hexahexagonal tiling
In geometry, the snub hexahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,6}.
Snub hexahexagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.3.6.3.6
Schläfli symbols{6,4}
sr{6,6}
Wythoff symbol| 6 6 2
Coxeter diagram
Symmetry group[6,6]+, (662)
[6+,4], (6*2)
DualOrder-6-6 floret hexagonal tiling
PropertiesVertex-transitive
Images
Drawn in chiral pairs, with edges missing between black triangles:
Symmetry
A higher symmetry coloring can be constructed from [6,4] symmetry as s{6,4}, . In this construction there is only one color of hexagon.
Related polyhedra and tiling
Uniform hexahexagonal tilings
Symmetry: [6,6], (*662)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
{6,6}
= h{4,6}
t{6,6}
= h2{4,6}
r{6,6}
{6,4}
t{6,6}
= h2{4,6}
{6,6}
= h{4,6}
rr{6,6}
r{6,4}
tr{6,6}
t{6,4}
Uniform duals
V66 V6.12.12 V6.6.6.6 V6.12.12 V66 V4.6.4.6 V4.12.12
Alternations
[1+,6,6]
(*663)
[6+,6]
(6*3)
[6,1+,6]
(*3232)
[6,6+]
(6*3)
[6,6,1+]
(*663)
[(6,6,2+)]
(2*33)
[6,6]+
(662)
= = =
h{6,6} s{6,6} hr{6,6} s{6,6} h{6,6} hrr{6,6} sr{6,6}
Uniform tetrahexagonal tilings
Symmetry: [6,4], (*642)
(with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries)
(And [(∞,3,∞,3)] (*3232) index 4 subsymmetry)
=
=
=
=
=
=
=
=
=
=
=
=
{6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4}
Uniform duals
V64 V4.12.12 V(4.6)2 V6.8.8 V46 V4.4.4.6 V4.8.12
Alternations
[1+,6,4]
(*443)
[6+,4]
(6*2)
[6,1+,4]
(*3222)
[6,4+]
(4*3)
[6,4,1+]
(*662)
[(6,4,2+)]
(2*32)
[6,4]+
(642)
=
=
=
=
=
=
h{6,4} s{6,4} hr{6,4} s{4,6} h{4,6} hrr{6,4} sr{6,4}
4n2 symmetry mutations of snub tilings: 3.3.n.3.n
Symmetry
4n2
Spherical Euclidean Compact hyperbolic Paracompact
222 322 442 552 662 772 882 ∞∞2
Snub
figures
Config. 3.3.2.3.2 3.3.3.3.3 3.3.4.3.4 3.3.5.3.5 3.3.6.3.6 3.3.7.3.7 3.3.8.3.8 3.3.∞.3.∞
Gyro
figures
Config. V3.3.2.3.2 V3.3.3.3.3 V3.3.4.3.4 V3.3.5.3.5 V3.3.6.3.6 V3.3.7.3.7 V3.3.8.3.8 V3.3.∞.3.∞
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.
See also
Wikimedia Commons has media related to Uniform tiling 3-3-6-3-6.
• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
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
|
Snub dodecahedron
In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.
Snub dodecahedron
(Click here for rotating model)
TypeArchimedean solid
Uniform polyhedron
ElementsF = 92, E = 150, V = 60 (χ = 2)
Faces by sides(20+60){3}+12{5}
Conway notationsD
Schläfli symbolssr{5,3} or $s{\begin{Bmatrix}5\\3\end{Bmatrix}}$
ht0,1,2{5,3}
Wythoff symbol| 2 3 5
Coxeter diagram
Symmetry groupI, 1/2H3, [5,3]+, (532), order 60
Rotation groupI, [5,3]+, (532), order 60
Dihedral angle3-3: 164°10′31″ (164.18°)
3-5: 152°55′53″ (152.93°)
ReferencesU29, C32, W18
PropertiesSemiregular convex chiral
Colored faces
3.3.3.3.5
(Vertex figure)
Pentagonal hexecontahedron
(dual polyhedron)
Net
The snub dodecahedron has 92 faces (the most of the 13 Archimedean solids): 12 are pentagons and the other 80 are equilateral triangles. It also has 150 edges, and 60 vertices.
It has two distinct forms, which are mirror images (or "enantiomorphs") of each other. The union of both forms is a compound of two snub dodecahedra, and the convex hull of both forms is a truncated icosidodecahedron.
Kepler first named it in Latin as dodecahedron simum in 1619 in his Harmonices Mundi. H. S. M. Coxeter, noting it could be derived equally from either the dodecahedron or the icosahedron, called it snub icosidodecahedron, with a vertical extended Schläfli symbol $s\scriptstyle {\begin{Bmatrix}5\\3\end{Bmatrix}}$ and flat Schläfli symbol sr{5,3}.
Cartesian coordinates
Let ξ ≈ 0.94315125924 be the real zero of the cubic polynomial x3 + 2x2 − φ2, where φ is the golden ratio. Let the point p be given by
$p={\begin{pmatrix}\phi ^{2}-\phi ^{2}\xi \\-\phi ^{3}+\phi \xi +2\phi \xi ^{2}\\\xi \end{pmatrix}}$.
Let the rotation matrices M1 and M2 be given by
$M_{1}={\begin{pmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{pmatrix}}$
and
$M_{2}={\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix}}.$
M1 represents the rotation around the axis (0,1,φ) through an angle of 2π/5 counterclockwise, while M2 being a cyclic shift of (x,y,z) represents the rotation around the axis (1,1,1) through an angle of 2π/3. Then the 60 vertices of the snub dodecahedron are the 60 images of point p under repeated multiplication by M1 and/or M2, iterated to convergence. (The matrices M1 and M2 generate the 60 rotation matrices corresponding to the 60 rotational symmetries of a regular icosahedron.) The coordinates of the vertices are integral linear combinations of 1, φ, ξ, φξ, ξ2 and φξ2. The edge length equals
$2\xi {\sqrt {1-\xi }}\approx 0.449\,750\,618\,41.$
Negating all coordinates gives the mirror image of this snub dodecahedron.
As a volume, the snub dodecahedron consists of 80 triangular and 12 pentagonal pyramids. The volume V3 of one triangular pyramid is given by:
$V_{3}={\frac {1}{3}}\phi \left(3\xi ^{2}-\phi ^{2}\right)\approx 0.027\,274\,068\,85,$
and the volume V5 of one pentagonal pyramid by:
$V_{5}={\frac {1}{3}}(3\phi +1)\left(\phi +3-2\xi -3\xi ^{2}\right)\xi ^{3}\approx 0.103\,349\,665\,04.$
The total volume is
$80V_{3}+12V_{5}\approx 3.422\,121\,488\,76.$
The circumradius equals
${\sqrt {4\xi ^{2}-\phi ^{2}}}\approx 0.969\,589\,192\,65.$
The midradius equals ξ. This gives an interesting geometrical interpretation of the number ξ. The 20 "icosahedral" triangles of the snub dodecahedron described above are coplanar with the faces of a regular icosahedron. The midradius of this "circumscribed" icosahedron equals 1. This means that ξ is the ratio between the midradii of a snub dodecahedron and the icosahedron in which it is inscribed.
The triangle–triangle dihedral angle is given by
$\theta _{33}=180^{\circ }-\arccos \left({\frac {2}{3}}\xi +{\frac {1}{3}}\right)\approx 164.175\,366\,056\,03^{\circ }.$
The triangle–pentagon dihedral angle is given by
$\theta _{35}=180^{\circ }-\arccos {\sqrt {\frac {-(4\phi +8)\xi ^{2}-(4\phi +8)\xi +12\phi +19}{15}}}\approx 152.929\,920\,275\,84^{\circ }.$
Metric properties
For a snub dodecahedron whose edge length is 1, the surface area is
$A=20{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}\approx 55.286\,744\,958\,445\,15.$
Its volume is
$V={\frac {(3\phi +1)\xi ^{2}+(3\phi +1)\xi -\phi /6-2}{\sqrt {3\xi ^{2}-\phi ^{2}}}}\approx 37.616\,649\,962\,733\,36.$
Alternatively, this volume may be written as
$V={\frac {5+5{\sqrt {5}}}{6{\sqrt {3}}}}{\sqrt {{18+6{\sqrt {5}}}+{a\left({3+3{\sqrt {5}}}\,{+}\,{a}\right)}}}+{\frac {5+3{\sqrt {5}}}{24{\sqrt {2}}}}{\sqrt {72+{\left({5+{\sqrt {5}}}\right)}a\left({3+3{\sqrt {5}}}\,{+}\,a\right)}}\approx 37.616\,649\,962\,733\,36,$
where,
$a={\sqrt[{3}]{54(1+{\sqrt {5}})+6{\sqrt {102+162{\sqrt {5}}}}}}+{\sqrt[{3}]{54(1+{\sqrt {5}})-6{\sqrt {102+162{\sqrt {5}}}}}}\approx 10.293\,368\,998\,184\,21.$
Its circumradius is
$R={\frac {1}{2}}{\sqrt {\frac {2-\xi }{1-\xi }}}\approx 2.155\,837\,375.$
Its midradius is
$r={\frac {1}{2}}{\sqrt {\frac {1}{1-\xi }}}\approx 2.097\,053\,835\,25.$
There are two inscribed spheres, one touching the triangular faces, and one, slightly smaller, touching the pentagonal faces. Their radii are, respectively:
$r_{3}={\frac {\phi {\sqrt {3}}}{6\xi }}{\sqrt {\frac {1}{1-\xi }}}\approx 2.077\,089\,659\,74$
and
$r_{5}={\frac {1}{2}}{\sqrt {\phi ^{2}\xi ^{2}+3\phi ^{2}\xi +{\frac {11}{5}}\phi +{\frac {12}{5}}}}\approx 1.980\,915\,947\,28.$
The four positive real roots of the sextic equation in R2
$4096R^{12}-27648R^{10}+47104R^{8}-35776R^{6}+13872R^{4}-2696R^{2}+209=0$
are the circumradii of the snub dodecahedron (U29), great snub icosidodecahedron (U57), great inverted snub icosidodecahedron (U69), and great retrosnub icosidodecahedron (U74).
The snub dodecahedron has the highest sphericity of all Archimedean solids. If sphericity is defined as the ratio of volume squared over surface area cubed, multiplied by a constant of 36π (where this constant makes the sphericity of a sphere equal to 1), the sphericity of the snub dodecahedron is about 0.947.[1]
Orthogonal projections
The snub dodecahedron has two especially symmetric orthogonal projections as shown below, centered on two types of faces: triangles and pentagons, corresponding to the A2 and H2 Coxeter planes.
Orthogonal projections
Centered by Face
Triangle
Face
Pentagon
Edge
Solid
Wireframe
Projective
symmetry
[3] [5]+ [2]
Dual
Geometric relations
Dodecahedron, rhombicosidodecahedron and snub dodecahedron (animated expansion and twisting)
Uniform alternation of a truncated icosidodecahedron
The snub dodecahedron can be generated by taking the twelve pentagonal faces of the dodecahedron and pulling them outward so they no longer touch. At a proper distance this can create the rhombicosidodecahedron by filling in square faces between the divided edges and triangle faces between the divided vertices. But for the snub form, pull the pentagonal faces out slightly less, only add the triangle faces and leave the other gaps empty (the other gaps are rectangles at this point). Then apply an equal rotation to the centers of the pentagons and triangles, continuing the rotation until the gaps can be filled by two equilateral triangles. (The fact that the proper amount to pull the faces out is less in the case of the snub dodecahedron can be seen in either of two ways: the circumradius of the snub dodecahedron is smaller than that of the icosidodecahedron; or, the edge length of the equilateral triangles formed by the divided vertices increases when the pentagonal faces are rotated.)
The snub dodecahedron can also be derived from the truncated icosidodecahedron by the process of alternation. Sixty of the vertices of the truncated icosidodecahedron form a polyhedron topologically equivalent to one snub dodecahedron; the remaining sixty form its mirror-image. The resulting polyhedron is vertex-transitive but not uniform.
Alternatively, combining the vertices of the snub dodecahedron given by the Cartesian coordinates (above) and its mirror will form a semiregular truncated icosidodecahedron. The comparisons between these regular and semiregular polyhedrons is shown in the figure to the right.
Cartesian coordinates for the vertices of this alternative snub dodecahedron are obtained by selecting sets of 12 (of 24 possible even permutations contained in the five sets of truncated icosidodecahedron Cartesian coordinates). The alternations are those with an odd number of minus signs in these three sets:
Overlay of regular and semiregular truncated icosidodecahedra and snub dodecahedra
(±1/φ, ±1/φ, ±(3 + φ)),
(±1/φ, ±φ2, ±(−1 + 3φ)),
(±(2φ − 1), ±2, ±(2 + φ))
and an even number of minus signs in the these two sets:
(±2/φ, ±φ, ±(1 + 2φ)),
(±φ, ±3, ±2φ)
where φ = 1 + √5/2 is the golden ratio. The mirrors of both the regular truncated icosidodecahedron and this alternative snub dodecahedron are obtained by switching the even and odd references to both sign and position permutations.
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
This semiregular polyhedron is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram . These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n = 6, and hyperbolic plane for any higher n. The series can be considered to begin with n = 2, with one set of faces degenerated into digons.
n32 symmetry mutations of snub tilings: 3.3.3.3.n
Symmetry
n32
Spherical Euclidean Compact hyperbolic Paracomp.
232 332 432 532 632 732 832 ∞32
Snub
figures
Config. 3.3.3.3.2 3.3.3.3.3 3.3.3.3.4 3.3.3.3.5 3.3.3.3.6 3.3.3.3.7 3.3.3.3.8 3.3.3.3.∞
Gyro
figures
Config. V3.3.3.3.2 V3.3.3.3.3 V3.3.3.3.4 V3.3.3.3.5 V3.3.3.3.6 V3.3.3.3.7 V3.3.3.3.8 V3.3.3.3.∞
Snub dodecahedral graph
Snub dodecahedral graph
5-fold symmetry Schlegel diagram
Vertices60
Edges150
Automorphisms60
PropertiesHamiltonian, regular
Table of graphs and parameters
In the mathematical field of graph theory, a snub dodecahedral graph is the graph of vertices and edges of the snub dodecahedron, one of the Archimedean solids. It has 60 vertices and 150 edges, and is an Archimedean graph.[2]
See also
• Planar polygon to polyhedron transformation animation
• ccw and cw spinning snub dodecahedron
References
1. How Spherical Are the Archimedean Solids and Their Duals? P. K. Aravind, The College Mathematics Journal, Vol. 42, No. 2 (March 2011), pp. 98-107
2. Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
• Jayatilake, Udaya (March 2005). "Calculations on face and vertex regular polyhedra". Mathematical Gazette. 89 (514): 76–81. doi:10.1017/S0025557200176818. S2CID 125675814.
• 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)
• Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2.
External links
• Eric W. Weisstein, Snub dodecahedron (Archimedean solid) at MathWorld.
• Weisstein, Eric W. "Snub dodecahedral graph". MathWorld.
• Klitzing, Richard. "3D convex uniform polyhedra s3s5s - snid".
• Editable printable net of a Snub Dodecahedron with interactive 3D view
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
• Mark S. Adams and Menno T. Kosters. Volume Solutions to the Snub Dodecahedron
Archimedean solids
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)
Catalan duals
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)
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
|
Snub icosidodecadodecahedron
In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U46. It has 104 faces (80 triangles, 12 pentagons, and 12 pentagrams), 180 edges, and 60 vertices.[1] As the name indicates, it belongs to the family of snub polyhedra.
Snub icosidodecadodecahedron
TypeUniform star polyhedron
ElementsF = 104, E = 180
V = 60 (χ = −16)
Faces by sides(20+60){3}+12{5}+12{5/2}
Coxeter diagram
Wythoff symbol| 5/3 3 5
Symmetry groupI, [5,3]+, 532
Index referencesU46, C58, W112
Dual polyhedronMedial hexagonal hexecontahedron
Vertex figure
3.3.3.5.3.5/3
Bowers acronymSided
The circumradius of the snub icosidodecadodecahedron with unit edge length is
${\frac {1}{2}}{\sqrt {\frac {2\rho -1}{\rho -1}}},$
where ρ is the plastic constant, or the unique real root of ρ3 = ρ + 1.[2]
Cartesian coordinates
Cartesian coordinates for the vertices of a snub icosidodecadodecahedron are all the even permutations of
(±2α, ±2γ, ±2β),
(±(α+β/τ+γτ), ±(-ατ+β+γ/τ), ±(α/τ+βτ-γ)),
(±(-α/τ+βτ+γ), ±(-α+β/τ-γτ), ±(ατ+β-γ/τ)),
(±(-α/τ+βτ-γ), ±(α-β/τ-γτ), ±(ατ+β+γ/τ)) and
(±(α+β/τ-γτ), ±(ατ-β+γ/τ), ±(α/τ+βτ+γ))
with an even number of plus signs, where τ = (1+√5)/2 is the golden ratio; ρ is the plastic constant, or the unique real solution to ρ3=ρ+1;
α = ρ+1 = ρ3;
β = τ2ρ4+τ; and
γ = ρ2+τρ.
Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.[3]
Related polyhedra
Medial hexagonal hexecontahedron
Medial hexagonal hexecontahedron
TypeStar polyhedron
Face
ElementsF = 60, E = 180
V = 104 (χ = −16)
Symmetry groupI, [5,3]+, 532
Index referencesDU46
dual polyhedronSnub icosidodecadodecahedron
The medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.
See also
• List of uniform polyhedra
References
1. Maeder, Roman. "46: snub icosidodecadodecahedron". MathConsult.{{cite web}}: CS1 maint: url-status (link)
2. Weisstein, Eric W. "Snub icosidodecadodecahedron". MathWorld.
3. Skilling, John (1975), "The complete set of uniform polyhedra", Philosophical Transactions of the Royal Society A, 278 (1278): 111–135, doi:10.1098/rsta.1975.0022.
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208
External links
• Weisstein, Eric W. "Medial hexagonal hexecontahedron". MathWorld.
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
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Compound of five icosahedra
The compound of five icosahedra is uniform polyhedron compound. It's composed of 5 icosahedra, rotated around a common axis. It has icosahedral symmetry Ih.
Compound of five icosahedra
TypeUniform compound
IndexUC47
Polyhedra5 icosahedra
Faces40+60 Triangles
Edges150
Vertices60
Symmetry groupicosahedral (Ih)
Subgroup restricting to one constituentpyritohedral (Th)
The triangles in this compound decompose into two orbits under action of the symmetry group: 40 of the triangles lie in coplanar pairs in icosahedral planes, while the other 60 lie in unique planes.
Cartesian coordinates
Cartesian coordinates for the vertices of this compound are all the cyclic permutations of
(0, ±2, ±2τ)
(±τ−1, ±1, ±(1+τ2))
(±τ, ±τ2, ±(2τ−1))
where τ = (1+√5)/2 is the golden ratio (sometimes written φ).
References
• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (3): 447–457, doi:10.1017/S0305004100052440, MR 0397554.
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Snub 24-cell honeycomb
In four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation (or honeycomb) by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 paper of semiregular polytopes. It is not semiregular by Gosset's definition of regular facets, but all of its cells (ridges) are regular, either tetrahedra or icosahedra.
Snub 24-cell honeycomb
(No image)
TypeUniform 4-honeycomb
Schläfli symbolss{3,4,3,3}
sr{3,3,4,3}
2sr{4,3,3,4}
2sr{4,3,31,1}
s{31,1,1,1}
Coxeter diagrams
=
4-face typesnub 24-cell
16-cell
5-cell
Cell type{3,3}
{3,5}
Face typetriangle {3}
Vertex figure
Irregular decachoron
Symmetries[3+,4,3,3]
[3,4,(3,3)+]
[4,(3,3)+,4]
[4,(3,31,1)+]
[31,1,1,1]+
PropertiesVertex transitive, nonWythoffian
It can be seen as an alternation of a truncated 24-cell honeycomb, and can be represented by Schläfli symbol s{3,4,3,3}, s{31,1,1,1}, and 3 other snub constructions.
It is defined by an irregular decachoron vertex figure (10-celled 4-polytope), faceted by four snub 24-cells, one 16-cell, and five 5-cells. The vertex figure can be seen topologically as a modified tetrahedral prism, where one of the tetrahedra is subdivided at mid-edges into a central octahedron and four corner tetrahedra. Then the four side-facets of the prism, the triangular prisms become tridiminished icosahedra.
Symmetry constructions
There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored snub 24-cell, 16-cell, and 5-cell facets. In all cases, four snub 24-cells, five 5-cells, and one 16-cell meet at each vertex, but the vertex figures have different symmetry generators.
Symmetry Coxeter
Schläfli
Facets (on vertex figure)
Snub 24-cell
(4)
16-cell
(1)
5-cell
(5)
[3+,4,3,3]
s{3,4,3,3}
4:
[3,4,(3,3)+]
sr{3,3,4,3}
3:
1:
[[4,(3,3)+,4]]
2sr{4,3,3,4}
2,2:
[(31,1,3)+,4]
2sr{4,3,31,1}
1,1:
2:
[31,1,1,1]+
s{31,1,1,1}
1,1,1,1:
See also
Regular and uniform honeycombs in 4-space:
• Tesseractic honeycomb
• 16-cell honeycomb
• 24-cell honeycomb
• Truncated 24-cell honeycomb
• 5-cell honeycomb
• Truncated 5-cell honeycomb
• Omnitruncated 5-cell honeycomb
References
• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• 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 133
• Klitzing, Richard. "4D Euclidean tesselations"., o4s3s3s4o, s3s3s *b3s4o, s3s3s *b3s *b3s, o3o3o4s3s, s3s3s4o3o - sadit - O133
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
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Icosahedral prism
In geometry, an icosahedral prism is a convex uniform 4-polytope (four-dimensional polytope). This 4-polytope has 22 polyhedral cells: 2 icosahedra connected by 20 triangular prisms. It has 70 faces: 30 squares and 40 triangles. It has 72 edges and 24 vertices.
Icosahedral prism
TypePrismatic uniform 4-polytope
Uniform index59
Schläfli symbolt{2,3,5} or {3,5}×{}
s{3,4}×{}
sr{3,3}×{}
Coxeter-Dynkin
Cells2 (3.3.3.3.3)
20 (3.4.4)
Faces30 {4}
40 {3}
Edges72
Vertices24
Vertex figure
pentagonal pyramids
DualDodecahedral bipyramid
Symmetry group[5,3,2], order 240
[3+,4,2], order 48
[(3,3)+,2], order 24
Propertiesconvex
It can be constructed by creating two coinciding icosahedra in 3-space, and translating each copy in opposite perpendicular directions in 4-space until their separation equals their edge length.
It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids or Archimedean solids.
Net
Schlegel diagram
Only one icosahedral cell shown
Orthographic projection
Alternate names
1. Icosahedral dyadic prism Norman W. Johnson
2. Ipe for icosahedral prism/hyperprism (Jonathan Bowers)
3. Snub tetrahedral prism/hyperprism
Related polytopes
• Snub tetrahedral antiprism - $s\left\{{\begin{array}{l}3\\3\\2\end{array}}\right\}$ = ht0,1,2,3{3,3,2} or , a related nonuniform 4-polytope
External links
• 6. Convex uniform prismatic polychora - Model 59, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora) x o3o5x - ipe".
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Snub octaoctagonal tiling
In geometry, the snub octaoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{8,8}.
Snub octaoctagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.3.8.3.8
Schläfli symbols{8,4}
sr{8,8}
Wythoff symbol| 8 8 2
Coxeter diagram
or
Symmetry group[8,8]+, (882)
[8+,4], (8*2)
DualOrder-8-8 floret hexagonal tiling
PropertiesVertex-transitive
Images
Drawn in chiral pairs, with edges missing between black triangles:
Symmetry
A higher symmetry coloring can be constructed from [8,4] symmetry as s{8,4}, . In this construction there is only one color of octagon.
Related polyhedra and tiling
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 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
4n2 symmetry mutations of snub tilings: 3.3.n.3.n
Symmetry
4n2
Spherical Euclidean Compact hyperbolic Paracompact
222 322 442 552 662 772 882 ∞∞2
Snub
figures
Config. 3.3.2.3.2 3.3.3.3.3 3.3.4.3.4 3.3.5.3.5 3.3.6.3.6 3.3.7.3.7 3.3.8.3.8 3.3.∞.3.∞
Gyro
figures
Config. V3.3.2.3.2 V3.3.3.3.3 V3.3.4.3.4 V3.3.5.3.5 V3.3.6.3.6 V3.3.7.3.7 V3.3.8.3.8 V3.3.∞.3.∞
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.
See also
Wikimedia Commons has media related to Uniform tiling 3-3-8-3-8.
• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
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
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Snub triheptagonal tiling
In geometry, the order-3 snub heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles and one heptagon on each vertex. It has Schläfli symbol of sr{7,3}. The snub tetraheptagonal tiling is another related hyperbolic tiling with Schläfli symbol sr{7,4}.
Snub triheptagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.3.3.3.7
Schläfli symbolsr{7,3} or $s{\begin{Bmatrix}7\\3\end{Bmatrix}}$
Wythoff symbol| 7 3 2
Coxeter diagram or
Symmetry group[7,3]+, (732)
DualOrder-7-3 floret pentagonal tiling
PropertiesVertex-transitive Chiral
Images
Drawn in chiral pairs, with edges missing between black triangles:
Dual tiling
The dual tiling is called an order-7-3 floret pentagonal tiling, and is related to the floret pentagonal tiling.
Related polyhedra and tilings
This semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram . These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons.
n32 symmetry mutations of snub tilings: 3.3.3.3.n
Symmetry
n32
Spherical Euclidean Compact hyperbolic Paracomp.
232 332 432 532 632 732 832 ∞32
Snub
figures
Config. 3.3.3.3.2 3.3.3.3.3 3.3.3.3.4 3.3.3.3.5 3.3.3.3.6 3.3.3.3.7 3.3.3.3.8 3.3.3.3.∞
Gyro
figures
Config. V3.3.3.3.2 V3.3.3.3.3 V3.3.3.3.4 V3.3.3.3.5 V3.3.3.3.6 V3.3.3.3.7 V3.3.3.3.8 V3.3.3.3.∞
From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.
Uniform heptagonal/triangular tilings
Symmetry: [7,3], (*732) [7,3]+, (732)
{7,3} t{7,3} r{7,3} t{3,7} {3,7} rr{7,3} tr{7,3} sr{7,3}
Uniform duals
V73 V3.14.14 V3.7.3.7 V6.6.7 V37 V3.4.7.4 V4.6.14 V3.3.3.3.7
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.
See also
Wikimedia Commons has media related to Uniform tiling 3-3-3-3-7.
• Snub hexagonal tiling
• Floret pentagonal tiling
• Order-3 heptagonal tiling
• Tilings of regular polygons
• List of uniform planar tilings
• Kagome lattice
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
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Snub pentapentagonal tiling
In geometry, the snub pentapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{5,5}, constructed from two regular pentagons and three equilateral triangles around every vertex.
Snub pentapentagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.3.5.3.5
Schläfli symbols{5,4}
sr{5,5}
Wythoff symbol| 5 5 2
Coxeter diagram
or
Symmetry group[5+,4], (5*2)
[5,5]+, (552)
DualOrder-5-5 floret pentagonal tiling
PropertiesVertex-transitive
Images
Drawn in chiral pairs, with edges missing between black triangles:
Symmetry
A double symmetry coloring can be constructed from [5,4] symmetry with only one color pentagon. It has Schläfli symbol s{5,4}, and Coxeter diagram .
Related tilings
Uniform pentapentagonal tilings
Symmetry: [5,5], (*552) [5,5]+, (552)
=
=
=
=
=
=
=
=
Order-5 pentagonal tiling
{5,5}
Truncated order-5 pentagonal tiling
t{5,5}
Order-4 pentagonal tiling
r{5,5}
Truncated order-5 pentagonal tiling
2t{5,5} = t{5,5}
Order-5 pentagonal tiling
2r{5,5} = {5,5}
Tetrapentagonal tiling
rr{5,5}
Truncated order-4 pentagonal tiling
tr{5,5}
Snub pentapentagonal tiling
sr{5,5}
Uniform duals
Order-5 pentagonal tiling
V5.5.5.5.5
V5.10.10 Order-5 square tiling
V5.5.5.5
V5.10.10 Order-5 pentagonal tiling
V5.5.5.5.5
V4.5.4.5 V4.10.10 V3.3.5.3.5
Uniform pentagonal/square tilings
Symmetry: [5,4], (*542) [5,4]+, (542) [5+,4], (5*2) [5,4,1+], (*552)
{5,4} t{5,4} r{5,4} 2t{5,4}=t{4,5} 2r{5,4}={4,5} rr{5,4} tr{5,4} sr{5,4} s{5,4} h{4,5}
Uniform duals
V54 V4.10.10 V4.5.4.5 V5.8.8 V45 V4.4.5.4 V4.8.10 V3.3.4.3.5 V3.3.5.3.5 V55
4n2 symmetry mutations of snub tilings: 3.3.n.3.n
Symmetry
4n2
Spherical Euclidean Compact hyperbolic Paracompact
222 322 442 552 662 772 882 ∞∞2
Snub
figures
Config. 3.3.2.3.2 3.3.3.3.3 3.3.4.3.4 3.3.5.3.5 3.3.6.3.6 3.3.7.3.7 3.3.8.3.8 3.3.∞.3.∞
Gyro
figures
Config. V3.3.2.3.2 V3.3.3.3.3 V3.3.4.3.4 V3.3.5.3.5 V3.3.6.3.6 V3.3.7.3.7 V3.3.8.3.8 V3.3.∞.3.∞
See also
Wikimedia Commons has media related to Uniform tiling 3-3-5-3-5.
• Square tiling
• 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
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Wikipedia
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Snub polyhedron
In geometry, a snub polyhedron is a polyhedron obtained by performing a snub operation: alternating a corresponding omnitruncated or truncated polyhedron, depending on the definition. Some, but not all, authors include antiprisms as snub polyhedra, as they are obtained by this construction from a degenerate "polyhedron" with only two faces (a dihedron).
Polyhedron
Class Number and properties
Platonic solids
(5, convex, regular)
Archimedean solids
(13, convex, uniform)
Kepler–Poinsot polyhedra
(4, regular, non-convex)
Uniform polyhedra
(75, uniform)
Prismatoid:
prisms, antiprisms etc.
(4 infinite uniform classes)
Polyhedra tilings (11 regular, in the plane)
Quasi-regular polyhedra
(8)
Johnson solids (92, convex, non-uniform)
Pyramids and Bipyramids (infinite)
Stellations Stellations
Polyhedral compounds (5 regular)
Deltahedra (Deltahedra,
equilateral triangle faces)
Snub polyhedra
(12 uniform, not mirror image)
Zonohedron (Zonohedra,
faces have 180°symmetry)
Dual polyhedron
Self-dual polyhedron (infinite)
Catalan solid (13, Archimedean dual)
Chiral snub polyhedra do not always have reflection symmetry and hence sometimes have two enantiomorphous (left- and right-handed) forms which are reflections of each other. Their symmetry groups are all point groups.
For example, the snub cube:
Snub polyhedra have Wythoff symbol | p q r and by extension, vertex configuration 3.p.3.q.3.r. Retrosnub polyhedra (a subset of the snub polyhedron, containing the great icosahedron, small retrosnub icosicosidodecahedron, and great retrosnub icosidodecahedron) still have this form of Wythoff symbol, but their vertex configurations are instead ${\frac {(3.-p.3.-q.3.-r)}{2}}.$
List of snub polyhedra
Uniform
There are 12 uniform snub polyhedra, not including the antiprisms, the icosahedron as a snub tetrahedron, the great icosahedron as a retrosnub tetrahedron and the great disnub dirhombidodecahedron, also known as Skilling's figure.
When the Schwarz triangle of the snub polyhedron is isosceles, the snub polyhedron is not chiral. This is the case for the antiprisms, the icosahedron, the great icosahedron, the small snub icosicosidodecahedron, and the small retrosnub icosicosidodecahedron.
In the pictures of the snub derivation (showing a distorted snub polyhedron, topologically identical to the uniform version, arrived at from geometrically alternating the parent uniform omnitruncated polyhedron) where green is not present, the faces derived from alternation are coloured red and yellow, while the snub triangles are blue. Where green is present (only for the snub icosidodecadodecahedron and great snub dodecicosidodecahedron), the faces derived from alternation are red, yellow, and blue, while the snub triangles are green.
Snub polyhedron Image Original omnitruncated polyhedron Image Snub derivation Symmetry group Wythoff symbol
Vertex description
Icosahedron (snub tetrahedron) Truncated octahedron Ih (Th) | 3 3 2
3.3.3.3.3
Great icosahedron (retrosnub tetrahedron) Truncated octahedron Ih (Th) | 2 3/2 3/2
(3.3.3.3.3)/2
Snub cube
or snub cuboctahedron
Truncated cuboctahedron O | 4 3 2
3.3.3.3.4
Snub dodecahedron
or snub icosidodecahedron
Truncated icosidodecahedron I | 5 3 2
3.3.3.3.5
Small snub icosicosidodecahedron Doubly covered truncated icosahedron Ih | 3 3 5/2
3.3.3.3.3.5/2
Snub dodecadodecahedron Small rhombidodecahedron with extra 12{10/2} faces I | 5 5/2 2
3.3.5/2.3.5
Snub icosidodecadodecahedron Icositruncated dodecadodecahedron I | 5 3 5/3
3.5/3.3.3.3.5
Great snub icosidodecahedron Rhombicosahedron with extra 12{10/2} faces I | 3 5/2 2
3.3.5/2.3.3
Inverted snub dodecadodecahedron Truncated dodecadodecahedron I | 5 2 5/3
3.5/3.3.3.3.5
Great snub dodecicosidodecahedron Great dodecicosahedron with extra 12{10/2} faces no image yet I | 3 5/2 5/3
3.5/3.3.5/2.3.3
Great inverted snub icosidodecahedron Great truncated icosidodecahedron I | 3 2 5/3
3.5/3.3.3.3
Small retrosnub icosicosidodecahedron Doubly covered truncated icosahedron no image yet Ih | 5/2 3/2 3/2
(3.3.3.3.3.5/2)/2
Great retrosnub icosidodecahedron Great rhombidodecahedron with extra 20{6/2} faces no image yet I | 2 5/3 3/2
(3.3.3.5/2.3)/2
Great dirhombicosidodecahedron — — — Ih | 3/2 5/3 3 5/2
(4.3/2.4.5/3.4.3.4.5/2)/2
Great disnub dirhombidodecahedron — — — Ih | (3/2) 5/3 (3) 5/2
(3/2.3/2.3/2.4.5/3.4.3.3.3.4.5/2.4)/2
Notes:
• The icosahedron, snub cube and snub dodecahedron are the only three convex ones. They are obtained by snubification of the truncated octahedron, truncated cuboctahedron and the truncated icosidodecahedron - the three convex truncated quasiregular polyhedra.
• The only snub polyhedron with the chiral octahedral group of symmetries is the snub cube.
• Only the icosahedron and the great icosahedron are also regular polyhedra. They are also deltahedra.
• Only the icosahedron, great icosahedron, small snub icosicosidodecahedron, small retrosnub icosicosidodecahedron, great dirhombicosidodecahedron, and great disnub dirhombidodecahedron also have reflective symmetries.
There is also the infinite set of antiprisms. They are formed from prisms, which are truncated hosohedra, degenerate regular polyhedra. Those up to hexagonal are listed below. In the pictures showing the snub derivation, the faces derived from alternation (of the prism bases) are coloured red, and the snub triangles are coloured yellow. The exception is the tetrahedron, for which all the faces are derived as red snub triangles, as alternating the square bases of the cube results in degenerate digons as faces.
Snub polyhedron Image Original omnitruncated polyhedron Image Snub derivation Symmetry group Wythoff symbol
Vertex description
Tetrahedron Cube Td (D2d) | 2 2 2
3.3.3
Octahedron Hexagonal prism Oh (D3d) | 3 2 2
3.3.3.3
Square antiprism Octagonal prism D4d | 4 2 2
3.4.3.3
Pentagonal antiprism Decagonal prism D5d | 5 2 2
3.5.3.3
Pentagrammic antiprism Doubly covered pentagonal prism D5h | 5/2 2 2
3.5/2.3.3
Pentagrammic crossed-antiprism Decagrammic prism D5d | 2 2 5/3
3.5/3.3.3
Hexagonal antiprism Dodecagonal prism D6d | 6 2 2
3.6.3.3
Notes:
• Two of these polyhedra may be constructed from the first two snub polyhedra in the list starting with the icosahedron: the pentagonal antiprism is a parabidiminished icosahedron and a pentagrammic crossed-antiprism is a parabidiminished great icosahedron, also known as a parabireplenished great icosahedron.
Non-uniform
Two Johnson solids are snub polyhedra: the snub disphenoid and the snub square antiprism. Neither is chiral.
Snub polyhedron Image Original polyhedron Image Symmetry group
Snub disphenoid Disphenoid D2d
Snub square antiprism Square antiprism D4d
References
• Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J. C. P. (1954), "Uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 246 (916): 401–450, doi:10.1098/rsta.1954.0003, ISSN 0080-4614, JSTOR 91532, MR 0062446, S2CID 202575183
• Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.
• Skilling, J. (1975), "The complete set of uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 278 (1278): 111–135, doi:10.1098/rsta.1975.0022, ISSN 0080-4614, JSTOR 74475, MR 0365333, S2CID 122634260
• Mäder, R. E. Uniform Polyhedra. Mathematica J. 3, 48-57, 1993.
Polyhedron operators
Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations
t0{p,q}
{p,q}
t01{p,q}
t{p,q}
t1{p,q}
r{p,q}
t12{p,q}
2t{p,q}
t2{p,q}
2r{p,q}
t02{p,q}
rr{p,q}
t012{p,q}
tr{p,q}
ht0{p,q}
h{q,p}
ht12{p,q}
s{q,p}
ht012{p,q}
sr{p,q}
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Wikipedia
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Snub rhombicuboctahedron
The snub rhombicuboctahedron is a polyhedron, constructed as a truncated rhombicuboctahedron. It has 74 faces: 18 squares, and 56 triangles. It can also be called the Conway snub cuboctahedron in but will be confused with the Coxeter snub cuboctahedron, the snub cube.
Snub rhombicuboctahedron
Schläfli symbolsrr{4,3} = $sr{\begin{Bmatrix}4\\3\end{Bmatrix}}$
Conway notationsaC
Faces74:
8+48 {3}
6+12 {4}
Edges120
Vertices48
Symmetry groupO, [4,3]+, (432) order 24
Dual polyhedronPentagonal tetracontoctahedron
Propertiesconvex, chiral
Related polyhedra
The snub rhombicuboctahedron can be seen in sequence of operations from the cuboctahedron.
Name Cubocta-
hedron
Truncated
cubocta-
hedron
Snub
cubocta-
hedron
Truncated
rhombi-
cubocta-
hedron
Snub
rhombi-
cubocta-
hedron
Coxeter CO (rC) tCO (trC) sCO (srC) trCO (trrC) srCO (htrrC)
Conway aC taC = bC sC taaC = baC saC
Image
Conway jC mC gC maC gaC
Dual
See also
• Expanded cuboctahedron
• Truncated rhombicosidodecahedron
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
External links
• George Hart's Conway interpreter: generates polyhedra in VRML, taking Conway notation as input
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Wikipedia
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Snub square antiprism
In geometry, the snub square antiprism is one of the Johnson solids (J85). A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]
Snub square antiprism
TypeJohnson
J84 – J85 – J86
Faces8+16 triangles
2 squares
Edges40
Vertices16
Vertex configuration8(35)
8(34.4)
Symmetry groupD4d
Dual polyhedron-
Propertiesconvex
Net
It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids, although it is a relative of the icosahedron that has fourfold symmetry instead of threefold.
Construction
The snub square antiprism is constructed as its name suggests, a square antiprism which is snubbed, and represented as ss{2,8}, with s{2,8} as a square antiprism.[2] It can be constructed in Conway polyhedron notation as sY4 (snub square pyramid).[3]
It can also be constructed as a square gyrobianticupolae, connecting two anticupolae with gyrated orientations.
Cartesian coordinates
Let k ≈ 0.82354 be the positive root of the cubic polynomial
$9x^{3}+3{\sqrt {3}}\left(5-{\sqrt {2}}\right)x^{2}-3\left(5-2{\sqrt {2}}\right)x-17{\sqrt {3}}+7{\sqrt {6}}.$
Furthermore, let h ≈ 1.35374 be defined by
$h={\frac {{\sqrt {2}}+8+2{\sqrt {3}}k-3\left(2+{\sqrt {2}}\right)k^{2}}{4{\sqrt {3-3k^{2}}}}}.$
Then, Cartesian coordinates of a snub square antiprism with edge length 2 are given by the union of the orbits of the points
$(1,1,h),\,\left(1+{\sqrt {3}}k,0,h-{\sqrt {3-3k^{2}}}\right)$
under the action of the group generated by a rotation around the z-axis by 90° and by a rotation by 180° around a straight line perpendicular to the z-axis and making an angle of 22.5° with the x-axis.[4]
We may then calculate the surface area of a snub square antiprism of edge length a as
$A=\left(2+6{\sqrt {3}}\right)a^{2}\approx 12.39230a^{2},$[5]
and its volume as
$V=\xi a^{3},$
where ξ ≈ 3.60122 is the greatest real root of the polynomial
$531441x^{12}-85726026x^{8}-48347280x^{6}+11588832x^{4}+4759488x^{2}-892448.$[6]
Snub antiprisms
Similarly constructed, the ss{2,6} is a snub triangular antiprism (a lower symmetry octahedron), and result as a regular icosahedron. A snub pentagonal antiprism, ss{2,10}, or higher n-antiprisms can be similar constructed, but not as a convex polyhedron with equilateral triangles. The preceding Johnson solid, the snub disphenoid also fits constructionally as ss{2,4}, but one has to retain two degenerate digonal faces (drawn in red) in the digonal antiprism.
Snub antiprisms
Symmetry D2d, [2+,4], (2*2) D3d, [2+,6], (2*3) D4d, [2+,8], (2*4) D5d, [2+,10], (2*5)
Antiprisms
s{2,4}
A2
(v:4; e:8; f:6)
s{2,6}
A3
(v:6; e:12; f:8)
s{2,8}
A4
(v:8; e:16; f:10)
s{2,10}
A5
(v:10; e:20; f:12)
Truncated
antiprisms
ts{2,4}
tA2
(v:16;e:24;f:10)
ts{2,6}
tA3
(v:24; e:36; f:14)
ts{2,8}
tA4
(v:32; e:48; f:18)
ts{2,10}
tA5
(v:40; e:60; f:22)
Symmetry D2, [2,2]+, (222) D3, [3,2]+, (322) D4, [4,2]+, (422) D5, [5,2]+, (522)
Snub
antiprisms
J84 Icosahedron J85 Concave
sY3 = HtA3 sY4 = HtA4 sY5 = HtA5
ss{2,4}
(v:8; e:20; f:14)
ss{2,6}
(v:12; e:30; f:20)
ss{2,8}
(v:16; e:40; f:26)
ss{2,10}
(v:20; e:50; f:32)
References
1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
2. Snub Anti-Prisms
3. "PolyHédronisme".
4. Timofeenko, A. V. (2009). "The non-Platonic and non-Archimedean noncomposite polyhedra". Journal of Mathematical Science. 162 (5): 725. doi:10.1007/s10958-009-9655-0. S2CID 120114341.
5. Wolfram Research, Inc. (2020). "Wolfram|Alpha Knowledgebase". Champaign, IL. PolyhedronData[{"Johnson", 85}, "SurfaceArea"] {{cite journal}}: Cite journal requires |journal= (help)
6. Wolfram Research, Inc. (2020). "Wolfram|Alpha Knowledgebase". Champaign, IL. MinimalPolynomial[PolyhedronData[{"Johnson", 85}, "Volume"], x] {{cite journal}}: Cite journal requires |journal= (help)
External links
• Eric W. Weisstein, Snub square antiprism (Johnson solid) at MathWorld.
Johnson solids
Pyramids, cupolae and rotundae
• square pyramid
• pentagonal pyramid
• triangular cupola
• square cupola
• pentagonal cupola
• pentagonal rotunda
Modified pyramids
• elongated triangular pyramid
• elongated square pyramid
• elongated pentagonal pyramid
• gyroelongated square pyramid
• gyroelongated pentagonal pyramid
• triangular bipyramid
• pentagonal bipyramid
• elongated triangular bipyramid
• elongated square bipyramid
• elongated pentagonal bipyramid
• gyroelongated square bipyramid
Modified cupolae and rotundae
• elongated triangular cupola
• elongated square cupola
• elongated pentagonal cupola
• elongated pentagonal rotunda
• gyroelongated triangular cupola
• gyroelongated square cupola
• gyroelongated pentagonal cupola
• gyroelongated pentagonal rotunda
• gyrobifastigium
• triangular orthobicupola
• square orthobicupola
• square gyrobicupola
• pentagonal orthobicupola
• pentagonal gyrobicupola
• pentagonal orthocupolarotunda
• pentagonal gyrocupolarotunda
• pentagonal orthobirotunda
• elongated triangular orthobicupola
• elongated triangular gyrobicupola
• elongated square gyrobicupola
• elongated pentagonal orthobicupola
• elongated pentagonal gyrobicupola
• elongated pentagonal orthocupolarotunda
• elongated pentagonal gyrocupolarotunda
• elongated pentagonal orthobirotunda
• elongated pentagonal gyrobirotunda
• gyroelongated triangular bicupola
• gyroelongated square bicupola
• gyroelongated pentagonal bicupola
• gyroelongated pentagonal cupolarotunda
• gyroelongated pentagonal birotunda
Augmented prisms
• augmented triangular prism
• biaugmented triangular prism
• triaugmented triangular prism
• augmented pentagonal prism
• biaugmented pentagonal prism
• augmented hexagonal prism
• parabiaugmented hexagonal prism
• metabiaugmented hexagonal prism
• triaugmented hexagonal prism
Modified Platonic solids
• augmented dodecahedron
• parabiaugmented dodecahedron
• metabiaugmented dodecahedron
• triaugmented dodecahedron
• metabidiminished icosahedron
• tridiminished icosahedron
• augmented tridiminished icosahedron
Modified Archimedean solids
• augmented truncated tetrahedron
• augmented truncated cube
• biaugmented truncated cube
• augmented truncated dodecahedron
• parabiaugmented truncated dodecahedron
• metabiaugmented truncated dodecahedron
• triaugmented truncated dodecahedron
• gyrate rhombicosidodecahedron
• parabigyrate rhombicosidodecahedron
• metabigyrate rhombicosidodecahedron
• trigyrate rhombicosidodecahedron
• diminished rhombicosidodecahedron
• paragyrate diminished rhombicosidodecahedron
• metagyrate diminished rhombicosidodecahedron
• bigyrate diminished rhombicosidodecahedron
• parabidiminished rhombicosidodecahedron
• metabidiminished rhombicosidodecahedron
• gyrate bidiminished rhombicosidodecahedron
• tridiminished rhombicosidodecahedron
Elementary solids
• snub disphenoid
• snub square antiprism
• sphenocorona
• augmented sphenocorona
• sphenomegacorona
• hebesphenomegacorona
• disphenocingulum
• bilunabirotunda
• triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)
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Wikipedia
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Cubic honeycomb
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.
Cubic honeycomb
TypeRegular honeycomb
FamilyHypercube honeycomb
Indexing[1] J11,15, A1
W1, G22
Schläfli symbol{4,3,4}
Coxeter diagram
Cell type{4,3}
Face typesquare {4}
Vertex figure
octahedron
Space group
Fibrifold notation
Pm3m (221)
4−:2
Coxeter group${\tilde {C}}_{3}$, [4,3,4]
Dualself-dual
Cell:
PropertiesVertex-transitive, regular
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.
Related honeycombs
It is part of a multidimensional family of hypercube honeycombs, with Schläfli symbols of the form {4,3,...,3,4}, starting with the square tiling, {4,4} in the plane.
It is one of 28 uniform honeycombs using convex uniform polyhedral cells.
Isometries of simple cubic lattices
Simple cubic lattices can be distorted into lower symmetries, represented by lower crystal systems:
Crystal system Monoclinic
Triclinic
Orthorhombic Tetragonal Rhombohedral Cubic
Unit cell Parallelepiped Rectangular cuboid Square cuboid Trigonal
trapezohedron
Cube
Point group
Order
Rotation subgroup
[ ], (*)
Order 2
[ ]+, (1)
[2,2], (*222)
Order 8
[2,2]+, (222)
[4,2], (*422)
Order 16
[4,2]+, (422)
[3], (*33)
Order 6
[3]+, (33)
[4,3], (*432)
Order 48
[4,3]+, (432)
Diagram
Space group
Rotation subgroup
Pm (6)
P1 (1)
Pmmm (47)
P222 (16)
P4/mmm (123)
P422 (89)
R3m (160)
R3 (146)
Pm3m (221)
P432 (207)
Coxeter notation - [∞]a×[∞]b×[∞]c [4,4]a×[∞]c - [4,3,4]a
Coxeter diagram - -
Uniform colorings
There is a large number of uniform colorings, derived from different symmetries. These include:
Coxeter notation
Space group
Coxeter diagram Schläfli symbol Partial
honeycomb
Colors by letters
[4,3,4]
Pm3m (221)
=
{4,3,4} 1: aaaa/aaaa
[4,31,1] = [4,3,4,1+]
Fm3m (225)
= {4,31,1} 2: abba/baab
[4,3,4]
Pm3m (221)
t0,3{4,3,4} 4: abbc/bccd
[[4,3,4]]
Pm3m (229)
t0,3{4,3,4} 4: abbb/bbba
[4,3,4,2,∞]
or
{4,4}×t{∞} 2: aaaa/bbbb
[4,3,4,2,∞] t1{4,4}×{∞} 2: abba/abba
[∞,2,∞,2,∞] t{∞}×t{∞}×{∞} 4: abcd/abcd
[∞,2,∞,2,∞] = [4,(3,4)*] = t{∞}×t{∞}×t{∞} 8: abcd/efgh
Projections
The cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements. The highest (hexagonal) symmetry form projects into a triangular tiling. A square symmetry projection forms a square tiling.
Orthogonal projections
Symmetry p6m (*632) p4m (*442) pmm (*2222)
Solid
Frame
Related polytopes and honeycombs
It is related to the regular 4-polytope tesseract, Schläfli symbol {4,3,3}, which exists in 4-space, and only has 3 cubes around each edge. It's also related to the order-5 cubic honeycomb, Schläfli symbol {4,3,5}, of hyperbolic space with 5 cubes around each edge.
It is in a sequence of polychora and honeycombs with octahedral vertex figures.
{p,3,4} regular honeycombs
Space S3 E3 H3
Form Finite Affine Compact Paracompact Noncompact
Name {3,3,4}
{4,3,4}
{5,3,4}
{6,3,4}
{7,3,4}
{8,3,4}
... {∞,3,4}
Image
Cells
{3,3}
{4,3}
{5,3}
{6,3}
{7,3}
{8,3}
{∞,3}
It in a sequence of regular polytopes and honeycombs with cubic cells.
{4,3,p} regular honeycombs
Space S3 E3 H3
Form Finite Affine Compact Paracompact Noncompact
Name
{4,3,3}
{4,3,4}
{4,3,5}
{4,3,6}
{4,3,7}
{4,3,8}
... {4,3,∞}
Image
Vertex
figure
{3,3}
{3,4}
{3,5}
{3,6}
{3,7}
{3,8}
{3,∞}
{p,3,p} regular honeycombs
Space S3 Euclidean E3 H3
Form Finite Affine Compact Paracompact Noncompact
Name {3,3,3} {4,3,4} {5,3,5} {6,3,6} {7,3,7} {8,3,8} ...{∞,3,∞}
Image
Cells
{3,3}
{4,3}
{5,3}
{6,3}
{7,3}
{8,3}
{∞,3}
Vertex
figure
{3,3}
{3,4}
{3,5}
{3,6}
{3,7}
{3,8}
{3,∞}
Related polytopes
The cubic honeycomb has a lower symmetry as a runcinated cubic honeycomb, with two sizes of cubes. A double symmetry construction can be constructed by placing a small cube into each large cube, resulting in a nonuniform honeycomb with cubes, square prisms, and rectangular trapezoprisms (a cube with D2d symmetry). Its vertex figure is a triangular pyramid with its lateral faces augmented by tetrahedra.
Dual cell
The resulting honeycomb can be alternated to produce another nonuniform honeycomb with regular tetrahedra, two kinds of tetragonal disphenoids, triangular pyramids, and sphenoids. Its vertex figure has C3v symmetry and has 26 triangular faces, 39 edges, and 15 vertices.
Related Euclidean tessellations
The [4,3,4], , Coxeter group generates 15 permutations of uniform tessellations, 9 with distinct geometry including the alternated cubic honeycomb. The expanded cubic honeycomb (also known as the runcinated cubic honeycomb) is geometrically identical to the cubic honeycomb.
C3 honeycombs
Space
group
Fibrifold Extended
symmetry
Extended
diagram
Order Honeycombs
Pm3m
(221)
4−:2 [4,3,4] ×1 1, 2, 3, 4,
5, 6
Fm3m
(225)
2−:2 [1+,4,3,4]
↔ [4,31,1]
↔
Half 7, 11, 12, 13
I43m
(217)
4o:2 [[(4,3,4,2+)]] Half × 2 (7),
Fd3m
(227)
2+:2 [[1+,4,3,4,1+]]
↔ [[3[4]]]
↔
Quarter × 2 10,
Im3m
(229)
8o:2 [[4,3,4]] ×2
(1), 8, 9
The [4,31,1], , Coxeter group generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb.
B3 honeycombs
Space
group
Fibrifold Extended
symmetry
Extended
diagram
Order Honeycombs
Fm3m
(225)
2−:2 [4,31,1]
↔ [4,3,4,1+]
↔
×1 1, 2, 3, 4
Fm3m
(225)
2−:2 <[1+,4,31,1]>
↔ <[3[4]]>
↔
×2 (1), (3)
Pm3m
(221)
4−:2 <[4,31,1]> ×2
5, 6, 7, (6), 9, 10, 11
This honeycomb is one of five distinct uniform honeycombs[2] constructed by the ${\tilde {A}}_{3}$ Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:
A3 honeycombs
Space
group
Fibrifold Square
symmetry
Extended
symmetry
Extended
diagram
Extended
group
Honeycomb diagrams
F43m
(216)
1o:2 a1 [3[4]] ${\tilde {A}}_{3}$ (None)
Fm3m
(225)
2−:2 d2 <[3[4]]>
↔ [4,31,1]
↔
${\tilde {A}}_{3}$×21
↔ ${\tilde {B}}_{3}$
1, 2
Fd3m
(227)
2+:2 g2 [[3[4]]]
or [2+[3[4]]]
↔
${\tilde {A}}_{3}$×22 3
Pm3m
(221)
4−:2 d4 <2[3[4]]>
↔ [4,3,4]
↔
${\tilde {A}}_{3}$×41
↔ ${\tilde {C}}_{3}$
4
I3
(204)
8−o r8 [4[3[4]]]+
↔ [[4,3+,4]]
↔
½${\tilde {A}}_{3}$×8
↔ ½${\tilde {C}}_{3}$×2
(*)
Im3m
(229)
8o:2 [4[3[4]]]
↔ [[4,3,4]]
${\tilde {A}}_{3}$×8
↔ ${\tilde {C}}_{3}$×2
5
Rectified cubic honeycomb
Rectified cubic honeycomb
TypeUniform honeycomb
Schläfli symbolr{4,3,4} or t1{4,3,4}
r{4,31,1}
2r{4,31,1}
r{3[4]}
Coxeter diagrams
=
=
= = =
Cellsr{4,3}
{3,4}
Facestriangle {3}
square {4}
Vertex figure
square prism
Space group
Fibrifold notation
Pm3m (221)
4−:2
Coxeter group${\tilde {C}}_{3}$, [4,3,4]
Dualoblate octahedrille
Cell:
PropertiesVertex-transitive, edge-transitive
The rectified cubic honeycomb or rectified cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of octahedra and cuboctahedra in a ratio of 1:1, with a square prism vertex figure.
John Horton Conway calls this honeycomb a cuboctahedrille, and its dual an oblate octahedrille.
Projections
The rectified cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
Orthogonal projections
Symmetry p6m (*632) p4m (*442) pmm (*2222)
Solid
Frame
Symmetry
There are four uniform colorings for the cells of this honeycomb with reflective symmetry, listed by their Coxeter group, and Wythoff construction name, and the Coxeter diagram below.
Symmetry [4,3,4]
${\tilde {C}}_{3}$
[1+,4,3,4]
[4,31,1], ${\tilde {B}}_{3}$
[4,3,4,1+]
[4,31,1], ${\tilde {B}}_{3}$
[1+,4,3,4,1+]
[3[4]], ${\tilde {A}}_{3}$
Space groupPm3m
(221)
Fm3m
(225)
Fm3m
(225)
F43m
(216)
Coloring
Coxeter
diagram
Vertex figure
Vertex
figure
symmetry
D4h
[4,2]
(*224)
order 16
D2h
[2,2]
(*222)
order 8
C4v
[4]
(*44)
order 8
C2v
[2]
(*22)
order 4
This honeycomb can be divided on trihexagonal tiling planes, using the hexagon centers of the cuboctahedra, creating two triangular cupolae. This scaliform honeycomb is represented by Coxeter diagram , and symbol s3{2,6,3}, with coxeter notation symmetry [2+,6,3].
.
Related polytopes
A double symmetry construction can be made by placing octahedra on the cuboctahedra, resulting in a nonuniform honeycomb with two kinds of octahedra (regular octahedra and triangular antiprisms). The vertex figure is a square bifrustum. The dual is composed of elongated square bipyramids.
Dual cell
Truncated cubic honeycomb
Truncated cubic honeycomb
TypeUniform honeycomb
Schläfli symbolt{4,3,4} or t0,1{4,3,4}
t{4,31,1}
Coxeter diagrams
=
Cell typet{4,3}
{3,4}
Face typetriangle {3}
octagon {8}
Vertex figure
isosceles square pyramid
Space group
Fibrifold notation
Pm3m (221)
4−:2
Coxeter group${\tilde {C}}_{3}$, [4,3,4]
DualPyramidille
Cell:
PropertiesVertex-transitive
The truncated cubic honeycomb or truncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of truncated cubes and octahedra in a ratio of 1:1, with an isosceles square pyramid vertex figure.
John Horton Conway calls this honeycomb a truncated cubille, and its dual pyramidille.
Projections
The truncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
Orthogonal projections
Symmetry p6m (*632) p4m (*442) pmm (*2222)
Solid
Frame
Symmetry
There is a second uniform coloring by reflectional symmetry of the Coxeter groups, the second seen with alternately colored truncated cubic cells.
Construction Bicantellated alternate cubic Truncated cubic honeycomb
Coxeter group [4,31,1], ${\tilde {B}}_{3}$ [4,3,4], ${\tilde {C}}_{3}$
=<[4,31,1]>
Space groupFm3mPm3m
Coloring
Coxeter diagram =
Vertex figure
Related polytopes
A double symmetry construction can be made by placing octahedra on the truncated cubes, resulting in a nonuniform honeycomb with two kinds of octahedra (regular octahedra and triangular antiprisms) and two kinds of tetrahedra (tetragonal disphenoids and digonal disphenoids). The vertex figure is an octakis square cupola.
Vertex figure
Dual cell
Bitruncated cubic honeycomb
Bitruncated cubic honeycomb
TypeUniform honeycomb
Schläfli symbol2t{4,3,4}
t1,2{4,3,4}
Coxeter-Dynkin diagram
Cellst{3,4}
Facessquare {4}
hexagon {6}
Edge figureisosceles triangle {3}
Vertex figure
tetragonal disphenoid
Symmetry group
Fibrifold notation
Coxeter notation
Im3m (229)
8o:2
[[4,3,4]]
Coxeter group${\tilde {C}}_{3}$, [4,3,4]
DualOblate tetrahedrille
Disphenoid tetrahedral honeycomb
Cell:
PropertiesVertex-transitive, edge-transitive, cell-transitive
The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra (or, equivalently, bitruncated cubes). It has four truncated octahedra around each vertex, in a tetragonal disphenoid vertex figure. Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive. It is one of 28 uniform honeycombs.
John Horton Conway calls this honeycomb a truncated octahedrille in his Architectonic and catoptric tessellation list, with its dual called an oblate tetrahedrille, also called a disphenoid tetrahedral honeycomb. Although a regular tetrahedron can not tessellate space alone, this dual has identical disphenoid tetrahedron cells with isosceles triangle faces.
Projections
The bitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements. The highest (hexagonal) symmetry form projects into a nonuniform rhombitrihexagonal tiling. A square symmetry projection forms two overlapping truncated square tiling, which combine together as a chamfered square tiling.
Orthogonal projections
Symmetry p6m (*632) p4m (*442) pmm (*2222)
Solid
Frame
Symmetry
The vertex figure for this honeycomb is a disphenoid tetrahedron, and it is also the Goursat tetrahedron (fundamental domain) for the ${\tilde {A}}_{3}$ Coxeter group. This honeycomb has four uniform constructions, with the truncated octahedral cells having different Coxeter groups and Wythoff constructions. These uniform symmetries can be represented by coloring differently the cells in each construction.
Five uniform colorings by cell
Space groupIm3m (229)Pm3m (221)Fm3m (225)F43m (216)Fd3m (227)
Fibrifold8o:24−:22−:21o:22+:2
Coxeter group ${\tilde {C}}_{3}$×2
[[4,3,4]]
=[4[3[4]]]
=
${\tilde {C}}_{3}$
[4,3,4]
=[2[3[4]]]
=
${\tilde {B}}_{3}$
[4,31,1]
=<[3[4]]>
=
${\tilde {A}}_{3}$
[3[4]]
${\tilde {A}}_{3}$×2
[[3[4]]]
=[[3[4]]]
Coxeter diagram
truncated octahedra 1
1:1
:
2:1:1
::
1:1:1:1
:::
1:1
:
Vertex figure
Vertex
figure
symmetry
[2+,4]
(order 8)
[2]
(order 4)
[ ]
(order 2)
[ ]+
(order 1)
[2]+
(order 2)
Image
Colored by
cell
Related polytopes
Nonuniform variants with [4,3,4] symmetry and two types of truncated octahedra can be doubled by placing the two types of truncated octahedra to produce a nonuniform honeycomb with truncated octahedra and hexagonal prisms (as ditrigonal trapezoprisms). Its vertex figure is a C2v-symmetric triangular bipyramid.
This honeycomb can then be alternated to produce another nonuniform honeycomb with pyritohedral icosahedra, octahedra (as triangular antiprisms), and tetrahedra (as sphenoids). Its vertex figure has C2v symmetry and consists of 2 pentagons, 4 rectangles, 4 isosceles triangles (divided into two sets of 2), and 4 scalene triangles.
Alternated bitruncated cubic honeycomb
Alternated bitruncated cubic honeycomb
TypeConvex honeycomb
Schläfli symbol2s{4,3,4}
2s{4,31,1}
sr{3[4]}
Coxeter diagrams
=
=
=
Cells{3,3}
s{3,3}
Facestriangle {3}
Vertex figure
Coxeter group[[4,3+,4]], ${\tilde {C}}_{3}$
DualTen-of-diamonds honeycomb
Cell:
PropertiesVertex-transitive, non-uniform
The alternated bitruncated cubic honeycomb or bisnub cubic honeycomb is non-uniform, with the highest symmetry construction reflecting an alternation of the uniform bitruncated cubic honeycomb. A lower-symmetry construction involves regular icosahedra paired with golden icosahedra (with 8 equilateral triangles paired with 12 golden triangles). There are three constructions from three related Coxeter diagrams: , , and . These have symmetry [4,3+,4], [4,(31,1)+] and [3[4]]+ respectively. The first and last symmetry can be doubled as [[4,3+,4]] and [[3[4]]]+.
This honeycomb is represented in the boron atoms of the α-rhombohedral crystal. The centers of the icosahedra are located at the fcc positions of the lattice.[3]
Five uniform colorings
Space groupI3 (204)Pm3 (200)Fm3 (202)Fd3 (203)F23 (196)
Fibrifold8−o4−2−2o+1o
Coxeter group[[4,3+,4]][4,3+,4][4,(31,1)+][[3[4]]]+[3[4]]+
Coxeter diagram
Order double full half quarter
double
quarter
Cantellated cubic honeycomb
Cantellated cubic honeycomb
TypeUniform honeycomb
Schläfli symbolrr{4,3,4} or t0,2{4,3,4}
rr{4,31,1}
Coxeter diagram
=
Cellsrr{4,3}
r{4,3}
{}x{4}
Vertex figure
wedge
Space group
Fibrifold notation
Pm3m (221)
4−:2
Coxeter group[4,3,4], ${\tilde {C}}_{3}$
Dualquarter oblate octahedrille
Cell:
PropertiesVertex-transitive
The cantellated cubic honeycomb or cantellated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of rhombicuboctahedra, cuboctahedra, and cubes in a ratio of 1:1:3, with a wedge vertex figure.
John Horton Conway calls this honeycomb a 2-RCO-trille, and its dual quarter oblate octahedrille.
Images
It is closely related to the perovskite structure, shown here with cubic symmetry, with atoms placed at the center of the cells of this honeycomb.
Projections
The cantellated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
Orthogonal projections
Symmetry p6m (*632) p4m (*442) pmm (*2222)
Solid
Frame
Symmetry
There is a second uniform colorings by reflectional symmetry of the Coxeter groups, the second seen with alternately colored rhombicuboctahedral cells.
Vertex uniform colorings by cell
Construction Truncated cubic honeycomb Bicantellated alternate cubic
Coxeter group [4,3,4], ${\tilde {C}}_{3}$
=<[4,31,1]>
[4,31,1], ${\tilde {B}}_{3}$
Space groupPm3mFm3m
Coxeter diagram
Coloring
Vertex figure
Vertex
figure
symmetry
[ ]
order 2
[ ]+
order 1
Related polytopes
A double symmetry construction can be made by placing cuboctahedra on the rhombicuboctahedra, which results in the rectified cubic honeycomb, by taking the triangular antiprism gaps as regular octahedra, square antiprism pairs and zero-height tetragonal disphenoids as components of the cuboctahedron. Other variants result in cuboctahedra, square antiprisms, octahedra (as triangular antipodiums), and tetrahedra (as tetragonal disphenoids), with a vertex figure topologically equivalent to a cube with a triangular prism attached to one of its square faces.
Quarter oblate octahedrille
The dual of the cantellated cubic honeycomb is called a quarter oblate octahedrille, a catoptric tessellation with Coxeter diagram , containing faces from two of four hyperplanes of the cubic [4,3,4] fundamental domain.
It has irregular triangle bipyramid cells which can be seen as 1/12 of a cube, made from the cube center, 2 face centers, and 2 vertices.
Cantitruncated cubic honeycomb
Cantitruncated cubic honeycomb
TypeUniform honeycomb
Schläfli symboltr{4,3,4} or t0,1,2{4,3,4}
tr{4,31,1}
Coxeter diagram
=
Cellstr{4,3}
t{3,4}
{}x{4}
Facessquare {4}
hexagon {6}
octagon {8}
Vertex figure
mirrored sphenoid
Coxeter group[4,3,4], ${\tilde {C}}_{3}$
Symmetry group
Fibrifold notation
Pm3m (221)
4−:2
Dualtriangular pyramidille
Cells:
PropertiesVertex-transitive
The cantitruncated cubic honeycomb or cantitruncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space, made up of truncated cuboctahedra, truncated octahedra, and cubes in a ratio of 1:1:3, with a mirrored sphenoid vertex figure.
John Horton Conway calls this honeycomb a n-tCO-trille, and its dual triangular pyramidille.
Images
Four cells exist around each vertex:
Projections
The cantitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
Orthogonal projections
Symmetry p6m (*632) p4m (*442) pmm (*2222)
Solid
Frame
Symmetry
Cells can be shown in two different symmetries. The linear Coxeter diagram form can be drawn with one color for each cell type. The bifurcating diagram form can be drawn with two types (colors) of truncated cuboctahedron cells alternating.
Construction Cantitruncated cubic Omnitruncated alternate cubic
Coxeter group [4,3,4], ${\tilde {C}}_{3}$
=<[4,31,1]>
[4,31,1], ${\tilde {B}}_{3}$
Space groupPm3m (221)Fm3m (225)
Fibrifold4−:22−:2
Coloring
Coxeter diagram
Vertex figure
Vertex
figure
symmetry
[ ]
order 2
[ ]+
order 1
Triangular pyramidille
The dual of the cantitruncated cubic honeycomb is called a triangular pyramidille, with Coxeter diagram, . This honeycomb cells represents the fundamental domains of ${\tilde {B}}_{3}$ symmetry.
A cell can be as 1/24 of a translational cube with vertices positioned: taking two corner, ne face center, and the cube center. The edge colors and labels specify how many cells exist around the edge.
Related polyhedra and honeycombs
It is related to a skew apeirohedron with vertex configuration 4.4.6.6, with the octagons and some of the squares removed. It can be seen as constructed by augmenting truncated cuboctahedral cells, or by augmenting alternated truncated octahedra and cubes.
Two views
Related polytopes
A double symmetry construction can be made by placing truncated octahedra on the truncated cuboctahedra, resulting in a nonuniform honeycomb with truncated octahedra, hexagonal prisms (as ditrigonal trapezoprisms), cubes (as square prisms), triangular prisms (as C2v-symmetric wedges), and tetrahedra (as tetragonal disphenoids). Its vertex figure is topologically equivalent to the octahedron.
Vertex figure
Dual cell
Alternated cantitruncated cubic honeycomb
Alternated cantitruncated cubic honeycomb
TypeConvex honeycomb
Schläfli symbolsr{4,3,4}
sr{4,31,1}
Coxeter diagrams
=
Cellss{4,3}
s{3,3}
{3,3}
Facestriangle {3}
square {4}
Vertex figure
Coxeter group[(4,3)+,4]
Dual
Cell:
PropertiesVertex-transitive, non-uniform
The alternated cantitruncated cubic honeycomb or snub rectified cubic honeycomb contains three types of cells: snub cubes, icosahedra (with Th symmetry), tetrahedra (as tetragonal disphenoids), and new tetrahedral cells created at the gaps.
Although it is not uniform, constructionally it can be given as Coxeter diagrams or .
Despite being non-uniform, there is a near-miss version with two edge lengths shown below, one of which is around 4.3% greater than the other. The snub cubes in this case are uniform, but the rest of the cells are not.
Cantic snub cubic honeycomb
Orthosnub cubic honeycomb
TypeConvex honeycomb
Schläfli symbol2s0{4,3,4}
Coxeter diagrams
Cellss2{3,4}
s{3,3}
{}x{3}
Facestriangle {3}
square {4}
Vertex figure
Coxeter group[4+,3,4]
DualCell:
PropertiesVertex-transitive, non-uniform
The cantic snub cubic honeycomb is constructed by snubbing the truncated octahedra in a way that leaves only rectangles from the cubes (square prisms). It is not uniform but it can be represented as Coxeter diagram . It has rhombicuboctahedra (with Th symmetry), icosahedra (with Th symmetry), and triangular prisms (as C2v-symmetry wedges) filling the gaps.[4]
Related polytopes
A double symmetry construction can be made by placing icosahedra on the rhombicuboctahedra, resulting in a nonuniform honeycomb with icosahedra, octahedra (as triangular antiprisms), triangular prisms (as C2v-symmetric wedges), and square pyramids.
Vertex figure
Dual cell
Runcitruncated cubic honeycomb
Runcitruncated cubic honeycomb
TypeUniform honeycomb
Schläfli symbolt0,1,3{4,3,4}
Coxeter diagrams
Cellsrr{4,3}
t{4,3}
{}x{8}
{}x{4}
Facestriangle {3}
square {4}
octagon {8}
Vertex figure
isosceles-trapezoidal pyramid
Coxeter group[4,3,4], ${\tilde {C}}_{3}$
Space group
Fibrifold notation
Pm3m (221)
4−:2
Dualsquare quarter pyramidille
Cell
PropertiesVertex-transitive
The runcitruncated cubic honeycomb or runcitruncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of rhombicuboctahedra, truncated cubes, octagonal prisms, and cubes in a ratio of 1:1:3:3, with an isosceles-trapezoidal pyramid vertex figure.
Its name is derived from its Coxeter diagram, with three ringed nodes representing 3 active mirrors in the Wythoff construction from its relation to the regular cubic honeycomb.
John Horton Conway calls this honeycomb a 1-RCO-trille, and its dual square quarter pyramidille.
Projections
The runcitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
Orthogonal projections
Symmetry p6m (*632) p4m (*442) pmm (*2222)
Solid
Frame
Related skew apeirohedron
Two related uniform skew apeirohedrons exists with the same vertex arrangement, seen as boundary cells from a subset of cells. One has triangles and squares, and the other triangles, squares, and octagons.
Square quarter pyramidille
The dual to the runcitruncated cubic honeycomb is called a square quarter pyramidille, with Coxeter diagram . Faces exist in 3 of 4 hyperplanes of the [4,3,4], ${\tilde {C}}_{3}$ Coxeter group.
Cells are irregular pyramids and can be seen as 1/24 of a cube, using one corner, one mid-edge point, two face centers, and the cube center.
Related polytopes
A double symmetry construction can be made by placing rhombicuboctahedra on the truncated cubes, resulting in a nonuniform honeycomb with rhombicuboctahedra, octahedra (as triangular antiprisms), cubes (as square prisms), two kinds of triangular prisms (both C2v-symmetric wedges), and tetrahedra (as digonal disphenoids). Its vertex figure is topologically equivalent to the augmented triangular prism.
Vertex figure
Dual cell
Omnitruncated cubic honeycomb
Omnitruncated cubic honeycomb
TypeUniform honeycomb
Schläfli symbolt0,1,2,3{4,3,4}
Coxeter diagram
Cellstr{4,3}
{}x{8}
Facessquare {4}
hexagon {6}
octagon {8}
Vertex figure
phyllic disphenoid
Symmetry group
Fibrifold notation
Coxeter notation
Im3m (229)
8o:2
[[4,3,4]]
Coxeter group[4,3,4], ${\tilde {C}}_{3}$
Dualeighth pyramidille
Cell
PropertiesVertex-transitive
The omnitruncated cubic honeycomb or omnitruncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of truncated cuboctahedra and octagonal prisms in a ratio of 1:3, with a phyllic disphenoid vertex figure.
John Horton Conway calls this honeycomb a b-tCO-trille, and its dual eighth pyramidille.
Projections
The omnitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
Orthogonal projections
Symmetry p6m (*632) p4m (*442) pmm (*2222)
Solid
Frame
Symmetry
Cells can be shown in two different symmetries. The Coxeter diagram form has two colors of truncated cuboctahedra and octagonal prisms. The symmetry can be doubled by relating the first and last branches of the Coxeter diagram, which can be shown with one color for all the truncated cuboctahedral and octagonal prism cells.
Two uniform colorings
Symmetry ${\tilde {C}}_{3}$, [4,3,4] ${\tilde {C}}_{3}$×2, [[4,3,4]]
Space groupPm3m (221)Im3m (229)
Fibrifold4−:28o:2
Coloring
Coxeter diagram
Vertex figure
Related polyhedra
Two related uniform skew apeirohedron exist with the same vertex arrangement. The first has octagons removed, and vertex configuration 4.4.4.6. It can be seen as truncated cuboctahedra and octagonal prisms augmented together. The second can be seen as augmented octagonal prisms, vertex configuration 4.8.4.8.
4.4.4.6
4.8.4.8
Related polytopes
Nonuniform variants with [4,3,4] symmetry and two types of truncated cuboctahedra can be doubled by placing the two types of truncated cuboctahedra on each other to produce a nonuniform honeycomb with truncated cuboctahedra, octagonal prisms, hexagonal prisms (as ditrigonal trapezoprisms), and two kinds of cubes (as rectangular trapezoprisms and their C2v-symmetric variants). Its vertex figure is an irregular triangular bipyramid.
Vertex figure
Dual cell
This honeycomb can then be alternated to produce another nonuniform honeycomb with snub cubes, square antiprisms, octahedra (as triangular antiprisms), and three kinds of tetrahedra (as tetragonal disphenoids, phyllic disphenoids, and irregular tetrahedra).
Vertex figure
Alternated omnitruncated cubic honeycomb
Alternated omnitruncated cubic honeycomb
TypeConvex honeycomb
Schläfli symbolht0,1,2,3{4,3,4}
Coxeter diagram
Cellss{4,3}
s{2,4}
{3,3}
Facestriangle {3}
square {4}
Vertex figure
Symmetry[[4,3,4]]+
DualDual alternated omnitruncated cubic honeycomb
PropertiesVertex-transitive, non-uniform
An alternated omnitruncated cubic honeycomb or omnisnub cubic honeycomb can be constructed by alternation of the omnitruncated cubic honeycomb, although it can not be made uniform, but it can be given Coxeter diagram: and has symmetry [[4,3,4]]+. It makes snub cubes from the truncated cuboctahedra, square antiprisms from the octagonal prisms, and creates new tetrahedral cells from the gaps.
Dual alternated omnitruncated cubic honeycomb
Dual alternated omnitruncated cubic honeycomb
TypeDual alternated uniform honeycomb
Schläfli symboldht0,1,2,3{4,3,4}
Coxeter diagram
Cell
Vertex figurespentagonal icositetrahedron
tetragonal trapezohedron
tetrahedron
Symmetry[[4,3,4]]+
DualAlternated omnitruncated cubic honeycomb
PropertiesCell-transitive
A dual alternated omnitruncated cubic honeycomb is a space-filling honeycomb constructed as the dual of the alternated omnitruncated cubic honeycomb.
24 cells fit around a vertex, making a chiral octahedral symmetry that can be stacked in all 3-dimensions:
Individual cells have 2-fold rotational symmetry. In 2D orthogonal projection, this looks like a mirror symmetry.
Cell views
Net
Runcic cantitruncated cubic honeycomb
Runcic cantitruncated cubic honeycomb
TypeConvex honeycomb
Schläfli symbolsr3{4,3,4}
Coxeter diagrams
Cellss2{3,4}
s{4,3}
{}x{4}
{}x{3}
Facestriangle {3}
square {4}
Vertex figure
Coxeter group[4,3+,4]
DualCell:
PropertiesVertex-transitive, non-uniform
The runcic cantitruncated cubic honeycomb or runcic cantitruncated cubic cellulation is constructed by removing alternating long rectangles from the octagons and is not uniform, but it can be represented as Coxeter diagram . It has rhombicuboctahedra (with Th symmetry), snub cubes, two kinds of cubes: square prisms and rectangular trapezoprisms (topologically equivalent to a cube but with D2d symmetry), and triangular prisms (as C2v-symmetry wedges) filling the gaps.
Biorthosnub cubic honeycomb
Biorthosnub cubic honeycomb
TypeConvex honeycomb
Schläfli symbol2s0,3{4,3,4}
Coxeter diagrams
Cellss2{3,4}
{}x{4}
Facestriangle {3}
square {4}
Vertex figure
(Tetragonal antiwedge)
Coxeter group[[4,3+,4]]
DualCell:
PropertiesVertex-transitive, non-uniform
The biorthosnub cubic honeycomb is constructed by removing alternating long rectangles from the octagons orthogonally and is not uniform, but it can be represented as Coxeter diagram . It has rhombicuboctahedra (with Th symmetry) and two kinds of cubes: square prisms and rectangular trapezoprisms (topologically equivalent to a cube but with D2d symmetry).
Truncated square prismatic honeycomb
Truncated square prismatic honeycomb
TypeUniform honeycomb
Schläfli symbolt{4,4}×{∞} or t0,1,3{4,4,2,∞}
tr{4,4}×{∞} or t0,1,2,3{4,4,∞}
Coxeter-Dynkin diagram
Cells{}x{8}
{}x{4}
Facessquare {4}
octagon {8}
Coxeter group[4,4,2,∞]
DualTetrakis square prismatic tiling
Cell:
PropertiesVertex-transitive
The truncated square prismatic honeycomb or tomo-square prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of octagonal prisms and cubes in a ratio of 1:1.
It is constructed from a truncated square tiling extruded into prisms.
It is one of 28 convex uniform honeycombs.
Snub square prismatic honeycomb
Snub square prismatic honeycomb
TypeUniform honeycomb
Schläfli symbols{4,4}×{∞}
sr{4,4}×{∞}
Coxeter-Dynkin diagram
Cells{}x{4}
{}x{3}
Facestriangle {3}
square {4}
Coxeter group[4+,4,2,∞]
[(4,4)+,2,∞]
DualCairo pentagonal prismatic honeycomb
Cell:
PropertiesVertex-transitive
The snub square prismatic honeycomb or simo-square prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of cubes and triangular prisms in a ratio of 1:2.
It is constructed from a snub square tiling extruded into prisms.
It is one of 28 convex uniform honeycombs.
Snub square antiprismatic honeycomb
Snub square antiprismatic honeycomb
TypeConvex honeycomb
Schläfli symbolht1,2,3{4,4,2,∞}
ht0,1,2,3{4,4,∞}
Coxeter-Dynkin diagram
Cellss{2,4}
{3,3}
Facestriangle {3}
square {4}
Vertex figure
Symmetry[4,4,2,∞]+
PropertiesVertex-transitive, non-uniform
A snub square antiprismatic honeycomb can be constructed by alternation of the truncated square prismatic honeycomb, although it can not be made uniform, but it can be given Coxeter diagram: and has symmetry [4,4,2,∞]+. It makes square antiprisms from the octagonal prisms, tetrahedra (as tetragonal disphenoids) from the cubes, and two tetrahedra from the triangular bipyramids.
See also
Wikimedia Commons has media related to Cubic honeycomb.
• Architectonic and catoptric tessellation
• Alternated cubic honeycomb
• List of regular polytopes
• Order-5 cubic honeycomb A hyperbolic cubic honeycomb with 5 cubes per edge
• Snub (geometry)
• Voxel
References
1. For cross-referencing, they are given with list indices from Andreini (1-22), Williams(1-2,9-19), Johnson (11-19, 21-25, 31-34, 41-49, 51-52, 61-65), and Grünbaum(1-28).
2. , A000029 6-1 cases, skipping one with zero marks
3. Williams, 1979, p 199, Figure 5-38.
4. cantic snub cubic honeycomb
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292-298, includes all the nonprismatic forms)
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
• George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
• Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
• 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 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
• A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.
• Klitzing, Richard. "3D Euclidean Honeycombs x4o3o4o - chon - O1".
• Uniform Honeycombs in 3-Space: 01-Chon
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
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Wikipedia
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Snub square tiling
In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. Its Schläfli symbol is s{4,4}.
Snub square tiling
TypeSemiregular tiling
Vertex configuration
3.3.4.3.4
Schläfli symbols{4,4}
sr{4,4} or $s{\begin{Bmatrix}4\\4\end{Bmatrix}}$
Wythoff symbol| 4 4 2
Coxeter diagram
or
Symmetryp4g, [4+,4], (4*2)
Rotation symmetryp4, [4,4]+, (442)
Bowers acronymSnasquat
DualCairo pentagonal tiling
PropertiesVertex-transitive
Conway calls it a snub quadrille, constructed by a snub operation applied to a square tiling (quadrille).
There are 3 regular and 8 semiregular tilings in the plane.
Uniform colorings
There are two distinct uniform colorings of a snub square tiling. (Naming the colors by indices around a vertex (3.3.4.3.4): 11212, 11213.)
Coloring
11212
11213
Symmetry 4*2, [4+,4], (p4g) 442, [4,4]+, (p4)
Schläfli symbol s{4,4} sr{4,4}
Wythoff symbol | 4 4 2
Coxeter diagram
Circle packing
The snub square tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing (kissing number).[1]
Wythoff construction
The snub square tiling can be constructed as a snub operation from the square tiling, or as an alternate truncation from the truncated square tiling.
An alternate truncation deletes every other vertex, creating a new triangular faces at the removed vertices, and reduces the original faces to half as many sides. In this case starting with a truncated square tiling with 2 octagons and 1 square per vertex, the octagon faces into squares, and the square faces degenerate into edges and 2 new triangles appear at the truncated vertices around the original square.
If the original tiling is made of regular faces the new triangles will be isosceles. Starting with octagons which alternate long and short edge lengths, derived from a regular dodecagon, will produce a snub tiling with perfect equilateral triangle faces.
Example:
Regular octagons alternately truncated
→
(Alternate
truncation)
Isosceles triangles (Nonuniform tiling)
Nonregular octagons alternately truncated
→
(Alternate
truncation)
Equilateral triangles
Related tilings
• A snub operator applied twice to the square tiling, while it doesn't have regular faces, is made of square with irregular triangles and pentagons.
• A related isogonal tiling that combines pairs of triangles into rhombi
• A 2-isogonal tiling can be made by combining 2 squares and 3 triangles into heptagons.
• The Cairo pentagonal tiling is dual to the snub square tiling.
Related k-uniform tilings
This tiling is related to the elongated triangular tiling which also has 3 triangles and two squares on a vertex, but in a different order, 3.3.3.4.4. The two vertex figures can be mixed in many k-uniform tilings.[2][3]
Related tilings of triangles and squares
snub square elongated triangular 2-uniform 3-uniform
p4g, (4*2) p2, (2222) p2, (2222) cmm, (2*22) p2, (2222)
[32434]
[3342]
[3342; 32434]
[3342; 32434]
[2: 3342; 32434]
[3342; 2: 32434]
Related topological series of polyhedra and tiling
The snub square tiling is third in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.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.∞
The snub square tiling is third in a series of snub polyhedra and tilings with vertex figure 3.3.n.3.n.
4n2 symmetry mutations of snub tilings: 3.3.n.3.n
Symmetry
4n2
Spherical Euclidean Compact hyperbolic Paracompact
222 322 442 552 662 772 882 ∞∞2
Snub
figures
Config. 3.3.2.3.2 3.3.3.3.3 3.3.4.3.4 3.3.5.3.5 3.3.6.3.6 3.3.7.3.7 3.3.8.3.8 3.3.∞.3.∞
Gyro
figures
Config. V3.3.2.3.2 V3.3.3.3.3 V3.3.4.3.4 V3.3.5.3.5 V3.3.6.3.6 V3.3.7.3.7 V3.3.8.3.8 V3.3.∞.3.∞
Uniform tilings based on square tiling symmetry
Symmetry: [4,4], (*442) [4,4]+, (442) [4,4+], (4*2)
{4,4} t{4,4} r{4,4} t{4,4} {4,4} rr{4,4} tr{4,4} sr{4,4} s{4,4}
Uniform duals
V4.4.4.4 V4.8.8 V4.4.4.4 V4.8.8 V4.4.4.4 V4.4.4.4 V4.8.8 V3.3.4.3.4
See also
Wikimedia Commons has media related to Uniform tiling 3-3-4-3-4 (snub square tiling).
• List of uniform planar tilings
• Snub (geometry)
• Snub square prismatic honeycomb
• Tilings of regular polygons
• Elongated triangular tiling
References
1. Order in Space: A design source book, Keith Critchlow, p.74-75, circle pattern C
2. Chavey, D. (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings". Computers & Mathematics with Applications. 17: 147–165. doi:10.1016/0898-1221(89)90156-9.
3. "Uniform Tilings". Archived from the original on 2006-09-09. Retrieved 2006-09-09.
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
• Klitzing, Richard. "2D Euclidean tilings s4s4s - snasquat - O10".
• 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. ISBN 0-486-23729-X. p38
• Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–56, dual p. 115
External links
• Weisstein, Eric W. "Semiregular tessellation". 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
• 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
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Wikipedia
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Runcinated tesseracts
In four-dimensional geometry, a runcinated tesseract (or runcinated 16-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular tesseract.
Tesseract
Runcinated tesseract
(Runcinated 16-cell)
16-cell
Runcitruncated tesseract
(Runcicantellated 16-cell)
Runcitruncated 16-cell
(Runcicantellated tesseract)
Omnitruncated tesseract
(Omnitruncated 16-cell)
Orthogonal projections in B4 Coxeter plane
There are 4 variations of runcinations of the tesseract including with permutations truncations and cantellations.
Runcinated tesseract
Runcinated tesseract
Schlegel diagram with 16 tetrahedra
Type Uniform 4-polytope
Schläfli symbol t0,3{4,3,3}
Coxeter diagrams
Cells 80 16 3.3.3
32 3.4.4
32 4.4.4
Faces 208 64 {3}
144 {4}
Edges 192
Vertices 64
Vertex figure
Equilateral-triangular antipodium
Symmetry group B4, [3,3,4], order 384
Properties convex
Uniform index 14 15 16
The runcinated tesseract or (small) disprismatotesseractihexadecachoron has 16 tetrahedra, 32 cubes, and 32 triangular prisms. Each vertex is shared by 4 cubes, 3 triangular prisms and one tetrahedron.
Construction
The runcinated tesseract may be constructed by expanding the cells of a tesseract radially, and filling in the gaps with tetrahedra (vertex figures), cubes (face prisms), and triangular prisms (edge figure prisms). The same process applied to a 16-cell also yields the same figure.
Cartesian coordinates
The Cartesian coordinates of the vertices of the runcinated tesseract with edge length 2 are all permutations of:
$\left(\pm 1,\ \pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}})\right)$
Images
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]
Schlegel diagrams
Wireframe
Wireframe with 16 tetrahedra.
Wireframe with 32 triangular prisms.
Structure
Eight of the cubical cells are connected to the other 24 cubical cells via all 6 square faces. The other 24 cubical cells are connected to the former 8 cells via only two opposite square faces; the remaining 4 faces are connected to the triangular prisms. The triangular prisms are connected to the tetrahedra via their triangular faces.
The runcinated tesseract can be dissected into 2 cubic cupolae and a rhombicuboctahedral prism between them. This dissection can be seen analogous to the 3D rhombicuboctahedron being dissected into two square cupola and a central octagonal prism.
cubic cupola
rhombicuboctahedral prism
Projections
The cube-first orthographic projection of the runcinated tesseract into 3-dimensional space has a (small) rhombicuboctahedral envelope. The images of its cells are laid out within this envelope as follows:
• The nearest and farthest cube from the 4d viewpoint projects to a cubical volume in the center of the envelope.
• Six cuboidal volumes connect this central cube to the 6 axial square faces of the rhombicuboctahedron. These are the images of 12 of the cubical cells (each pair of cubes share an image).
• The 18 square faces of the envelope are the images of the other cubical cells.
• The 12 wedge-shaped volumes connecting the edges of the central cube to the non-axial square faces of the envelope are the images of 24 of the triangular prisms (a pair of cells per image).
• The 8 triangular faces of the envelope are the images of the remaining 8 triangular prisms.
• Finally, the 8 tetrahedral volumes connecting the vertices of the central cube to the triangular faces of the envelope are the images of the 16 tetrahedra (again, a pair of cells per image).
This layout of cells in projection is analogous to the layout of the faces of the (small) rhombicuboctahedron under projection to 2 dimensions. The rhombicuboctahedron is also constructed from the cube or the octahedron in an analogous way to the runcinated tesseract. Hence, the runcinated tesseract may be thought of as the 4-dimensional analogue of the rhombicuboctahedron.
Runcitruncated tesseract
Runcitruncated tesseract
Schlegel diagram
centered on a truncated cube,
with cuboctahedral cells shown
Type Uniform 4-polytope
Schläfli symbol t0,1,3{4,3,3}
Coxeter diagrams
Cells 80 8 3.4.4
16 3.4.3.4
24 4.4.8
32 3.4.4
Faces 368 128 {3}
192 {4}
48 {8}
Edges 480
Vertices 192
Vertex figure
Rectangular pyramid
Symmetry group B4, [3,3,4], order 384
Properties convex
Uniform index 18 19 20
The runcitruncated tesseract, runcicantellated 16-cell, or prismatorhombated hexadecachoron is bounded by 80 cells: 8 truncated cubes, 16 cuboctahedra, 24 octagonal prisms, and 32 triangular prisms.
Construction
The runcitruncated tesseract may be constructed from the truncated tesseract by expanding the truncated cube cells outward radially, and inserting octagonal prisms between them. In the process, the tetrahedra expand into cuboctahedra, and triangular prisms fill in the remaining gaps.
The Cartesian coordinates of the vertices of the runcitruncated tesseract having an edge length of 2 is given by all permutations of:
$\left(\pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+2{\sqrt {2}})\right)$
Projections
In the truncated cube first parallel projection of the runcitruncated tesseract into 3-dimensional space, the projection image is laid out as follows:
• The projection envelope is a non-uniform (small) rhombicuboctahedron, with 6 square faces and 12 rectangular faces.
• Two of the truncated cube cells project to a truncated cube in the center of the projection envelope.
• Six octagonal prisms connect this central truncated cube to the square faces of the envelope. These are the images of 12 of the octagonal prism cells, two cells to each image.
• The remaining 12 octagonal prisms are projected to the rectangular faces of the envelope.
• The 6 square faces of the envelope are the images of the remaining 6 truncated cube cells.
• Twelve right-angle triangular prisms connect the inner octagonal prisms. These are the images of 24 of the triangular prism cells. The remaining 8 triangular prisms project onto the triangular faces of the envelope.
• The 8 remaining volumes lying between the triangular faces of the envelope and the inner truncated cube are the images of the 16 cuboctahedral cells, a pair of cells to each image.
Images
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]
Stereographic projection with its 128 blue triangular faces and its 192 green quad faces.
Runcitruncated 16-cell
Runcitruncated 16-cell
Schlegel diagrams centered on
rhombicuboctahedron and truncated tetrahedron
Type Uniform 4-polytope
Schläfli symbol t0,1,3{3,3,4}
Coxeter diagram
Cells 80 8 3.4.4.4
16 3.6.6
24 4.4.4
32 4.4.6
Faces 368 64 {3}
240 {4}
64 {6}
Edges 480
Vertices 192
Vertex figure
Trapezoidal pyramid
Symmetry group B4, [3,3,4], order 384
Properties convex
Uniform index 19 20 21
The runcitruncated 16-cell, runcicantellated tesseract, or prismatorhombated tesseract is bounded by 80 cells: 8 rhombicuboctahedra, 16 truncated tetrahedra, 24 cubes, and 32 hexagonal prisms.
Construction
The runcitruncated 16-cell may be constructed by contracting the small rhombicuboctahedral cells of the cantellated tesseract radially, and filling in the spaces between them with cubes. In the process, the octahedral cells expand into truncated tetrahedra (half of their triangular faces are expanded into hexagons by pulling apart the edges), and the triangular prisms expand into hexagonal prisms (each with its three original square faces joined, as before, to small rhombicuboctahedra, and its three new square faces joined to cubes).
The vertices of a runcitruncated 16-cell having an edge length of 2 is given by all permutations of the following Cartesian coordinates:
$\left(\pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+2{\sqrt {2}})\right)$
Images
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]
Structure
The small rhombicuboctahedral cells are joined via their 6 axial square faces to the cubical cells, and joined via their 12 non-axial square faces to the hexagonal prisms. The cubical cells are joined to the rhombicuboctahedra via 2 opposite faces, and joined to the hexagonal prisms via the remaining 4 faces. The hexagonal prisms are connected to the truncated tetrahedra via their hexagonal faces, and to the rhombicuboctahedra via 3 of their square faces each, and to the cubes via the other 3 square faces. The truncated tetrahedra are joined to the rhombicuboctahedra via their triangular faces, and the hexagonal prisms via their hexagonal faces.
Projections
The following is the layout of the cells of the runcitruncated 16-cell under the parallel projection, small rhombicuboctahedron first, into 3-dimensional space:
• The projection envelope is a truncated cuboctahedron.
• Six of the small rhombicuboctahedra project onto the 6 octagonal faces of this envelope, and the other two project to a small rhombicuboctahedron lying at the center of this envelope.
• The 6 cuboidal volumes connecting the axial square faces of the central small rhombicuboctahedron to the center of the octagons correspond with the image of 12 of the cubical cells (each pair of the twelve share the same image).
• The remaining 12 cubical cells project onto the 12 square faces of the great rhombicuboctahedral envelope.
• The 8 volumes connecting the hexagons of the envelope to the triangular faces of the central rhombicuboctahedron are the images of the 16 truncated tetrahedra.
• The remaining 12 spaces connecting the non-axial square faces of the central small rhombicuboctahedron to the square faces of the envelope are the images of 24 of the hexagonal prisms.
• Finally, the last 8 hexagonal prisms project onto the hexagonal faces of the envelope.
This layout of cells is similar to the layout of the faces of the great rhombicuboctahedron under the projection into 2-dimensional space. Hence, the runcitruncated 16-cell may be thought of as one of the 4-dimensional analogues of the great rhombicuboctahedron. The other analogue is the omnitruncated tesseract.
Omnitruncated tesseract
Omnitruncated tesseract
Schlegel diagram,
centered on truncated cuboctahedron,
truncated octahedral cells shown
Type Uniform 4-polytope
Schläfli symbol t0,1,2,3{3,3,4}
Coxeter diagram
Cells 80 8 4.6.8
16 4.6.6
24 4.4.8
32 4.4.6
Faces 464 288 {4}
128 {6}
48 {8}
Edges 768
Vertices 384
Vertex figure
Chiral scalene tetrahedron
Symmetry group B4, [3,3,4], order 384
Properties convex
Uniform index 20 21 22
The omnitruncated tesseract, omnitruncated 16-cell, or great disprismatotesseractihexadecachoron is bounded by 80 cells: 8 truncated cuboctahedra, 16 truncated octahedra, 24 octagonal prisms, and 32 hexagonal prisms.
Construction
The omnitruncated tesseract can be constructed from the cantitruncated tesseract by radially displacing the truncated cuboctahedral cells so that octagonal prisms can be inserted between their octagonal faces. As a result, the triangular prisms expand into hexagonal prisms, and the truncated tetrahedra expand into truncated octahedra.
The Cartesian coordinates of the vertices of an omnitruncated tesseract having an edge length of 2 are given by all permutations of coordinates and sign of:
$\left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}}\right)$
Structure
The truncated cuboctahedra cells are joined to the octagonal prisms via their octagonal faces, the truncated octahedra via their hexagonal faces, and the hexagonal prisms via their square faces. The octagonal prisms are joined to the hexagonal prisms and the truncated octahedra via their square faces, and the hexagonal prisms are joined to the truncated octahedra via their hexagonal faces.
Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time. Edges exist at 4 symmetry positions. Squares exist at 3 positions, hexagons 2 positions, and octagons one. Finally the 4 types of cells exist centered on the 4 corners of the fundamental simplex.[1]
B4 k-facefkf0f1f2f3k-figure Notes
( ) f0 38411111111111111Tetrahedron B4 = 384
A1{ } f1 2192***1110001110Scalene triangle B4/A1 = 192
A1{ } 2*192**1001101101 B4/A1 = 192
A1{ } 2**192*0101011011 B4/A1 = 192
A1{ } 2***1920010110111 B4/A1 = 192
A2{6} f2 6330064*****1100{ } B4/A2 = 64
A1A1{4} 42020*96****1010 B4/A1A1 = 96
A1A1{4} 42002**96***0110 B4/A1A1 = 96
A2{6} 60330***64**1001 B4/A2 = 64
A1A1{4} 40202****96*0101 B4/A1A1 = 96
B2{8} 80044*****480011 B4/B2 = 48
A3tr{3,3} f3 24121212046040016***( ) B4/A3 = 16
A2A1{6}×{ } 126606203030*32** B4/A2A1 = 32
B2A1{8}×{ } 168088044002**24* B4/B2A1 = 24
B3tr{4,3} 4802424240008126***8 B4/B3 = 8
Projections
In the truncated cuboctahedron first parallel projection of the omnitruncated tesseract into 3 dimensions, the images of its cells are laid out as follows:
• The projection envelope is in the shape of a non-uniform truncated cuboctahedron.
• Two of the truncated cuboctahedra project to the center of the projection envelope.
• The remaining 6 truncated cuboctahedra project to the (non-regular) octagonal faces of the envelope. These are connected to the central truncated cuboctahedron via 6 octagonal prisms, which are the images of the octagonal prism cells, a pair to each image.
• The 8 hexagonal faces of the envelope are the images of 8 of the hexagonal prisms.
• The remaining hexagonal prisms are projected to 12 non-regular hexagonal prism images, lying where a cube's edges would be. Each image corresponds to two cells.
• Finally, the 8 volumes between the hexagonal faces of the projection envelope and the hexagonal faces of the central truncated cuboctahedron are the images of the 16 truncated octahedra, two cells to each image.
This layout of cells in projection is similar to that of the runcitruncated 16-cell, which is analogous to the layout of faces in the octagon-first projection of the truncated cuboctahedron into 2 dimensions. Thus, the omnitruncated tesseract may be thought of as another analogue of the truncated cuboctahedron in 4 dimensions.
Images
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]
Perspective projections
Perspective projection centered on one of the truncated cuboctahedral cells, highlighted in yellow. Six of the surrounding octagonal prisms rendered in blue, and the remaining cells in green. Cells obscured from 4D viewpoint culled for clarity's sake.
Perspective projection centered on one of the truncated octahedral cells, highlighted in yellow. Four of the surrounding hexagonal prisms are shown in blue, with 4 more truncated octahedra on the other side of these prisms also shown in yellow. Cells obscured from 4D viewpoint culled for clarity's sake. Some of the other hexagonal and octagonal prisms may be discerned from this view as well.
Stereographic projections
Centered on truncated cuboctahedron
Centered on truncated octahedron
Net
Omnitruncated tesseract
Dual to omnitruncated tesseract
Full snub tesseract
The full snub tesseract or omnisnub tesseract, defined as an alternation of the omnitruncated tesseract, can not be made uniform, but it can be given Coxeter diagram , and symmetry [4,3,3]+, and constructed from 8 snub cubes, 16 icosahedra, 24 square antiprisms, 32 octahedra (as triangular antiprisms), and 192 tetrahedra filling the gaps at the deleted vertices. It has 272 cells, 944 faces, 864 edges, and 192 vertices.[2]
Bialternatosnub 16-cell
The bialternatosnub 16-cell or runcic snub rectified 16-cell, constructed by removing alternating long rectangles from the octagons, is also not uniform. Like the omnisnub tesseract, it has a highest symmetry construction of order 192, with 8 rhombicuboctahedra (with Th symmetry), 16 icosahedra (with T symmetry), 24 rectangular trapezoprisms (topologically equivalent to a cube but with D2d symmetry), 32 triangular prisms, with 96 triangular prisms (as Cs-symmetry wedges) filling the gaps.[3]
A variant with regular icosahedra and uniform triangular prisms has two edge lengths in the ratio of 1 : 2, and occurs as a vertex-faceting of the scaliform runcic snub 24-cell.
Related uniform polytopes
B4 symmetry polytopes
Name tesseract rectified
tesseract
truncated
tesseract
cantellated
tesseract
runcinated
tesseract
bitruncated
tesseract
cantitruncated
tesseract
runcitruncated
tesseract
omnitruncated
tesseract
Coxeter
diagram
=
=
Schläfli
symbol
{4,3,3} t1{4,3,3}
r{4,3,3}
t0,1{4,3,3}
t{4,3,3}
t0,2{4,3,3}
rr{4,3,3}
t0,3{4,3,3} t1,2{4,3,3}
2t{4,3,3}
t0,1,2{4,3,3}
tr{4,3,3}
t0,1,3{4,3,3} t0,1,2,3{4,3,3}
Schlegel
diagram
B4
Name 16-cell rectified
16-cell
truncated
16-cell
cantellated
16-cell
runcinated
16-cell
bitruncated
16-cell
cantitruncated
16-cell
runcitruncated
16-cell
omnitruncated
16-cell
Coxeter
diagram
=
=
=
=
=
=
Schläfli
symbol
{3,3,4} t1{3,3,4}
r{3,3,4}
t0,1{3,3,4}
t{3,3,4}
t0,2{3,3,4}
rr{3,3,4}
t0,3{3,3,4} t1,2{3,3,4}
2t{3,3,4}
t0,1,2{3,3,4}
tr{3,3,4}
t0,1,3{3,3,4} t0,1,2,3{3,3,4}
Schlegel
diagram
B4
Notes
1. Klitzing, Richard. "x3x3x4x - gidpith".
2. Klitzing, Richard. "s3s3s4s".
3. Klitzing, Richard. "s3s3s4x".
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, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
• 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 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]
• 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)
• 2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell) - Model 15, 19, 20, and 21, George Olshevsky.
• http://www.polytope.de/nr17.html
• Klitzing, Richard. "4D uniform polytopes (polychora)". x3o3o4x - sidpith, x3o3x4x - proh, x3x3o4x - prit, x3x3x4x - gidpith
External links
• H4 uniform polytopes with coordinates: t03{4,3,3} t013{3,3,4} t013{4,3,3} t0123{4,3,3}
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
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Wikipedia
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Truncated octahedral prism
In 4-dimensional geometry, a truncated octahedral prism or omnitruncated tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 16 cells (2 truncated octahedra connected by 6 cubes, 8 hexagonal prisms.) It has 64 faces (48 squares and 16 hexagons), and 96 edges and 48 vertices.
Truncated octahedral prism
TypePrismatic uniform 4-polytope
Uniform index54
Schläfli symbolt0,1,3{3,4,2} or t{3,4}×{}
t0,1,2,3{3,3,2} or tr{3,3}×{}
Coxeter-Dynkin
Cells16:
2 4.6.6
6 {4,3}
8 {}x{6}
Faces64:
48 {4}
16 {6}
Edges96
Vertices48
Vertex figure
Isosceles-triangular pyramid
Symmetry group[3,4,2], order 96
[3,3,2], order 48
Dual polytopeTetrakis hexahedral bipyramid
Propertiesconvex
It has two symmetry constructions, one from the truncated octahedron, and one as an omnitruncation of the tetrahedron.
It is one of 18 uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids and Archimedean solids.
Images
Net
Schlegel diagram
Alternative names
• Truncated octahedral dyadic prism (Norman W. Johnson)
• Truncated octahedral hyperprism
• Tope (Jonathan Bowers: for truncated octahedral prism)
Related polytopes
The snub tetrahedral prism (also called an icosahedral prism), , sr{3,3}×{ }, is related to this polytope just like a snub tetrahedron (icosahedron), is the alternation of the truncated octahedron in its tetrahedral symmetry . The snub tetrahedral prism has symmetry [(3,3)+,2], order 24, although as an icosahedral prism, its full symmetry is [5,3,2], order 240.
Also related, the full snub tetrahedral antiprism or omnisnub tetrahedral antiprism is defined as an alternation of an omnitruncated tetrahedral prism, represented by $s\left\{{\begin{array}{l}3\\3\\2\end{array}}\right\}$ = ht0,1,2,3{3,3,2}, or , although it cannot be constructed as a uniform 4-polytope. It can also be seen as an alternated truncated octahedral prism or pyritohedral icosahedral antiprism, . It has 2 icosahedra connected by 6 tetrahedra and 8 octahedra, with 24 irregular tetrahedra in the alternated gaps. In total it has 40 cells, 112 triangular faces, 96 edges, and 24 vertices. It has [4,(3,2)+] symmetry, order 48, and also [3,3,2]+ symmetry, order 24.
A construction exists with two regular icosahedra in snub positions with two edge lengths in a ratio of around 0.831 : 1.
Vertex figure for the omnisnub tetrahedral antiprism
See also
• Truncated 16-cell,
External links
• 6. Convex uniform prismatic polychora - Model 54, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora) x x3x3x - tope".
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Wikipedia
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Snub tetraheptagonal tiling
In geometry, the snub tetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{7,4}.
Snub tetraheptagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.3.4.3.7
Schläfli symbolsr{7,4} or $s{\begin{Bmatrix}7\\4\end{Bmatrix}}$
Wythoff symbol| 7 4 2
Coxeter diagram
Symmetry group[7,4]+, (742)
DualOrder-7-4 floret pentagonal tiling
PropertiesVertex-transitive Chiral
Images
Drawn in chiral pairs, with edges missing between black triangles:
Dual tiling
The dual is called an order-7-4 floret pentagonal tiling, defined by face configuration V3.3.4.3.7.
Related polyhedra and tiling
The snub tetraheptagonal tiling is sixth in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.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.∞
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
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.
See also
Wikimedia Commons has media related to Uniform tiling 3-3-4-3-7.
• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
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
|
Snub tetrahexagonal tiling
In geometry, the snub tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,4}.
Snub tetrahexagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.3.4.3.6
Schläfli symbolsr{6,4} or $s{\begin{Bmatrix}6\\4\end{Bmatrix}}$
Wythoff symbol| 6 4 2
Coxeter diagram or
Symmetry group[6,4]+, (642)
DualOrder-6-4 floret pentagonal tiling
PropertiesVertex-transitive Chiral
Images
Drawn in chiral pairs, with edges missing between black triangles:
Related polyhedra and tiling
The snub tetrahexagonal tiling is fifth in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.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.∞
Uniform tetrahexagonal tilings
Symmetry: [6,4], (*642)
(with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries)
(And [(∞,3,∞,3)] (*3232) index 4 subsymmetry)
=
=
=
=
=
=
=
=
=
=
=
=
{6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4}
Uniform duals
V64 V4.12.12 V(4.6)2 V6.8.8 V46 V4.4.4.6 V4.8.12
Alternations
[1+,6,4]
(*443)
[6+,4]
(6*2)
[6,1+,4]
(*3222)
[6,4+]
(4*3)
[6,4,1+]
(*662)
[(6,4,2+)]
(2*32)
[6,4]+
(642)
=
=
=
=
=
=
h{6,4} s{6,4} hr{6,4} s{4,6} h{4,6} hrr{6,4} sr{6,4}
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.
See also
Wikimedia Commons has media related to Uniform tiling 3-3-4-3-6.
• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
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
|
Snub tetraoctagonal tiling
In geometry, the snub tetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{8,4}.
Snub tetraoctagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.3.4.3.8
Schläfli symbolsr{8,4} or $s{\begin{Bmatrix}8\\4\end{Bmatrix}}$
Wythoff symbol| 8 4 2
Coxeter diagram
Symmetry group[8,4]+, (842)
DualOrder-8-4 floret pentagonal tiling
PropertiesVertex-transitive Chiral
Images
Drawn in chiral pairs, with edges missing between black triangles:
Related polyhedra and tiling
The snub tetraoctagonal tiling is seventh in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.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.∞
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
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.
See also
Wikimedia Commons has media related to Uniform tiling 3-3-4-3-8.
• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
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
|
Snub tetraapeirogonal tiling
In geometry, the snub tetraapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{∞,4}.
Snub tetraapeirogonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.3.4.3.∞
Schläfli symbolsr{∞,4} or $s{\begin{Bmatrix}\infty \\4\end{Bmatrix}}$
Wythoff symbol| ∞ 4 2
Coxeter diagram or
Symmetry group[∞,4]+, (∞42)
DualOrder-4-infinite floret pentagonal tiling
PropertiesVertex-transitive Chiral
Images
Drawn in chiral pairs, with edges missing between black triangles:
Related polyhedra and tiling
The snub tetrapeirogonal tiling is last in an infinite series of snub polyhedra and tilings with vertex figure 3.3.4.3.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.∞
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.∞
See also
Wikimedia Commons has media related to Uniform tiling 3-3-4-3-i.
• 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
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
|
Snub order-6 square tiling
In geometry, the snub order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of s{(4,4,3)} or s{4,6}.
Snub order-6 square tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.3.3.4.3.4
Schläfli symbols(4,4,3)
s{4,6}
Wythoff symbol| 4 4 3
Coxeter diagram
Symmetry group[(4,4,3)]+, (443)
[6,4+], (4*3)
DualOrder-4-4-3 snub dual tiling
PropertiesVertex-transitive
Images
Symmetry
The symmetry is doubled as a snub order-6 square tiling, with only one color of square. It has Schläfli symbol of s{4,6}.
Related polyhedra and tiling
The vertex figure 3.3.3.4.3.4 does not uniquely generate a uniform hyperbolic tiling. Another with quadrilateral fundamental domain (3 2 2 2) and 2*32 symmetry is generated by :
Uniform (4,4,3) tilings
Symmetry: [(4,4,3)] (*443) [(4,4,3)]+
(443)
[(4,4,3+)]
(3*22)
[(4,1+,4,3)]
(*3232)
h{6,4}
t0(4,4,3)
h2{6,4}
t0,1(4,4,3)
{4,6}1/2
t1(4,4,3)
h2{6,4}
t1,2(4,4,3)
h{6,4}
t2(4,4,3)
r{6,4}1/2
t0,2(4,4,3)
t{4,6}1/2
t0,1,2(4,4,3)
s{4,6}1/2
s(4,4,3)
hr{4,6}1/2
hr(4,3,4)
h{4,6}1/2
h(4,3,4)
q{4,6}
h1(4,3,4)
Uniform duals
V(3.4)4 V3.8.4.8 V(4.4)3 V3.8.4.8 V(3.4)4 V4.6.4.6 V6.8.8 V3.3.3.4.3.4 V(4.4.3)2 V66 V4.3.4.6.6
Uniform tetrahexagonal tilings
Symmetry: [6,4], (*642)
(with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries)
(And [(∞,3,∞,3)] (*3232) index 4 subsymmetry)
=
=
=
=
=
=
=
=
=
=
=
=
{6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4}
Uniform duals
V64 V4.12.12 V(4.6)2 V6.8.8 V46 V4.4.4.6 V4.8.12
Alternations
[1+,6,4]
(*443)
[6+,4]
(6*2)
[6,1+,4]
(*3222)
[6,4+]
(4*3)
[6,4,1+]
(*662)
[(6,4,2+)]
(2*32)
[6,4]+
(642)
=
=
=
=
=
=
h{6,4} s{6,4} hr{6,4} s{4,6} h{4,6} hrr{6,4} sr{6,4}
See also
• Square tiling
• Uniform tilings in hyperbolic plane
• List of regular polytopes
Footnotes
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
|
Snub order-8 triangular tiling
In geometry, the snub tritetratrigonal tiling or snub order-8 triangular tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbols of s{(3,4,3)} and s{3,8}.
Snub order-8 triangular tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.3.3.3.3.4
Schläfli symbols{3,8}
s(4,3,3)
Wythoff symbol| 4 3 3
Coxeter diagram
Symmetry group[8,3+], (3*4)
[(4,3,3)]+, (433)
DualOrder-4-3-3 snub dual tiling
PropertiesVertex-transitive
Images
Drawn in chiral pairs:
Symmetry
The alternated construction from the truncated order-8 triangular tiling has 2 colors of triangles and achiral symmetry. It has Schläfli symbol of s{3,8}.
Related polyhedra and tiling
Uniform (4,3,3) tilings
Symmetry: [(4,3,3)], (*433) [(4,3,3)]+, (433)
h{8,3}
t0(4,3,3)
r{3,8}1/2
t0,1(4,3,3)
h{8,3}
t1(4,3,3)
h2{8,3}
t1,2(4,3,3)
{3,8}1/2
t2(4,3,3)
h2{8,3}
t0,2(4,3,3)
t{3,8}1/2
t0,1,2(4,3,3)
s{3,8}1/2
s(4,3,3)
Uniform duals
V(3.4)3 V3.8.3.8 V(3.4)3 V3.6.4.6 V(3.3)4 V3.6.4.6 V6.6.8 V3.3.3.3.3.4
Uniform octagonal/triangular tilings
Symmetry: [8,3], (*832) [8,3]+
(832)
[1+,8,3]
(*443)
[8,3+]
(3*4)
{8,3} t{8,3} r{8,3} t{3,8} {3,8} rr{8,3}
s2{3,8}
tr{8,3} sr{8,3} h{8,3} h2{8,3} s{3,8}
or
or
Uniform duals
V83 V3.16.16 V3.8.3.8 V6.6.8 V38 V3.4.8.4 V4.6.16 V34.8 V(3.4)3 V8.6.6 V35.4
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.
See also
Wikimedia Commons has media related to Uniform tiling 3-3-3-3-3-4.
• Square tiling
• Uniform tilings in hyperbolic plane
• List of regular polytopes
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
|
Snub triapeirogonal tiling
In geometry, the snub triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of sr{∞,3}.
Snub triapeirogonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.3.3.3.∞
Schläfli symbolsr{∞,3} or $s{\begin{Bmatrix}\infty \\3\end{Bmatrix}}$
Wythoff symbol| ∞ 3 2
Coxeter diagram or
Symmetry group[∞,3]+, (∞32)
DualOrder-3-infinite floret pentagonal tiling
PropertiesVertex-transitive Chiral
Images
Drawn in chiral pairs, with edges missing between black triangles:
The dual tiling:
Related polyhedra and tiling
This hyperbolic tiling is topologically related as a part of sequence of uniform snub polyhedra with vertex configurations (3.3.3.3.n), and [n,3] Coxeter group symmetry.
n32 symmetry mutations of snub tilings: 3.3.3.3.n
Symmetry
n32
Spherical Euclidean Compact hyperbolic Paracomp.
232 332 432 532 632 732 832 ∞32
Snub
figures
Config. 3.3.3.3.2 3.3.3.3.3 3.3.3.3.4 3.3.3.3.5 3.3.3.3.6 3.3.3.3.7 3.3.3.3.8 3.3.3.3.∞
Gyro
figures
Config. V3.3.3.3.2 V3.3.3.3.3 V3.3.3.3.4 V3.3.3.3.5 V3.3.3.3.6 V3.3.3.3.7 V3.3.3.3.8 V3.3.3.3.∞
Paracompact uniform tilings in [∞,3] family
Symmetry: [∞,3], (*∞32) [∞,3]+
(∞32)
[1+,∞,3]
(*∞33)
[∞,3+]
(3*∞)
=
=
=
=
or
=
or
=
{∞,3} t{∞,3} r{∞,3} t{3,∞} {3,∞} rr{∞,3} tr{∞,3} sr{∞,3} h{∞,3} h2{∞,3} s{3,∞}
Uniform duals
V∞3 V3.∞.∞ V(3.∞)2 V6.6.∞ V3∞ V4.3.4.∞ V4.6.∞ V3.3.3.3.∞ V(3.∞)3 V3.3.3.3.3.∞
See also
• 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
• 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
|
Snub triapeirotrigonal tiling
In geometry, the snub triapeirotrigonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of s{3,∞}.
Snub triapeirotrigonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.3.3.3.3.∞
Schläfli symbols{3,∞}
s(∞,3,3)
Wythoff symbol| ∞ 3 3
Coxeter diagram
Symmetry group[(∞,3,3)]+, (∞33)
DualOrder-i-3-3_t0 dual tiling
PropertiesVertex-transitive Chiral
Related polyhedra and tiling
Paracompact hyperbolic uniform tilings in [(∞,3,3)] family
Symmetry: [(∞,3,3)], (*∞33) [(∞,3,3)]+, (∞33)
(∞,∞,3) t0,1(∞,3,3) t1(∞,3,3) t1,2(∞,3,3) t2(∞,3,3) t0,2(∞,3,3) t0,1,2(∞,3,3) s(∞,3,3)
Dual tilings
V(3.∞)3 V3.∞.3.∞ V(3.∞)3 V3.6.∞.6 V(3.3)∞ V3.6.∞.6 V6.6.∞ V3.3.3.3.3.∞
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.
See also
Wikimedia Commons has media related to Uniform tiling 3-3-3-3-3-i.
• Square tiling
• Uniform tilings in hyperbolic plane
• List of regular polytopes
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
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Wikipedia
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Snub trioctagonal tiling
In geometry, the order-3 snub octagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles, one octagon on each vertex. It has Schläfli symbol of sr{8,3}.
Snub trioctagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.3.3.3.8
Schläfli symbolsr{8,3} or $s{\begin{Bmatrix}8\\3\end{Bmatrix}}$
Wythoff symbol| 8 3 2
Coxeter diagram or or
Symmetry group[8,3]+, (832)
DualOrder-8-3 floret pentagonal tiling
PropertiesVertex-transitive Chiral
Images
Drawn in chiral pairs, with edges missing between black triangles:
Related polyhedra and tilings
This semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram . These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons.
n32 symmetry mutations of snub tilings: 3.3.3.3.n
Symmetry
n32
Spherical Euclidean Compact hyperbolic Paracomp.
232 332 432 532 632 732 832 ∞32
Snub
figures
Config. 3.3.3.3.2 3.3.3.3.3 3.3.3.3.4 3.3.3.3.5 3.3.3.3.6 3.3.3.3.7 3.3.3.3.8 3.3.3.3.∞
Gyro
figures
Config. V3.3.3.3.2 V3.3.3.3.3 V3.3.3.3.4 V3.3.3.3.5 V3.3.3.3.6 V3.3.3.3.7 V3.3.3.3.8 V3.3.3.3.∞
From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 10 forms.
Uniform octagonal/triangular tilings
Symmetry: [8,3], (*832) [8,3]+
(832)
[1+,8,3]
(*443)
[8,3+]
(3*4)
{8,3} t{8,3} r{8,3} t{3,8} {3,8} rr{8,3}
s2{3,8}
tr{8,3} sr{8,3} h{8,3} h2{8,3} s{3,8}
or
or
Uniform duals
V83 V3.16.16 V3.8.3.8 V6.6.8 V38 V3.4.8.4 V4.6.16 V34.8 V(3.4)3 V8.6.6 V35.4
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.
See also
• Snub hexagonal tiling
• Floret pentagonal tiling
• Order-3 heptagonal tiling
• Tilings of regular polygons
• List of uniform planar tilings
• Kagome lattice
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
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Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.
Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense.
Motivation
In this section and throughout the article $\Omega $ is an open subset of $\mathbb {R} ^{n}.$
There are many criteria for smoothness of mathematical functions. The most basic criterion may be that of continuity. A stronger notion of smoothness is that of differentiability (because functions that are differentiable are also continuous) and a yet stronger notion of smoothness is that the derivative also be continuous (these functions are said to be of class $C^{1}$ — see Differentiability classes). Differentiable functions are important in many areas, and in particular for differential equations. In the twentieth century, however, it was observed that the space $C^{1}$ (or $C^{2}$, etc.) was not exactly the right space to study solutions of differential equations. The Sobolev spaces are the modern replacement for these spaces in which to look for solutions of partial differential equations.
Quantities or properties of the underlying model of the differential equation are usually expressed in terms of integral norms. A typical example is measuring the energy of a temperature or velocity distribution by an $L^{2}$-norm. It is therefore important to develop a tool for differentiating Lebesgue space functions.
The integration by parts formula yields that for every $u\in C^{k}(\Omega )$, where $k$ is a natural number, and for all infinitely differentiable functions with compact support $\varphi \in C_{c}^{\infty }(\Omega ),$
$\int _{\Omega }u\,D^{\alpha \!}\varphi \,dx=(-1)^{|\alpha |}\int _{\Omega }\varphi \,D^{\alpha \!}u\,dx,$
where $\alpha $ is a multi-index of order $|\alpha |=k$ and we are using the notation:
$D^{\alpha \!}f={\frac {\partial ^{|\alpha |}\!f}{\partial x_{1}^{\alpha _{1}}\dots \partial x_{n}^{\alpha _{n}}}}.$
The left-hand side of this equation still makes sense if we only assume $u$ to be locally integrable. If there exists a locally integrable function $v$, such that
$\int _{\Omega }u\,D^{\alpha \!}\varphi \;dx=(-1)^{|\alpha |}\int _{\Omega }v\,\varphi \;dx\qquad {\text{for all }}\varphi \in C_{c}^{\infty }(\Omega ),$
then we call $v$ the weak $\alpha $-th partial derivative of $u$. If there exists a weak $\alpha $-th partial derivative of $u$, then it is uniquely defined almost everywhere, and thus it is uniquely determined as an element of a Lebesgue space. On the other hand, if $u\in C^{k}(\Omega )$, then the classical and the weak derivative coincide. Thus, if $v$ is a weak $\alpha $-th partial derivative of $u$, we may denote it by $D^{\alpha }u:=v$.
For example, the function
$u(x)={\begin{cases}1+x&-1<x<0\\10&x=0\\1-x&0<x<1\\0&{\text{else}}\end{cases}}$
is not continuous at zero, and not differentiable at −1, 0, or 1. Yet the function
$v(x)={\begin{cases}1&-1<x<0\\-1&0<x<1\\0&{\text{else}}\end{cases}}$
satisfies the definition for being the weak derivative of $u(x),$ which then qualifies as being in the Sobolev space $W^{1,p}$ (for any allowed $p$, see definition below).
The Sobolev spaces $W^{k,p}(\Omega )$ combine the concepts of weak differentiability and Lebesgue norms.
Sobolev spaces with integer k
One-dimensional case
In the one-dimensional case the Sobolev space $W^{k,p}(\mathbb {R} )$ for $1\leq p\leq \infty $ is defined as the subset of functions $f$ in $L^{p}(\mathbb {R} )$ such that $f$ and its weak derivatives up to order $k$ have a finite Lp norm. As mentioned above, some care must be taken to define derivatives in the proper sense. In the one-dimensional problem it is enough to assume that the $(k{-}1)$-th derivative $f^{(k-1)}$ is differentiable almost everywhere and is equal almost everywhere to the Lebesgue integral of its derivative (this excludes irrelevant examples such as Cantor's function).
With this definition, the Sobolev spaces admit a natural norm,
$\|f\|_{k,p}=\left(\sum _{i=0}^{k}\left\|f^{(i)}\right\|_{p}^{p}\right)^{\frac {1}{p}}=\left(\sum _{i=0}^{k}\int \left|f^{(i)}(t)\right|^{p}\,dt\right)^{\frac {1}{p}}.$
One can extend this to the case $p=\infty $, with the norm then defined using the essential supremum by
$\|f\|_{k,\infty }=\max _{i=0,\ldots ,k}\left\|f^{(i)}\right\|_{\infty }=\max _{i=0,\ldots ,k}\left({\text{ess}}\,\sup _{t}\left|f^{(i)}(t)\right|\right).$
Equipped with the norm $\|\cdot \|_{k,p},W^{k,p}$ becomes a Banach space. It turns out that it is enough to take only the first and last in the sequence, i.e., the norm defined by
$\left\|f^{(k)}\right\|_{p}+\|f\|_{p}$
is equivalent to the norm above (i.e. the induced topologies of the norms are the same).
The case p = 2
Sobolev spaces with p = 2 are especially important because of their connection with Fourier series and because they form a Hilbert space. A special notation has arisen to cover this case, since the space is a Hilbert space:
$H^{k}=W^{k,2}.$
The space $H^{k}$ can be defined naturally in terms of Fourier series whose coefficients decay sufficiently rapidly, namely,
$H^{k}(\mathbb {T} )={\Big \{}f\in L^{2}(\mathbb {T} ):\sum _{n=-\infty }^{\infty }\left(1+n^{2}+n^{4}+\dots +n^{2k}\right)\left|{\widehat {f}}(n)\right|^{2}<\infty {\Big \}},$
where ${\widehat {f}}$ is the Fourier series of $f,$ and $\mathbb {T} $ denotes the 1-torus. As above, one can use the equivalent norm
$\|f\|_{k,2}^{2}=\sum _{n=-\infty }^{\infty }\left(1+|n|^{2}\right)^{k}\left|{\widehat {f}}(n)\right|^{2}.$
Both representations follow easily from Parseval's theorem and the fact that differentiation is equivalent to multiplying the Fourier coefficient by $in$.
Furthermore, the space $H^{k}$ admits an inner product, like the space $H^{0}=L^{2}.$ In fact, the $H^{k}$ inner product is defined in terms of the $L^{2}$ inner product:
$\langle u,v\rangle _{H^{k}}=\sum _{i=0}^{k}\left\langle D^{i}u,D^{i}v\right\rangle _{L^{2}}.$
The space $H^{k}$ becomes a Hilbert space with this inner product.
Other examples
In one dimension, some other Sobolev spaces permit a simpler description. For example, $W^{1,1}(0,1)$ is the space of absolutely continuous functions on (0, 1) (or rather, equivalence classes of functions that are equal almost everywhere to such), while $W^{1,\infty }(I)$ is the space of bounded Lipschitz functions on I, for every interval I. However, these properties are lost or not as simple for functions of more than one variable.
All spaces $W^{k,\infty }$ are (normed) algebras, i.e. the product of two elements is once again a function of this Sobolev space, which is not the case for $p<\infty .$ (E.g., functions behaving like |x|−1/3 at the origin are in $L^{2},$ but the product of two such functions is not in $L^{2}$).
Multidimensional case
The transition to multiple dimensions brings more difficulties, starting from the very definition. The requirement that $f^{(k-1)}$ be the integral of $f^{(k)}$ does not generalize, and the simplest solution is to consider derivatives in the sense of distribution theory.
A formal definition now follows. Let $k\in \mathbb {N} ,1\leqslant p\leqslant \infty .$ The Sobolev space $W^{k,p}(\Omega )$ is defined to be the set of all functions $f$ on $\Omega $ such that for every multi-index $\alpha $ with $|\alpha |\leqslant k,$ the mixed partial derivative
$f^{(\alpha )}={\frac {\partial ^{|\alpha |\!}f}{\partial x_{1}^{\alpha _{1}}\dots \partial x_{n}^{\alpha _{n}}}}$
exists in the weak sense and is in $L^{p}(\Omega ),$ i.e.
$\left\|f^{(\alpha )}\right\|_{L^{p}}<\infty .$
That is, the Sobolev space $W^{k,p}(\Omega )$ is defined as
$W^{k,p}(\Omega )=\left\{u\in L^{p}(\Omega ):D^{\alpha }u\in L^{p}(\Omega )\,\,\forall |\alpha |\leqslant k\right\}.$
The natural number $k$ is called the order of the Sobolev space $W^{k,p}(\Omega ).$
There are several choices for a norm for $W^{k,p}(\Omega ).$ The following two are common and are equivalent in the sense of equivalence of norms:
$\|u\|_{W^{k,p}(\Omega )}:={\begin{cases}\left(\sum _{|\alpha |\leqslant k}\left\|D^{\alpha }u\right\|_{L^{p}(\Omega )}^{p}\right)^{\frac {1}{p}}&1\leqslant p<\infty ;\\\max _{|\alpha |\leqslant k}\left\|D^{\alpha }u\right\|_{L^{\infty }(\Omega )}&p=\infty ;\end{cases}}$ ;\\\max _{|\alpha |\leqslant k}\left\|D^{\alpha }u\right\|_{L^{\infty }(\Omega )}&p=\infty ;\end{cases}}}
and
$\|u\|'_{W^{k,p}(\Omega )}:={\begin{cases}\sum _{|\alpha |\leqslant k}\left\|D^{\alpha }u\right\|_{L^{p}(\Omega )}&1\leqslant p<\infty ;\\\sum _{|\alpha |\leqslant k}\left\|D^{\alpha }u\right\|_{L^{\infty }(\Omega )}&p=\infty .\end{cases}}$ ;\\\sum _{|\alpha |\leqslant k}\left\|D^{\alpha }u\right\|_{L^{\infty }(\Omega )}&p=\infty .\end{cases}}}
With respect to either of these norms, $W^{k,p}(\Omega )$ is a Banach space. For $p<\infty ,W^{k,p}(\Omega )$ is also a separable space. It is conventional to denote $W^{k,2}(\Omega )$ by $H^{k}(\Omega )$ for it is a Hilbert space with the norm $\|\cdot \|_{W^{k,2}(\Omega )}$.[1]
Approximation by smooth functions
It is rather hard to work with Sobolev spaces relying only on their definition. It is therefore interesting to know that by the Meyers–Serrin theorem a function $u\in W^{k,p}(\Omega )$ can be approximated by smooth functions. This fact often allows us to translate properties of smooth functions to Sobolev functions. If $p$ is finite and $\Omega $ is open, then there exists for any $u\in W^{k,p}(\Omega )$ an approximating sequence of functions $u_{m}\in C^{\infty }(\Omega )$ such that:
$\left\|u_{m}-u\right\|_{W^{k,p}(\Omega )}\to 0.$
If $\Omega $ has Lipschitz boundary, we may even assume that the $u_{m}$ are the restriction of smooth functions with compact support on all of $\mathbb {R} ^{n}.$[2]
Examples
In higher dimensions, it is no longer true that, for example, $W^{1,1}$ contains only continuous functions. For example, $|x|^{-1}\in W^{1,1}(\mathbb {B} ^{3})$ where $\mathbb {B} ^{3}$ is the unit ball in three dimensions. For $k>n/p$, the space $W^{k,p}(\Omega )$ will contain only continuous functions, but for which $k$ this is already true depends both on $p$ and on the dimension. For example, as can be easily checked using spherical polar coordinates for the function $f:\mathbb {B} ^{n}\to \mathbb {R} \cup \{\infty \}$ defined on the n-dimensional ball we have:
$f(x)=|x|^{-\alpha }\in W^{k,p}(\mathbb {B} ^{n})\Longleftrightarrow \alpha <{\tfrac {n}{p}}-k.$
Intuitively, the blow-up of f at 0 "counts for less" when n is large since the unit ball has "more outside and less inside" in higher dimensions.
Absolutely continuous on lines (ACL) characterization of Sobolev functions
Let $1\leqslant p\leqslant \infty .$ If a function is in $W^{1,p}(\Omega ),$ then, possibly after modifying the function on a set of measure zero, the restriction to almost every line parallel to the coordinate directions in $\mathbb {R} ^{n}$ is absolutely continuous; what's more, the classical derivative along the lines that are parallel to the coordinate directions are in $L^{p}(\Omega ).$ Conversely, if the restriction of $f$ to almost every line parallel to the coordinate directions is absolutely continuous, then the pointwise gradient $\nabla f$ exists almost everywhere, and $f$ is in $W^{1,p}(\Omega )$ provided $f,|\nabla f|\in L^{p}(\Omega ).$ In particular, in this case the weak partial derivatives of $f$ and pointwise partial derivatives of $f$ agree almost everywhere. The ACL characterization of the Sobolev spaces was established by Otto M. Nikodym (1933); see (Maz'ya 2011, §1.1.3).
A stronger result holds when $p>n.$ A function in $W^{1,p}(\Omega )$ is, after modifying on a set of measure zero, Hölder continuous of exponent $\gamma =1-{\tfrac {n}{p}},$ by Morrey's inequality. In particular, if $p=\infty $ and $\Omega $ has Lipschitz boundary, then the function is Lipschitz continuous.
Functions vanishing at the boundary
See also: Trace operator
The Sobolev space $W^{1,2}(\Omega )$ is also denoted by $H^{1}\!(\Omega ).$ It is a Hilbert space, with an important subspace $H_{0}^{1}\!(\Omega )$ defined to be the closure of the infinitely differentiable functions compactly supported in $\Omega $ in $H^{1}\!(\Omega ).$ The Sobolev norm defined above reduces here to
$\|f\|_{H^{1}}=\left(\int _{\Omega }\!|f|^{2}\!+\!|\nabla \!f|^{2}\right)^{\!{\frac {1}{2}}}.$
When $\Omega $ has a regular boundary, $H_{0}^{1}\!(\Omega )$ can be described as the space of functions in $H^{1}\!(\Omega )$ that vanish at the boundary, in the sense of traces (see below). When $n=1,$ if $\Omega =(a,b)$ is a bounded interval, then $H_{0}^{1}(a,b)$ consists of continuous functions on $[a,b]$ of the form
$f(x)=\int _{a}^{x}f'(t)\,\mathrm {d} t,\qquad x\in [a,b]$
where the generalized derivative $f'$ is in $L^{2}(a,b)$ and has 0 integral, so that $f(b)=f(a)=0.$
When $\Omega $ is bounded, the Poincaré inequality states that there is a constant $C=C(\Omega )$ such that:
$\int _{\Omega }|f|^{2}\leqslant C^{2}\int _{\Omega }|\nabla f|^{2},\qquad f\in H_{0}^{1}(\Omega ).$
When $\Omega $ is bounded, the injection from $H_{0}^{1}\!(\Omega )$ to $L^{2}\!(\Omega ),$ is compact. This fact plays a role in the study of the Dirichlet problem, and in the fact that there exists an orthonormal basis of $L^{2}(\Omega )$ consisting of eigenvectors of the Laplace operator (with Dirichlet boundary condition).
Traces
See also: Trace operator
Sobolev spaces are often considered when investigating partial differential equations. It is essential to consider boundary values of Sobolev functions. If $u\in C(\Omega )$, those boundary values are described by the restriction $u|_{\partial \Omega }.$ However, it is not clear how to describe values at the boundary for $u\in W^{k,p}(\Omega ),$ as the n-dimensional measure of the boundary is zero. The following theorem[2] resolves the problem:
Trace theorem — Assume Ω is bounded with Lipschitz boundary. Then there exists a bounded linear operator $T:W^{1,p}(\Omega )\to L^{p}(\partial \Omega )$ such that
${\begin{aligned}Tu&=u|_{\partial \Omega }&&u\in W^{1,p}(\Omega )\cap C({\overline {\Omega }})\\\|Tu\|_{L^{p}(\partial \Omega )}&\leqslant c(p,\Omega )\|u\|_{W^{1,p}(\Omega )}&&u\in W^{1,p}(\Omega ).\end{aligned}}$
Tu is called the trace of u. Roughly speaking, this theorem extends the restriction operator to the Sobolev space $W^{1,p}(\Omega )$ for well-behaved Ω. Note that the trace operator T is in general not surjective, but for 1 < p < ∞ it maps continuously onto the Sobolev–Slobodeckij space $W^{1-{\frac {1}{p}},p}(\partial \Omega ).$
Intuitively, taking the trace costs 1/p of a derivative. The functions u in W1,p(Ω) with zero trace, i.e. Tu = 0, can be characterized by the equality
$W_{0}^{1,p}(\Omega )=\left\{u\in W^{1,p}(\Omega ):Tu=0\right\},$
where
$W_{0}^{1,p}(\Omega ):=\left\{u\in W^{1,p}(\Omega ):\exists \{u_{m}\}_{m=1}^{\infty }\subset C_{c}^{\infty }(\Omega ),\ {\text{such that}}\ u_{m}\to u\ {\textrm {in}}\ W^{1,p}(\Omega )\right\}.$
In other words, for Ω bounded with Lipschitz boundary, trace-zero functions in $W^{1,p}(\Omega )$ can be approximated by smooth functions with compact support.
Sobolev spaces with non-integer k
Bessel potential spaces
For a natural number k and 1 < p < ∞ one can show (by using Fourier multipliers[3][4]) that the space $W^{k,p}(\mathbb {R} ^{n})$ can equivalently be defined as
$W^{k,p}(\mathbb {R} ^{n})=H^{k,p}(\mathbb {R} ^{n}):={\Big \{}f\in L^{p}(\mathbb {R} ^{n}):{\mathcal {F}}^{-1}{\Big [}{\big (}1+|\xi |^{2}{\big )}^{\frac {k}{2}}{\mathcal {F}}f{\Big ]}\in L^{p}(\mathbb {R} ^{n}){\Big \}},$
with the norm
$\|f\|_{H^{k,p}(\mathbb {R} ^{n})}:=\left\|{\mathcal {F}}^{-1}{\Big [}{\big (}1+|\xi |^{2}{\big )}^{\frac {k}{2}}{\mathcal {F}}f{\Big ]}\right\|_{L^{p}(\mathbb {R} ^{n})}.$
This motivates Sobolev spaces with non-integer order since in the above definition we can replace k by any real number s. The resulting spaces
$H^{s,p}(\mathbb {R} ^{n}):=\left\{f\in {\mathcal {S}}'(\mathbb {R} ^{n}):{\mathcal {F}}^{-1}\left[{\big (}1+|\xi |^{2}{\big )}^{\frac {s}{2}}{\mathcal {F}}f\right]\in L^{p}(\mathbb {R} ^{n})\right\}$
are called Bessel potential spaces[5] (named after Friedrich Bessel). They are Banach spaces in general and Hilbert spaces in the special case p = 2.
For $s\geq 0,H^{s,p}(\Omega )$ is the set of restrictions of functions from $H^{s,p}(\mathbb {R} ^{n})$ to Ω equipped with the norm
$\|f\|_{H^{s,p}(\Omega )}:=\inf \left\{\|g\|_{H^{s,p}(\mathbb {R} ^{n})}:g\in H^{s,p}(\mathbb {R} ^{n}),g|_{\Omega }=f\right\}.$
Again, Hs,p(Ω) is a Banach space and in the case p = 2 a Hilbert space.
Using extension theorems for Sobolev spaces, it can be shown that also Wk,p(Ω) = Hk,p(Ω) holds in the sense of equivalent norms, if Ω is domain with uniform Ck-boundary, k a natural number and 1 < p < ∞. By the embeddings
$H^{k+1,p}(\mathbb {R} ^{n})\hookrightarrow H^{s',p}(\mathbb {R} ^{n})\hookrightarrow H^{s,p}(\mathbb {R} ^{n})\hookrightarrow H^{k,p}(\mathbb {R} ^{n}),\quad k\leqslant s\leqslant s'\leqslant k+1$
the Bessel potential spaces $H^{s,p}(\mathbb {R} ^{n})$ form a continuous scale between the Sobolev spaces $W^{k,p}(\mathbb {R} ^{n}).$ From an abstract point of view, the Bessel potential spaces occur as complex interpolation spaces of Sobolev spaces, i.e. in the sense of equivalent norms it holds that
$\left[W^{k,p}(\mathbb {R} ^{n}),W^{k+1,p}(\mathbb {R} ^{n})\right]_{\theta }=H^{s,p}(\mathbb {R} ^{n}),$
where:
$1\leqslant p\leqslant \infty ,\ 0<\theta <1,\ s=(1-\theta )k+\theta (k+1)=k+\theta .$
Sobolev–Slobodeckij spaces
Another approach to define fractional order Sobolev spaces arises from the idea to generalize the Hölder condition to the Lp-setting.[6] For $1\leqslant p<\infty ,\theta \in (0,1)$ and $f\in L^{p}(\Omega ),$ the Slobodeckij seminorm (roughly analogous to the Hölder seminorm) is defined by
$[f]_{\theta ,p,\Omega }:=\left(\int _{\Omega }\int _{\Omega }{\frac {|f(x)-f(y)|^{p}}{|x-y|^{\theta p+n}}}\;dx\;dy\right)^{\frac {1}{p}}.$
Let s > 0 be not an integer and set $\theta =s-\lfloor s\rfloor \in (0,1)$. Using the same idea as for the Hölder spaces, the Sobolev–Slobodeckij space[7] $W^{s,p}(\Omega )$ is defined as
$W^{s,p}(\Omega ):=\left\{f\in W^{\lfloor s\rfloor ,p}(\Omega ):\sup _{|\alpha |=\lfloor s\rfloor }[D^{\alpha }f]_{\theta ,p,\Omega }<\infty \right\}.$
It is a Banach space for the norm
$\|f\|_{W^{s,p}(\Omega )}:=\|f\|_{W^{\lfloor s\rfloor ,p}(\Omega )}+\sup _{|\alpha |=\lfloor s\rfloor }[D^{\alpha }f]_{\theta ,p,\Omega }.$
If $\Omega $ is suitably regular in the sense that there exist certain extension operators, then also the Sobolev–Slobodeckij spaces form a scale of Banach spaces, i.e. one has the continuous injections or embeddings
$W^{k+1,p}(\Omega )\hookrightarrow W^{s',p}(\Omega )\hookrightarrow W^{s,p}(\Omega )\hookrightarrow W^{k,p}(\Omega ),\quad k\leqslant s\leqslant s'\leqslant k+1.$
There are examples of irregular Ω such that $W^{1,p}(\Omega )$ is not even a vector subspace of $W^{s,p}(\Omega )$ for 0 < s < 1 (see Example 9.1 of [8])
From an abstract point of view, the spaces $W^{s,p}(\Omega )$ coincide with the real interpolation spaces of Sobolev spaces, i.e. in the sense of equivalent norms the following holds:
$W^{s,p}(\Omega )=\left(W^{k,p}(\Omega ),W^{k+1,p}(\Omega )\right)_{\theta ,p},\quad k\in \mathbb {N} ,s\in (k,k+1),\theta =s-\lfloor s\rfloor .$
Sobolev–Slobodeckij spaces play an important role in the study of traces of Sobolev functions. They are special cases of Besov spaces.[4]
Extension operators
If $\Omega $ is a domain whose boundary is not too poorly behaved (e.g., if its boundary is a manifold, or satisfies the more permissive "cone condition") then there is an operator A mapping functions of $\Omega $ to functions of $\mathbb {R} ^{n}$ such that:
1. Au(x) = u(x) for almost every x in $\Omega $ and
2. $A:W^{k,p}(\Omega )\to W^{k,p}(\mathbb {R} ^{n})$ is continuous for any 1 ≤ p ≤ ∞ and integer k.
We will call such an operator A an extension operator for $\Omega .$
Case of p = 2
Extension operators are the most natural way to define $H^{s}(\Omega )$ for non-integer s (we cannot work directly on $\Omega $ since taking Fourier transform is a global operation). We define $H^{s}(\Omega )$ by saying that $u\in H^{s}(\Omega )$ if and only if $Au\in H^{s}(\mathbb {R} ^{n}).$ Equivalently, complex interpolation yields the same $H^{s}(\Omega )$ spaces so long as $\Omega $ has an extension operator. If $\Omega $ does not have an extension operator, complex interpolation is the only way to obtain the $H^{s}(\Omega )$ spaces.
As a result, the interpolation inequality still holds.
Extension by zero
Like above, we define $H_{0}^{s}(\Omega )$ to be the closure in $H^{s}(\Omega )$ of the space $C_{c}^{\infty }(\Omega )$ of infinitely differentiable compactly supported functions. Given the definition of a trace, above, we may state the following
Theorem — Let $\Omega $ be uniformly Cm regular, m ≥ s and let P be the linear map sending u in $H^{s}(\Omega )$ to
$\left.\left(u,{\frac {du}{dn}},\dots ,{\frac {d^{k}u}{dn^{k}}}\right)\right|_{G}$
where d/dn is the derivative normal to G, and k is the largest integer less than s. Then $H_{0}^{s}$ is precisely the kernel of P.
If $u\in H_{0}^{s}(\Omega )$ we may define its extension by zero ${\tilde {u}}\in L^{2}(\mathbb {R} ^{n})$ in the natural way, namely
${\tilde {u}}(x)={\begin{cases}u(x)&x\in \Omega \\0&{\text{else}}\end{cases}}$
Theorem — Let $s>{\tfrac {1}{2}}.$ The map $u\mapsto {\tilde {u}}$ is continuous into $H^{s}(\mathbb {R} ^{n})$ if and only if s is not of the form $n+{\tfrac {1}{2}}$ for n an integer.
For f ∈ Lp(Ω) its extension by zero,
$Ef:={\begin{cases}f&{\textrm {on}}\ \Omega ,\\0&{\textrm {otherwise}}\end{cases}}$
is an element of $L^{p}(\mathbb {R} ^{n}).$ Furthermore,
$\|Ef\|_{L^{p}(\mathbb {R} ^{n})}=\|f\|_{L^{p}(\Omega )}.$
In the case of the Sobolev space W1,p(Ω) for 1 ≤ p ≤ ∞, extending a function u by zero will not necessarily yield an element of $W^{1,p}(\mathbb {R} ^{n}).$ But if Ω is bounded with Lipschitz boundary (e.g. ∂Ω is C1), then for any bounded open set O such that Ω⊂⊂O (i.e. Ω is compactly contained in O), there exists a bounded linear operator[2]
$E:W^{1,p}(\Omega )\to W^{1,p}(\mathbb {R} ^{n}),$
such that for each $u\in W^{1,p}(\Omega ):Eu=u$ a.e. on Ω, Eu has compact support within O, and there exists a constant C depending only on p, Ω, O and the dimension n, such that
$\|Eu\|_{W^{1,p}(\mathbb {R} ^{n})}\leqslant C\|u\|_{W^{1,p}(\Omega )}.$
We call $Eu$ an extension of $u$ to $\mathbb {R} ^{n}.$
Sobolev embeddings
Main article: Sobolev inequality
It is a natural question to ask if a Sobolev function is continuous or even continuously differentiable. Roughly speaking, sufficiently many weak derivatives (i.e. large k) result in a classical derivative. This idea is generalized and made precise in the Sobolev embedding theorem.
Write $W^{k,p}$ for the Sobolev space of some compact Riemannian manifold of dimension n. Here k can be any real number, and 1 ≤ p ≤ ∞. (For p = ∞ the Sobolev space $W^{k,\infty }$ is defined to be the Hölder space Cn,α where k = n + α and 0 < α ≤ 1.) The Sobolev embedding theorem states that if $k\geqslant m$ and $k-{\tfrac {n}{p}}\geqslant m-{\tfrac {n}{q}}$ then
$W^{k,p}\subseteq W^{m,q}$
and the embedding is continuous. Moreover, if $k>m$ and $k-{\tfrac {n}{p}}>m-{\tfrac {n}{q}}$ then the embedding is completely continuous (this is sometimes called Kondrachov's theorem or the Rellich–Kondrachov theorem). Functions in $W^{m,\infty }$ have all derivatives of order less than m continuous, so in particular this gives conditions on Sobolev spaces for various derivatives to be continuous. Informally these embeddings say that to convert an Lp estimate to a boundedness estimate costs 1/p derivatives per dimension.
There are similar variations of the embedding theorem for non-compact manifolds such as $\mathbb {R} ^{n}$ (Stein 1970). Sobolev embeddings on $\mathbb {R} ^{n}$ that are not compact often have a related, but weaker, property of cocompactness.
See also
• Sobolev mapping
• Souček space
Notes
1. Evans 2010, Chapter 5.2
2. Adams & Fournier 2003
3. Bergh & Löfström 1976
4. Triebel 1995
5. Bessel potential spaces with variable integrability have been independently introduced by Almeida & Samko (A. Almeida and S. Samko, "Characterization of Riesz and Bessel potentials on variable Lebesgue spaces", J. Function Spaces Appl. 4 (2006), no. 2, 113–144) and Gurka, Harjulehto & Nekvinda (P. Gurka, P. Harjulehto and A. Nekvinda: "Bessel potential spaces with variable exponent", Math. Inequal. Appl. 10 (2007), no. 3, 661–676).
6. Lunardi 1995
7. In the literature, fractional Sobolev-type spaces are also called Aronszajn spaces, Gagliardo spaces or Slobodeckij spaces, after the names of the mathematicians who introduced them in the 1950s: N. Aronszajn ("Boundary values of functions with finite Dirichlet integral", Techn. Report of Univ. of Kansas 14 (1955), 77–94), E. Gagliardo ("Proprietà di alcune classi di funzioni in più variabili", Ricerche Mat. 7 (1958), 102–137), and L. N. Slobodeckij ("Generalized Sobolev spaces and their applications to boundary value problems of partial differential equations", Leningrad. Gos. Ped. Inst. Učep. Zap. 197 (1958), 54–112).
8. Di Nezza, Eleonora; Palatucci, Giampiero; Valdinoci, Enrico (2012-07-01). "Hitchhikerʼs guide to the fractional Sobolev spaces". Bulletin des Sciences Mathématiques. 136 (5): 521–573. doi:10.1016/j.bulsci.2011.12.004. ISSN 0007-4497.
References
• Adams, Robert A.; Fournier, John (2003) [1975]. Sobolev Spaces. Pure and Applied Mathematics. Vol. 140 (2nd ed.). Boston, MA: Academic Press. ISBN 978-0-12-044143-3..
• Aubin, Thierry (1982), Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-5734-9, ISBN 978-0-387-90704-8, MR 0681859.
• Bergh, Jöran; Löfström, Jörgen (1976), Interpolation Spaces, An Introduction, Grundlehren der Mathematischen Wissenschaften, vol. 223, Springer-Verlag, pp. X + 207, ISBN 978-7-5062-6011-4, MR 0482275, Zbl 0344.46071
• Evans, Lawrence C. (2010) [1998]. Partial Differential Equations. Graduate Studies in Mathematics. Vol. 19 (2nd ed.). American Mathematical Society. p. 749. ISBN 978-0-8218-4974-3.
• Leoni, Giovanni (2009). A First Course in Sobolev Spaces. Graduate Studies in Mathematics. Vol. 105. American Mathematical Society. pp. xvi+607. ISBN 978-0-8218-4768-8. MR 2527916. Zbl 1180.46001.
• Maz'ja, Vladimir G. (1985), Sobolev Spaces, Springer Series in Soviet Mathematics, Berlin–Heidelberg–New York: Springer-Verlag, pp. xix+486, doi:10.1007/978-3-662-09922-3, ISBN 0-387-13589-8, MR 0817985, Zbl 0692.46023
• Maz'ya, Vladimir G.; Poborchi, Sergei V. (1997), Differentiable Functions on Bad Domains, Singapore–New Jersey–London–Hong Kong: World Scientific, pp. xx+481, ISBN 981-02-2767-1, MR 1643072, Zbl 0918.46033.
• Maz'ya, Vladimir G. (2011) [1985], Sobolev Spaces. With Applications to Elliptic Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, vol. 342 (2nd revised and augmented ed.), Berlin–Heidelberg–New York: Springer Verlag, pp. xxviii+866, doi:10.1007/978-3-642-15564-2, ISBN 978-3-642-15563-5, MR 2777530, Zbl 1217.46002.
• Lunardi, Alessandra (1995), Analytic semigroups and optimal regularity in parabolic problems, Basel: Birkhäuser Verlag.
• Nikodym, Otto (1933), "Sur une classe de fonctions considérée dans l'étude du problème de Dirichlet", Fund. Math., 21: 129–150, doi:10.4064/fm-21-1-129-150.
• Nikol'skii, S.M. (2001) [1994], "Imbedding theorems", Encyclopedia of Mathematics, EMS Press.
• Nikol'skii, S.M. (2001) [1994], "Sobolev space", Encyclopedia of Mathematics, EMS Press.
• Sobolev, S. L. (1963), "On a theorem of functional analysis", Eleven Papers on Analysis, American Mathematical Society Translations: Series 2, vol. 34, pp. 39–68, doi:10.1090/trans2/034/02, ISBN 9780821817346; translation of Mat. Sb., 4 (1938) pp. 471–497.
• Sobolev, S.L. (1963), Some applications of functional analysis in mathematical physics, Amer. Math. Soc..
• Stein, E (1970), Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, ISBN 0-691-08079-8.
• Triebel, H. (1995), Interpolation Theory, Function Spaces, Differential Operators, Heidelberg: Johann Ambrosius Barth.
• Ziemer, William P. (1989), Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-1015-3, hdl:10338.dmlcz/143849, ISBN 978-0-387-97017-2, MR 1014685.
External links
• Eleonora Di Nezza, Giampiero Palatucci, Enrico Valdinoci (2011). "Hitchhiker's guide to the fractional Sobolev spaces".
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Bochner space
In mathematics, Bochner spaces are a generalization of the concept of $L^{p}$ spaces to functions whose values lie in a Banach space which is not necessarily the space $\mathbb {R} $ or $\mathbb {C} $ of real or complex numbers.
The space $L^{p}(X)$ consists of (equivalence classes of) all Bochner measurable functions $f$ with values in the Banach space $X$ whose norm $\|f\|_{X}$ lies in the standard $L^{p}$ space. Thus, if $X$ is the set of complex numbers, it is the standard Lebesgue $L^{p}$ space.
Almost all standard results on $L^{p}$ spaces do hold on Bochner spaces too; in particular, the Bochner spaces $L^{p}(X)$ are Banach spaces for $1\leq p\leq \infty .$
Bochner spaces are named for the mathematician Salomon Bochner.
Definition
Given a measure space $(T,\Sigma ;\mu ),$ ;\mu ),} a Banach space $\left(X,\|\,\cdot \,\|_{X}\right)$ and $1\leq p\leq \infty ,$ the Bochner space $L^{p}(T;X)$ is defined to be the Kolmogorov quotient (by equality almost everywhere) of the space of all Bochner measurable functions $u:T\to X$ such that the corresponding norm is finite:
$\|u\|_{L^{p}(T;X)}:=\left(\int _{T}\|u(t)\|_{X}^{p}\,\mathrm {d} \mu (t)\right)^{1/p}<+\infty {\mbox{ for }}1\leq p<\infty ,$
$\|u\|_{L^{\infty }(T;X)}:=\mathrm {ess\,sup} _{t\in T}\|u(t)\|_{X}<+\infty .$
In other words, as is usual in the study of $L^{p}$ spaces, $L^{p}(T;X)$ is a space of equivalence classes of functions, where two functions are defined to be equivalent if they are equal everywhere except upon a $\mu $-measure zero subset of $T.$ As is also usual in the study of such spaces, it is usual to abuse notation and speak of a "function" in $L^{p}(T;X)$ rather than an equivalence class (which would be more technically correct).
Applications
Bochner spaces are often used in the functional analysis approach to the study of partial differential equations that depend on time, e.g. the heat equation: if the temperature $g(t,x)$ is a scalar function of time and space, one can write $(f(t))(x):=g(t,x)$ to make $f$ a family $f(t)$ (parametrized by time) of functions of space, possibly in some Bochner space.
Application to PDE theory
Very often, the space $T$ is an interval of time over which we wish to solve some partial differential equation, and $\mu $ will be one-dimensional Lebesgue measure. The idea is to regard a function of time and space as a collection of functions of space, this collection being parametrized by time. For example, in the solution of the heat equation on a region $\Omega $ in $\mathbb {R} ^{n}$ and an interval of time $[0,T],$ one seeks solutions
$u\in L^{2}\left([0,T];H_{0}^{1}(\Omega )\right)$
with time derivative
${\frac {\partial u}{\partial t}}\in L^{2}\left([0,T];H^{-1}(\Omega )\right).$
Here $H_{0}^{1}(\Omega )$ denotes the Sobolev Hilbert space of once-weakly differentiable functions with first weak derivative in $L^{2}(\Omega )$ that vanish at the boundary of Ω (in the sense of trace, or, equivalently, are limits of smooth functions with compact support in Ω); $H^{-1}(\Omega )$ denotes the dual space of $H_{0}^{1}(\Omega ).$
(The "partial derivative" with respect to time $t$ above is actually a total derivative, since the use of Bochner spaces removes the space-dependence.)
See also
• Bochner integral
• Bochner measurable function
• Vector measure
• Vector-valued functions – Function valued in a vector space; typically a real or complex onePages displaying short descriptions of redirect targets
• Weakly measurable function
References
• Evans, Lawrence C. (1998). Partial differential equations. Providence, RI: American Mathematical Society. ISBN 0-8218-0772-2.
Lp spaces
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• Polarization identity
• Pythagorean theorem
• Square-integrable function
$L^{\infty }$ spaces
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• Essential
• Uniform norm
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• Function space
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• Locally integrable function
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Inequalities
• Babenko–Beckner
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• Clarkson's
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• Minkowski
• Young's convolution
Results
• Marcinkiewicz interpolation theorem
• Plancherel theorem
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• Riesz representation
• Closed graph
• Uniform boundedness principle
• Kakutani fixed-point
• Krein–Milman
• Min–max
• Gelfand–Naimark
• Banach–Alaoglu
Operators
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• Transpose
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• Invariant subspace problem
• Mahler's conjecture
Applications
• Hardy space
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• Index theorem
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• Jones polynomial
• Topological quantum field theory
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Wikipedia
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Sobolev conjugate
The Sobolev conjugate of p for $1\leq p<n$, where n is space dimensionality, is
$p^{*}={\frac {pn}{n-p}}>p$
This is an important parameter in the Sobolev inequalities.
Motivation
A question arises whether u from the Sobolev space $W^{1,p}(\mathbb {R} ^{n})$ belongs to $L^{q}(\mathbb {R} ^{n})$ for some q > p. More specifically, when does $\|Du\|_{L^{p}(\mathbb {R} ^{n})}$ control $\|u\|_{L^{q}(\mathbb {R} ^{n})}$? It is easy to check that the following inequality
$\|u\|_{L^{q}(\mathbb {R} ^{n})}\leq C(p,q)\|Du\|_{L^{p}(\mathbb {R} ^{n})}\qquad \qquad (*)$
can not be true for arbitrary q. Consider $u(x)\in C_{c}^{\infty }(\mathbb {R} ^{n})$, infinitely differentiable function with compact support. Introduce $u_{\lambda }(x):=u(\lambda x)$. We have that:
${\begin{aligned}\|u_{\lambda }\|_{L^{q}(\mathbb {R} ^{n})}^{q}&=\int _{\mathbb {R} ^{n}}|u(\lambda x)|^{q}dx={\frac {1}{\lambda ^{n}}}\int _{\mathbb {R} ^{n}}|u(y)|^{q}dy=\lambda ^{-n}\|u\|_{L^{q}(\mathbb {R} ^{n})}^{q}\\\|Du_{\lambda }\|_{L^{p}(\mathbb {R} ^{n})}^{p}&=\int _{\mathbb {R} ^{n}}|\lambda Du(\lambda x)|^{p}dx={\frac {\lambda ^{p}}{\lambda ^{n}}}\int _{\mathbb {R} ^{n}}|Du(y)|^{p}dy=\lambda ^{p-n}\|Du\|_{L^{p}(\mathbb {R} ^{n})}^{p}\end{aligned}}$
The inequality (*) for $u_{\lambda }$ results in the following inequality for $u$
$\|u\|_{L^{q}(\mathbb {R} ^{n})}\leq \lambda ^{1-{\frac {n}{p}}+{\frac {n}{q}}}C(p,q)\|Du\|_{L^{p}(\mathbb {R} ^{n})}$
If $1-{\frac {n}{p}}+{\frac {n}{q}}\neq 0,$ then by letting $\lambda $ going to zero or infinity we obtain a contradiction. Thus the inequality (*) could only be true for
$q={\frac {pn}{n-p}}$,
which is the Sobolev conjugate.
See also
• Sergei Lvovich Sobolev
• conjugate index
References
• Lawrence C. Evans. Partial differential equations. Graduate Studies in Mathematics, Vol 19. American Mathematical Society. 1998. ISBN 0-8218-0772-2
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Wikipedia
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Sobolev mapping
In mathematics, a Sobolev mapping is a mapping between manifolds which has smoothness in some sense. Sobolev mappings appear naturally in manifold-constrained problems in the calculus of variations and partial differential equations, including the theory of harmonic maps.
Definition
Given Riemannian manifolds $M$ and $N$, which is assumed by Nash's smooth embedding theorem without loss of generality to be isometrically embedded into $\mathbb {R} ^{\nu }$ as [1][2]
$W^{s,p}(M,N):=\{u\in W^{s,p}(M,\mathbb {R} ^{\nu })\,\vert \,u(x)\in N{\text{ for almost every }}x\in M\}.$
First-order ($s=1$) Sobolev mappings can also be defined in the context of metric spaces.[3][4]
Approximation
The strong approximation problem consists in determining whether smooth mappings from $M$ to $N$ are dense in $W^{s,p}(M,N)$ with respect to the norm topology. When $sp>\dim M$, Morrey's inequality implies that Sobolev mappings are continuous and can thus be strongly approximated by smooth maps. When $sp=\dim M$, Sobolev mappings have vanishing mean oscillation[5] and can thus be approximated by smooth maps.[6]
When $sp>\dim M$, the question of density is related to obstruction theory: $C^{\infty }(M,N)$ is dense in $W^{1,p}(M,N)$ if and only if every continuous mapping on a from a $\lfloor p\rfloor $–dimensional triangulation of $M$ into $N$ is the restriction of a continuous map from $M$ to $N$.[7][2]
The problem of finding a sequence of weak approximation of maps in $W^{1,p}(M,N)$ is equivalent to the strong approximation when $p$ is not an integer.[7] When $p$ is an integer, a necessary condition is that the restriction to a $\lfloor p-1\rfloor $-dimensional triangulation of every continuous mapping from a $\lfloor p\rfloor $–dimensional triangulation of $M$ into $N$ coincides with the restriction a continuous map from $M$ to $N$.[2] When $p=2$, this condition is sufficient[8] For $W^{1,3}(M,\mathbb {S} ^{2})$ with $\dim M\geq 4$, this condition is not sufficient.[9]
Homotopy
The homotopy problem consists in describing and classifying the path-connected components of the space $W^{s,p}(M,N)$ endowed with the norm topology. When $0<s\leq 1$ and $\dim M\leq sp$, then the path-connected components of $W^{s,p}(M,N)$ are essentially the same as the path-connected components of $C(M,N)$: two maps in $W^{s,p}(M,N)\cap C(M,N)$ are connected by a path in $W^{s,p}(M,N)$ if and only if they are connected by a path in $C(M,N)$, any path-connected component of $W^{s,p}(M,N)$ and any path-connected component of $C(M,N)$ intersects $W^{s,p}(M,N)\cap C(M,N)$ non trivially.[10][11][12] When $\dim M>p$, two maps in $W^{1,p}(M,N)$ are connected by a continuous path in $W^{1,p}(M,N)$ if and only if their restrictions to a generic $\lfloor p-1\rfloor $-dimensional triangulation are homotopic.[2]: th. 1.1
Extension of traces
The classical trace theory states that any Sobolev map $u\in W^{1,p}(M,N)$ has a trace $Tu\in W^{1-1/p,p}(\partial M,N)$ and that when $N=\mathbb {R} $, the trace operator is onto. The proof of the surjectivity being based on an averaging argument, the result does not readily extend to Sobolev mappings. The trace operator is known to be onto when $\pi _{1}(N)\simeq \dotsb \pi _{\lfloor p-1\rfloor }(N)\simeq \{0\}$[13] or when $p\geq 3$, $\pi _{1}(N)$ is finite and $\pi _{2}(N)\simeq \dotsb \pi _{\lfloor p-1\rfloor }(N)\simeq \{0\}$.[14] The surjectivity of the trace operator fails if $\pi _{\lfloor p-1\rfloor }(N)\not \simeq \{0\}$ [13][15] or if $\pi _{\ell }(N)$ is infinite for some $\ell \in \{1,\dotsc ,\lfloor p-1\rfloor \}$.[14][16]
Lifting
Given a covering map $\pi :{\tilde {N}}\to N$ :{\tilde {N}}\to N} , the lifting problem asks whether any map $u\in W^{s,p}(M,N)$ can be written as $u=\pi \circ {\tilde {u}}$ for some ${\tilde {u}}\in W^{s,p}(M,{\tilde {N}})$, as it is the case for continuous or smooth $u$ and ${\tilde {u}}$ when $M$ is simply-connected in the classical lifting theory. If the domain $M$ is simply connected, any map $u\in W^{s,p}(M,N)$ can be written as $u=\pi \circ {\tilde {u}}$ for some ${\tilde {u}}\in W^{s,p}(M,N)$ when $sp\geq \dim M$,[17][18] when $s\geq 1$ and $2\leq sp<\dim M$[19][18] and when $N$ is compact, $0<s<1$ and $2\leq sp<\dim M$.[20] There is a topological obstruction to the lifting when $sp<2$ and an analytical obstruction when $1\leq sp<\dim M$.[17][18]
References
1. Mironescu, Petru (2007). "Sobolev maps on manifolds: degree, approximation, lifting" (PDF). Contemporary Mathematics. 446: 413–436. doi:10.1090/conm/446/08642. ISBN 9780821841907.
2. Hang, Fengbo; Lin, Fanghua (2003). "Topology of sobolev mappings, II". Acta Mathematica. 191 (1): 55–107. doi:10.1007/BF02392696. S2CID 121520479.
3. Chiron, David (August 2007). "On the definitions of Sobolev and BV spaces into singular spaces and the trace problem". Communications in Contemporary Mathematics. 09 (4): 473–513. doi:10.1142/S0219199707002502.
4. Hajłasz, Piotr (2009). "Sobolev Mappings between Manifolds and Metric Spaces". Sobolev Spaces in Mathematics I. International Mathematical Series. 8: 185–222. doi:10.1007/978-0-387-85648-3_7. ISBN 978-0-387-85647-6.
5. Brezis, H.; Nirenberg, L. (September 1995). "Degree theory and BMO; part I: Compact manifolds without boundaries". Selecta Mathematica. 1 (2): 197–263. doi:10.1007/BF01671566. S2CID 195270732.
6. Schoen, Richard; Uhlenbeck, Karen (1 January 1982). "A regularity theory for harmonic maps". Journal of Differential Geometry. 17 (2). doi:10.4310/jdg/1214436923.
7. Bethuel, Fabrice (1991). "The approximation problem for Sobolev maps between two manifolds". Acta Mathematica. 167: 153–206. doi:10.1007/BF02392449. S2CID 122996551.
8. Pakzad, M.R.; Rivière, T. (February 2003). "Weak density of smooth maps for the Dirichlet energy between manifolds". Geometric and Functional Analysis. 13 (1): 223–257. doi:10.1007/s000390300006. S2CID 121794503.
9. Bethuel, Fabrice (February 2020). "A counterexample to the weak density of smooth maps between manifolds in Sobolev spaces". Inventiones Mathematicae. 219 (2): 507–651. arXiv:1401.1649. Bibcode:2020InMat.219..507B. doi:10.1007/s00222-019-00911-3. S2CID 119627475.
10. Brezis, Haı̈m; Li, YanYan (September 2000). "Topology and Sobolev spaces". Comptes Rendus de l'Académie des Sciences - Series I - Mathematics. 331 (5): 365–370. Bibcode:2000CRASM.331..365B. doi:10.1016/S0764-4442(00)01656-6.
11. Brezis, Haim; Li, Yanyan (July 2001). "Topology and Sobolev Spaces". Journal of Functional Analysis. 183 (2): 321–369. doi:10.1006/jfan.2000.3736.
12. Bousquet, Pierre (February 2008). "Fractional Sobolev spaces and topology". Nonlinear Analysis: Theory, Methods & Applications. 68 (4): 804–827. doi:10.1016/j.na.2006.11.038.
13. Hardt, Robert; Lin, Fang-Hua (September 1987). "Mappings minimizing the Lp norm of the gradient". Communications on Pure and Applied Mathematics. 40 (5): 555–588. doi:10.1002/cpa.3160400503.
14. Mironescu, Petru; Van Schaftingen, Jean (9 July 2021). "Trace theory for Sobolev mappings into a manifold". Annales de la Faculté des sciences de Toulouse: Mathématiques. 30 (2): 281–299. arXiv:2001.02226. doi:10.5802/afst.1675. S2CID 210023485.
15. Bethuel, Fabrice; Demengel, Françoise (October 1995). "Extensions for Sobolev mappings between manifolds". Calculus of Variations and Partial Differential Equations. 3 (4): 475–491. doi:10.1007/BF01187897. S2CID 121749565.
16. Bethuel, Fabrice (March 2014). "A new obstruction to the extension problem for Sobolev maps between manifolds". Journal of Fixed Point Theory and Applications. 15 (1): 155–183. arXiv:1402.4614. doi:10.1007/s11784-014-0185-0. S2CID 119614310.
17. Bourgain, Jean; Brezis, Haim; Mironescu, Petru (December 2000). "Lifting in Sobolev spaces". Journal d'Analyse Mathématique. 80 (1): 37–86. doi:10.1007/BF02791533.
18. Bethuel, Fabrice; Chiron, David (2007). "Some questions related to the lifting problem in Sobolev spaces". Contemporary Mathematics. 446: 125–152. doi:10.1090/conm/446/08628. ISBN 9780821841907.
19. Bethuel, Fabrice; Zheng, Xiaomin (September 1988). "Density of smooth functions between two manifolds in Sobolev spaces". Journal of Functional Analysis. 80 (1): 60–75. doi:10.1016/0022-1236(88)90065-1.
20. Mironescu, Petru; Van Schaftingen, Jean (7 September 2021). "Lifting in compact covering spaces for fractional Sobolev mappings". Analysis & PDE. 14 (6): 1851–1871. arXiv:1907.01373. doi:10.2140/apde.2021.14.1851. S2CID 195776361.
Further reading
• https://mathoverflow.net/questions/108808/differential-of-a-sobolev-map-between-manifolds
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Wikipedia
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Sobolev spaces for planar domains
In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the theory of partial differential equations for solving the Dirichlet and Neumann boundary value problems for the Laplacian in a bounded domain in the plane with smooth boundary. The methods use the theory of bounded operators on Hilbert space. They can be used to deduce regularity properties of solutions and to solve the corresponding eigenvalue problems.
Sobolev spaces with boundary conditions
Let Ω ⊂ R2 be a bounded domain with smooth boundary. Since Ω is contained in a large square in R2, it can be regarded as a domain in T2 by identifying opposite sides of the square. The theory of Sobolev spaces on T2 can be found in Bers, John & Schechter (1979), an account which is followed in several later textbooks such as Warner (1983) and Griffiths & Harris (1994).
For k an integer, the (restricted) Sobolev space Hk
0
(Ω)
is defined as the closure of C∞
c
(Ω)
in the standard Sobolev space Hk(T2).
• H0
0
(Ω) = L2(Ω)
.
• Vanishing properties on boundary: For k > 0 the elements of Hk
0
(Ω)
are referred to as "L2 functions on Ω which vanish with their first k − 1 derivatives on ∂Ω."[1] In fact if f ∈ Ck(Ω) agrees with a function in Hk
0
(Ω)
, then g = ∂ αf is in C1. Let fn ∈ C∞
c
(Ω)
be such that fn → f in the Sobolev norm, and set gn = ∂ αfn . Thus gn → g in H1
0
(Ω)
. Hence for h ∈ C∞(T2) and D = a∂x + b∂y,
$\iint _{\Omega }\left(g(Dh)+(Dg)h\right)\,dx\,dy=\lim _{n\to 0}\iint _{\Omega }\left(g(Dh_{n})+(Dg)h_{n}\right)\,dx\,dy=0.$
By Green's theorem this implies
$\int _{\partial \Omega }gk=0,$
where
$k=h\cos \left(\mathbf {n} \cdot (a,b)\right),$
with n the unit normal to the boundary. Since such k form a dense subspace of L2(Ω), it follows that g = 0 on ∂Ω.
• Support properties: Let Ωc be the complement of Ω and define restricted Sobolev spaces analogously for Ωc. Both sets of spaces have a natural pairing with C∞(T2). The Sobolev space for Ω is the annihilator in the Sobolev space for T2 of C∞
c
(Ωc)
and that for Ωc is the annihilator of C∞
c
(Ω)
.[2] In fact this is proved by locally applying a small translation to move the domain inside itself and then smoothing by a smooth convolution operator.
Suppose g in Hk(T2) annihilates C∞
c
(Ωc)
. By compactness, there are finitely many open sets U0, U1, ... , UN covering Ω such that the closure of U0 is disjoint from ∂Ω and each Ui is an open disc about a boundary point zi such that in Ui small translations in the direction of the normal vector ni carry Ω into Ω. Add an open UN+1 with closure in Ωc to produce a cover of T2 and let ψi be a partition of unity subordinate to this cover. If translation by n is denoted by λn, then the functions
$g_{t}=\psi _{0}g+\sum _{i=1}^{N}\psi _{n}\lambda _{tn_{i}}g$
tend to g as t decreases to 0 and still lie in the annihilator, indeed they are in the annihilator for a larger domain than Ωc, the complement of which lies in Ω. Convolving by smooth functions of small support produces smooth approximations in the annihilator of a slightly smaller domain still with complement in Ω. These are necessarily smooth functions of compact support in Ω.
• Further vanishing properties on the boundary: The characterization in terms of annihilators shows that f ∈ Ck(Ω) lies in H k
0
(Ω)
if (and only if) it and its derivatives of order less than k vanish on ∂Ω.[3] In fact f can be extended to T2 by setting it to be 0 on Ωc. This extension F defines an element in Hk(T2) using the formula for the norm
$\|h\|_{(k)}^{2}=\sum _{j=0}^{k}{k \choose j}\left\|\partial _{x}^{j}\partial _{y}^{k-j}h\right\|^{2}.$
Moreover F satisfies (F, g) = 0 for g in C∞
c
(Ωc)
.
• Duality: For k ≥ 0, define H−k(Ω) to be the orthogonal complement of H−k
0
(Ωc)
in H−k(T2). Let Pk be the orthogonal projection onto H−k(Ω), so that Qk = I − Pk is the orthogonal projection onto H−k
0
(Ωc)
. When k = 0, this just gives H0(Ω) = L2(Ω). If f ∈ Hk
0
(Ωc)
and g ∈ H−k(T2), then
$(f,g)=(f,P_{k}g).$
This implies that under the pairing between Hk(T2) and H−k(T2), Hk
0
(Ωc)
and H−k(Ω) are each other's duals.
• Approximation by smooth functions: The image of C∞
c
(Ω)
is dense in H−k(Ω) for k ≤ 0. This is obvious for k = 0 since the sum C∞
c
(Ω)
+ C∞
c
(Ωc)
is dense in L2(T2). Density for k < 0 follows because the image of L2(T2) is dense in H−k(T2) and Pk annihilates C∞
c
(Ωc)
.
• Canonical isometries: The operator (I + ∆)k gives an isometry of H 2k
0
(Ω)
into H0(Ω) and of H k
0
(Ω)
onto H−k(Ω). In fact the first statement follows because it is true on T2. That (I + ∆)k is an isometry on H k
0
(Ω)
follows using the density of C∞
c
(Ω)
in H−k(Ω): for f, g ∈ C∞
c
(Ω)
we have:
${\begin{aligned}\left\|P_{k}(I+\Delta )^{k}f\right\|_{(-k)}&=\sup _{\|g\|_{(-k)}=1}\left|\left((I+\Delta )^{k}f,g\right)_{(-k)}\right|\\&=\sup _{\|g\|_{(-k)}=1}|(f,g)|\\&=\|f\|_{(k)}.\end{aligned}}$
Since the adjoint map between the duals can by identified with this map, it follows that (I + ∆)k is a unitary map.
Application to Dirichlet problem
See also: Dirichlet problem
Invertibility of ∆
The operator ∆ defines an isomorphism between H1
0
(Ω)
and H−1(Ω). In fact it is a Fredholm operator of index 0. The kernel of ∆ in H1(T2) consists of constant functions and none of these except zero vanish on the boundary of Ω. Hence the kernel of H1
0
(Ω)
is (0) and ∆ is invertible.
In particular the equation ∆f = g has a unique solution in H1
0
(Ω)
for g in H−1(Ω).
Eigenvalue problem
Let T be the operator on L2(Ω) defined by
$T=R_{1}\Delta ^{-1}R_{0},$
where R0 is the inclusion of L2(Ω) in H−1(Ω) and R1 of H1
0
(Ω)
in L2(Ω), both compact operators by Rellich's theorem. The operator T is compact and self-adjoint with (Tf, f ) > 0 for all f. By the spectral theorem, there is a complete orthonormal set of eigenfunctions fn in L2(Ω) with
$Tf_{n}=\mu _{n}f_{n},\qquad \mu _{n}>0,\mu _{n}\to 0.$
Since μn > 0, fn lies in H1
0
(Ω)
. Setting λn = μ−n, the fn are eigenfunctions of the Laplacian:
$\Delta f_{n}=\lambda _{n}f_{n},\qquad \lambda _{n}>0,\lambda _{n}\to \infty .$
Sobolev spaces without boundary condition
To determine the regularity properties of the eigenfunctions fn and solutions of
$\Delta f=u,\qquad u\in H^{-1}(\Omega ),f\in H_{0}^{1}(\Omega ),$
enlargements of the Sobolev spaces Hk
0
(Ω)
have to be considered. Let C∞(Ω−) be the space of smooth functions on Ω which with their derivatives extend continuously to Ω. By Borel's lemma, these are precisely the restrictions of smooth functions on T2. The Sobolev space Hk(Ω) is defined to the Hilbert space completion of this space for the norm
$\|f\|_{(k)}^{2}=\sum _{j=0}^{k}{k \choose j}\left\|\partial _{x}^{j}\partial _{y}^{k-j}f\right\|^{2}.$
This norm agrees with the Sobolev norm on C∞
c
(Ω)
so that Hk
0
(Ω)
can be regarded as a closed subspace of Hk(Ω). Unlike Hk
0
(Ω)
, Hk(Ω) is not naturally a subspace of Hk(T2), but the map restricting smooth functions from T2 to Ω is continuous for the Sobolev norm so extends by continuity to a map ρk : Hk(T2) → Hk(Ω).
• Invariance under diffeomorphism: Any diffeomorphism between the closures of two smooth domains induces an isomorphism between the Sobolev space. This is a simple consequence of the chain rule for derivatives.
• Extension theorem: The restriction of ρk to the orthogonal complement of its kernel defines an isomorphism onto Hk(Ω). The extension map Ek is defined to be the inverse of this map: it is an isomorphism (not necessarily norm preserving) of Hk(Ω) onto the orthogonal complement of Hk
0
(Ωc)
such that ρk ∘ Ek = I. On C∞
c
(Ω)
, it agrees with the natural inclusion map. Bounded extension maps Ek of this kind from Hk(Ω) to Hk(T2) were constructed first constructed by Hestenes and Lions. For smooth curves the Seeley extension theorem provides an extension which is continuous in all the Sobolev norms. A version of the extension which applies in the case where the boundary is just a Lipschitz curve was constructed by Calderón using singular integral operators and generalized by Stein (1970).
It is sufficient to construct an extension E for a neighbourhood of a closed annulus, since a collar around the boundary is diffeomorphic to an annulus I × T with I a closed interval in T. Taking a smooth bump function ψ with 0 ≤ ψ ≤ 1, equal to 1 near the boundary and 0 outside the collar, E(ψf ) + (1 − ψ) f will provide an extension on Ω. On the annulus, the problem reduces to finding an extension for Ck( I ) in Ck(T). Using a partition of unity the task of extending reduces to a neighbourhood of the end points of I. Assuming 0 is the left end point, an extension is given locally by
$E(f)=\sum _{m=0}^{k}a_{m}f\left(-{\frac {x}{m+1}}\right).$
Matching the first derivatives of order k or less at 0, gives
$\sum _{m=0}^{k}(-m-1)^{-k}a_{m}=1.$
This matrix equation is solvable because the determinant is non-zero by Vandermonde's formula. It is straightforward to check that the formula for E( f ), when appropriately modified with bump functions, leads to an extension which is continuous in the above Sobolev norm.[4]
• Restriction theorem: The restriction map ρk is surjective with ker ρk = Hk
0
(Ωc)
. This is an immediate consequence of the extension theorem and the support properties for Sobolev spaces with boundary condition.
• Duality: Hk(Ω) is naturally the dual of H−k0(Ω). Again this is an immediate consequence of the restriction theorem. Thus the Sobolev spaces form a chain:
$\cdots \subset H^{2}(\Omega )\subset H^{1}(\Omega )\subset H^{0}(\Omega )\subset H_{0}^{-1}(\Omega )\subset H_{0}^{-2}(\Omega )\subset \cdots $
The differentiation operators ∂x, ∂y carry each Sobolev space into the larger one with index 1 less.
• Sobolev embedding theorem: Hk+2(Ω) is contained in Ck(Ω−). This is an immediate consequence of the extension theorem and the Sobolev embedding theorem for Hk+2(T2).
• Characterization: Hk(Ω) consists of f in L2(Ω) = H0(Ω) such that all the derivatives ∂αf lie in L2(Ω) for |α| ≤ k. Here the derivatives are taken within the chain of Sobolev spaces above.[5] Since C∞
c
(Ω)
is weakly dense in Hk(Ω), this condition is equivalent to the existence of L2 functions fα such that
$\left(f_{\alpha },\varphi \right)=(-1)^{|\alpha |}\left(f,\partial ^{\alpha }\varphi \right),\qquad |\alpha |\leq k,\varphi \in C_{c}^{\infty }(\Omega ).$
To prove the characterization, note that if f is in Hk(Ω), then ∂αf lies in Hk−|α|(Ω) and hence in H0(Ω) = L2(Ω). Conversely the result is well known for the Sobolev spaces Hk(T2): the assumption implies that the (∂x − i∂y)k f is in L2(T2) and the corresponding condition on the Fourier coefficients of f shows that f lies in Hk(T2). Similarly the result can be proved directly for an annulus [−δ, δ] × T. In fact by the argument on T2 the restriction of f to any smaller annulus [−δ',δ'] × T lies in Hk: equivalently the restriction of the function fR (x, y) = f (Rx, y) lies in Hk for R > 1. On the other hand ∂α fR → ∂α f in L2 as R → 1, so that f must lie in Hk. The case for a general domain Ω reduces to these two cases since f can be written as f = ψf + (1 − ψ) f with ψ a bump function supported in Ω such that 1 − ψ is supported in a collar of the boundary.
• Regularity theorem: If f in L2(Ω) has both derivatives ∂x f and ∂y f in Hk(Ω) then f lies in Hk+1(Ω). This is an immediate consequence of the characterization of Hk(Ω) above. In fact if this is true even when satisfied at the level of distributions: if there are functions g, h in Hk(Ω) such that (g,φ) = (f, φx) and (h,φ) = (f,φy) for φ in C∞
c
(Ω)
, then f is in Hk+1(Ω).
• Rotations on an annulus: For an annulus I × T, the extension map to T2 is by construction equivariant with respect to rotations in the second variable,
$R_{t}f(x,y)=f(x,y+t).$
On T2 it is known that if f is in Hk, then the difference quotient δh f = h−1(Rh f − f ) → ∂y f in Hk−1; if the difference quotients are bounded in Hk then ∂yf lies in Hk. Both assertions are consequences of the formula:
${\widehat {\delta _{h}f}}(m,n)=h^{-1}(e^{-ihn}-1){\widehat {f}}(m,n)=-\int _{0}^{1}ine^{-inht}\,dt\,\,{\widehat {f}}(m,n).$
These results on T2 imply analogous results on the annulus using the extension.
Regularity for Dirichlet problem
Regularity for dual Dirichlet problem
If ∆u = f with u in H1
0
(Ω)
and f in Hk−1(Ω) with k ≥ 0, then u lies in Hk+1(Ω).
Take a decomposition u = ψu + (1 − ψ)u with ψ supported in Ω and 1 − ψ supported in a collar of the boundary. Standard Sobolev theory for T2 can be applied to ψu: elliptic regularity implies that it lies in Hk+1(T2) and hence Hk+1(Ω). v = (1 − ψ)u lies in H1
0
of a collar, diffeomorphic to an annulus, so it suffices to prove the result with Ω a collar and ∆ replaced by
${\begin{aligned}\Delta _{1}&=\Delta -[\Delta ,\psi ]\\&=\Delta +\left(p\partial _{x}+q\partial _{y}-\Delta \psi \right)\\&=\Delta +X.\end{aligned}}$
The proof[6] proceeds by induction on k, proving simultaneously the inequality
$\|u\|_{(k+1)}\leq C\|\Delta _{1}u\|_{(k-1)}+C\|u\|_{(k)},$
for some constant C depending only on k. It is straightforward to establish this inequality for k = 0, where by density u can be taken to be smooth of compact support in Ω:
${\begin{aligned}\|u\|_{(1)}^{2}&=|(\Delta u,u)|\\&\leq |(\Delta _{1}u,u)|+|(Xu,u)|\\&\leq \|\Delta _{1}u\|_{(-1)}\|u\|_{(1)}+C^{\prime }\|u\|_{(1)}\|u\|_{(0)}.\end{aligned}}$
The collar is diffeomorphic to an annulus. The rotational flow Rt on the annulus induces a flow St on the collar with corresponding vector field Y = r∂x + s∂y. Thus Y corresponds to the vector field ∂θ. The radial vector field on the annulus r∂r is a commuting vector field which on the collar gives a vector field Z = p∂x + q∂y proportional to the normal vector field. The vector fields Y and Z commute.
The difference quotients δhu can be formed for the flow St. The commutators [δh, ∆1] are second order differential operators from Hk+1(Ω) to Hk−1(Ω). Their operators norms are uniformly bounded for h near 0; for the computation can be carried out on the annulus where the commutator just replaces the coefficients of ∆1 by their difference quotients composed with Sh. On the other hand, v = δhu lies in H1
0
(Ω)
, so the inequalities for u apply equally well for v:
${\begin{aligned}\|\delta _{h}u\|_{(k+1)}&\leq C\|\Delta _{1}\delta _{h}u\|_{(k-1)}+C\|\delta _{h}u\|_{(k)}\\&\leq C\|\delta _{h}\Delta _{1}u\|_{(k-1)}+C\|[\delta _{h},\Delta _{1}]u\|_{(k-1)}+C\|\delta _{h}u\|_{(k)}\\&\leq C\|\Delta _{1}u\|_{(k)}+C^{\prime }\|u\|_{(k+1)}.\end{aligned}}$
The uniform boundedness of the difference quotients δhu implies that Yu lies in Hk+1(Ω) with
$\|Yu\|_{(k+1)}\leq C\|\Delta _{1}u\|_{(k)}+C^{\prime }\|u\|_{(k+1)}.$
It follows that Vu lies in Hk+1(Ω) where V is the vector field
$V={\frac {Y}{\sqrt {r^{2}+s^{2}}}}=a\partial _{x}+b\partial _{y},\qquad a^{2}+b^{2}=1.$
Moreover, Vu satisfies a similar inequality to Yu.
$\|Vu\|_{(k+1)}\leq C^{\prime \prime }\left(\|\Delta _{1}u\|_{(k)}+\|u\|_{(k+1)}\right).$
Let W be the orthogonal vector field
$W=-b\partial _{x}+a\partial _{y}.$
It can also be written as ξZ for some smooth nowhere vanishing function ξ on a neighbourhood of the collar.
It suffices to show that Wu lies in Hk+1(Ω). For then
$(V\pm iW)u=(a\mp ib)\left(\partial _{x}\pm i\partial _{y}\right)u,$
so that ∂xu and ∂yu lie in Hk+1(Ω) and u must lie in Hk+2(Ω).
To check the result on Wu, it is enough to show that VWu and W2u lie in Hk(Ω). Note that
${\begin{aligned}A&=\Delta -V^{2}-W^{2},\\B&=[V,W],\end{aligned}}$
are vector fields. But then
${\begin{aligned}W^{2}u&=\Delta u-V^{2}u-Au,\\VWu&=WVu+Bu,\end{aligned}}$
with all terms on the right hand side in Hk(Ω). Moreover, the inequalities for Vu show that
${\begin{aligned}\|Wu\|_{(k+1)}&\leq C\left(\|VWu\|_{(k)}+\left\|W^{2}u\right\|_{(k)}\right)\\&\leq C\left\|\left(\Delta -V^{2}-A\right)u\right\|_{(k)}+C\|(WV+B)u\|_{(k)}\\&\leq C_{1}\|\Delta _{1}u\|_{(k)}+C_{1}\|u\|_{(k+1)}.\end{aligned}}$
Hence
${\begin{aligned}\|u\|_{(k+2)}&\leq C\left(\|Vu\|_{(k+1)}+\|Wu\|_{(k+1)}\right)\\&\leq C^{\prime }\|\Delta _{1}u\|_{(k)}+C^{\prime }\|u\|_{(k+1)}.\end{aligned}}$
Smoothness of eigenfunctions
It follows by induction from the regularity theorem for the dual Dirichlet problem that the eigenfunctions of ∆ in H1
0
(Ω)
lie in C∞(Ω−). Moreover, any solution of ∆u = f with f in C∞(Ω−) and u in H1
0
(Ω)
must have u in C∞(Ω−). In both cases by the vanishing properties, the eigenfunctions and u vanish on the boundary of Ω.
Solving the Dirichlet problem
The dual Dirichlet problem can be used to solve the Dirichlet problem:
${\begin{cases}\Delta f|_{\Omega }=0\\f|_{\partial \Omega }=g&g\in C^{\infty }(\partial \Omega )\end{cases}}$
By Borel's lemma g is the restriction of a function G in C∞(Ω−). Let F be the smooth solution of ∆F = ∆G with F = 0 on ∂Ω. Then f = G − F solves the Dirichlet problem. By the maximal principle, the solution is unique.[7]
Application to smooth Riemann mapping theorem
The solution to the Dirichlet problem can be used to prove a strong form of the Riemann mapping theorem for simply connected domains with smooth boundary. The method also applies to a region diffeomorphic to an annulus.[8] For multiply connected regions with smooth boundary Schiffer & Hawley (1962) have given a method for mapping the region onto a disc with circular holes. Their method involves solving the Dirichlet problem with a non-linear boundary condition. They construct a function g such that:
• g is harmonic in the interior of Ω;
• On ∂Ω we have: ∂ng = κ − KeG, where κ is the curvature of the boundary curve, ∂n is the derivative in the direction normal to ∂Ω and K is constant on each boundary component.
Taylor (2011) gives a proof of the Riemann mapping theorem for a simply connected domain Ω with smooth boundary. Translating if necessary, it can be assumed that 0 ∈ Ω. The solution of the Dirichlet problem shows that there is a unique smooth function U(z) on Ω which is harmonic in Ω and equals −log|z| on ∂Ω. Define the Green's function by G(z) = log|z| + U(z). It vanishes on ∂Ω and is harmonic on Ω away from 0. The harmonic conjugate V of U is the unique real function on Ω such that U + iV is holomorphic. As such it must satisfy the Cauchy–Riemann equations:
${\begin{aligned}U_{x}&=-V_{y},\\U_{y}&=V_{x}.\end{aligned}}$
The solution is given by
$V(z)=\int _{0}^{z}-U_{y}dx+V_{x}dy,$
where the integral is taken over any path in Ω. It is easily verified that Vx and Vy exist and are given by the corresponding derivatives of U. Thus V is a smooth function on Ω, vanishing at 0. By the Cauchy-Riemann f = U + iV is smooth on Ω, holomorphic on Ω and f (0) = 0. The function H = arg z + V(z) is only defined up to multiples of 2π, but the function
$F(z)=e^{G(z)+iH(z)}=ze^{f(z)}$
is a holomorphic on Ω and smooth on Ω. By construction, F(0) = 0 and |F(z)| = 1 for z ∈ ∂Ω. Since z has winding number 1, so too does F(z). On the other hand, F(z) = 0 only for z = 0 where there is a simple zero. So by the argument principle F assumes every value in the unit disc, D, exactly once and F′ does not vanish inside Ω. To check that the derivative on the boundary curve is non-zero amounts to computing the derivative of eiH, i.e. the derivative of H should not vanish on the boundary curve. By the Cauchy-Riemann equations these tangential derivative are up to a sign the directional derivative in the direction of the normal to the boundary. But G vanishes on the boundary and is strictly negative in Ω since |F| = eG. The Hopf lemma implies that the directional derivative of G in the direction of the outward normal is strictly positive. So on the boundary curve, F has nowhere vanishing derivative. Since the boundary curve has winding number one, F defines a diffeomorphism of the boundary curve onto the unit circle. Accordingly, F : Ω → D is a smooth diffeomorphism, which restricts to a holomorphic map Ω → D and a smooth diffeomorphism between the boundaries.
Similar arguments can be applied to prove the Riemann mapping theorem for a doubly connected domain Ω bounded by simple smooth curves Ci (the inner curve) and Co (the outer curve). By translating we can assume 1 lies on the outer boundary. Let u be the smooth solution of the Dirichlet problem with U = 0 on the outer curve and −1 on the inner curve. By the maximum principle 0 < u(z) < 1 for z in Ω and so by the Hopf lemma the normal derivatives of u are negative on the outer curve and positive on the inner curve. The integral of −uydx + uydx over the boundary is zero by Stoke's theorem so the contributions from the boundary curves cancel. On the other hand, on each boundary curve the contribution is the integral of the normal derivative along the boundary. So there is a constant c > 0 such that U = cu satisfies
$\int _{C}\left(-U_{y}\,dx+U_{x}\,dy\right)=2\pi $
on each boundary curve. The harmonic conjugate V of U can again be defined by
$V(z)=\int _{1}^{z}-u_{y}\,dx+u_{x}\,dy$
and is well-defined up to multiples of 2π. The function
$\displaystyle {F(z)=e^{U(z)+iV(z)}}$
is smooth on Ω and holomorphic in Ω. On the outer curve |F| = 1 and on the inner curve |F| = e−c = r < 1. The tangential derivatives on the outer curves are nowhere vanishing by the Cauchy-Riemann equations, since the normal derivatives are nowhere vanishing. The normalization of the integrals implies that F restricts to a diffeomorphism between the boundary curves and the two concentric circles. Since the images of outer and inner curve have winding number 1 and 0 about any point in the annulus, an application of the argument principle implies that F assumes every value within the annulus r < |z| < 1 exactly once; since that includes multiplicities, the complex derivative of F is nowhere vanishing in Ω. This F is a smooth diffeomorphism of Ω onto the closed annulus r ≤ |z| ≤ 1, restricting to a holomorphic map in the interior and a smooth diffeomorphism on both boundary curves.
Trace map
The restriction map τ : C∞(T2) → C∞(T) = C∞(1 × T) extends to a continuous map Hk(T2) → Hk − ½(T) for k ≥ 1.[9] In fact
$\displaystyle {{\widehat {\tau f}}(n)=\sum _{m}{\widehat {f}}(m,n),}$
so the Cauchy–Schwarz inequality yields
${\begin{aligned}\left|{\widehat {\tau f}}(n)\right|^{2}\left(1+n^{2}\right)^{k-{\frac {1}{2}}}&\leq \left(\sum _{m}{\frac {\left(1+n^{2}\right)^{k-{\frac {1}{2}}}}{\left(1+m^{2}+n^{2}\right)^{k}}}\right)\left(\sum _{m}\left|{\widehat {f}}(m,n)\right|^{2}\left(1+m^{2}+n^{2}\right)^{k}\right)\\&\leq C_{k}\sum _{m}\left|{\widehat {f}}(m,n)\right|^{2}\left(1+m^{2}+n^{2}\right)^{k},\end{aligned}}$
where, by the integral test,
${\begin{aligned}C_{k}&=\sup _{n}\sum _{m}{\frac {\left(1+n^{2}\right)^{k-{\frac {1}{2}}}}{\left(1+m^{2}+n^{2}\right)^{k}}}<\infty ,\\c_{k}&=\inf _{n}\sum _{m}{\frac {\left(1+n^{2}\right)^{k-{\frac {1}{2}}}}{\left(1+m^{2}+n^{2}\right)^{k}}}>0.\end{aligned}}$
The map τ is onto since a continuous extension map E can be constructed from Hk − ½(T) to Hk(T2).[10][11] In fact set
${\widehat {Eg}}(m,n)=\lambda _{n}^{-1}{\widehat {g}}(n){\frac {\left(1+n^{2}\right)^{k-{\frac {1}{2}}}}{\left(1+n^{2}+m^{2}\right)^{k}}},$
where
$\lambda _{n}=\sum _{m}{\frac {\left(1+n^{2}\right)^{k-{\frac {1}{2}}}}{\left(1+m^{2}+n^{2}\right)^{k}}}.$
Thus ck < λn < Ck. If g is smooth, then by construction Eg restricts to g on 1 × T. Moreover, E is a bounded linear map since
${\begin{aligned}\|Eg\|_{(k)}^{2}&=\sum _{m,n}\left|{\widehat {Eg}}(m,n)\right|^{2}\left(1+m^{2}+n^{2}\right)\\&\leq c_{k}^{-2}\sum _{m,n}\left|{\widehat {g}}(n)\right|^{2}{\frac {\left(1+n^{2}\right)^{2k-1}}{\left(1+m^{2}+n^{2}\right)^{k}}}\\&\leq c_{k}^{-2}C_{k}\|g\|_{k-{\frac {1}{2}}}^{2}.\end{aligned}}$
It follows that there is a trace map τ of Hk(Ω) onto Hk − ½(∂Ω). Indeed, take a tubular neighbourhood of the boundary and a smooth function ψ supported in the collar and equal to 1 near the boundary. Multiplication by ψ carries functions into Hk of the collar, which can be identified with Hk of an annulus for which there is a trace map. The invariance under diffeomorphisms (or coordinate change) of the half-integer Sobolev spaces on the circle follows from the fact that an equivalent norm on Hk + ½(T) is given by[12]
$\|f\|_{[k+{\frac {1}{2}}]}^{2}=\|f\|_{(k)}^{2}+\int _{0}^{2\pi }\int _{0}^{2\pi }{\frac {\left|f^{(k)}(s)-f^{(k)}(t)\right|^{2}}{\left|e^{is}-e^{it}\right|^{2}}}\,ds\,dt.$
It is also a consequence of the properties of τ and E (the "trace theorem").[13] In fact any diffeomorphism f of T induces a diffeomorphism F of T2 by acting only on the second factor. Invariance of Hk(T2) under the induced map F* therefore implies invariance of Hk − ½(T) under f*, since f* = τ ∘ F* ∘ E.
Further consequences of the trace theorem are the two exact sequences[14][15]
$(0)\to H_{0}^{1}(\Omega )\to H^{1}(\Omega )\to H^{\frac {1}{2}}(\partial \Omega )\to (0)$
and
$(0)\to H_{0}^{2}(\Omega )\to H^{2}(\Omega )\to H^{\frac {3}{2}}(\partial \Omega )\oplus H^{\frac {1}{2}}(\partial \Omega )\to (0),$
where the last map takes f in H2(Ω) to f|∂Ω and ∂nf|∂Ω. There are generalizations of these sequences to Hk(Ω) involving higher powers of the normal derivative in the trace map:
$(0)\to H_{0}^{k}(\Omega )\to H^{k}(\Omega )\to \bigoplus _{j=1}^{k}H^{j-{\frac {1}{2}}}(\partial \Omega )\to (0).$
The trace map to Hj − ½(∂Ω) takes f to ∂k − j
n
f |∂Ω
Abstract formulation of boundary value problems
The Sobolev space approach to the Neumann problem cannot be phrased quite as directly as that for the Dirichlet problem. The main reason is that for a function f in H1(Ω), the normal derivative ∂nf |∂Ω cannot be a priori defined at the level of Sobolev spaces. Instead an alternative formulation of boundary value problems for the Laplacian Δ on a bounded region Ω in the plane is used. It employs Dirichlet forms, sesqulinear bilinear forms on H1(Ω), H1
0
(Ω)
or an intermediate closed subspace. Integration over the boundary is not involved in defining the Dirichlet form. Instead, if the Dirichlet form satisfies a certain positivity condition, termed coerciveness, solution can be shown to exist in a weak sense, so-called "weak solutions". A general regularity theorem than implies that the solutions of the boundary value problem must lie in H2(Ω), so that they are strong solutions and satisfy boundary conditions involving the restriction of a function and its normal derivative to the boundary. The Dirichlet problem can equally well be phrased in these terms, but because the trace map f |∂Ω is already defined on H1(Ω), Dirichlet forms do not need to be mentioned explicitly and the operator formulation is more direct. A unified discussion is given in Folland (1995) and briefly summarised below. It is explained how the Dirichlet problem, as discussed above, fits into this framework. Then a detailed treatment of the Neumann problem from this point of view is given following Taylor (2011).
The Hilbert space formulation of boundary value problems for the Laplacian Δ on a bounded region Ω in the plane proceeds from the following data:[16]
• A closed subspace H1
0
(Ω) ⊆ H ⊆ H1(Ω)
.
• A Dirichlet form for Δ given by a bounded Hermitian bilinear form D( f, g) defined for f, g ∈ H1(Ω) such that D( f, g) = (∆f, g) for f, g ∈ H1
0
(Ω)
.
• D is coercive, i.e. there is a positive constant C and a non-negative constant λ such that D( f, f ) ≥ C ( f, f )(1) − λ( f, f ).
A weak solution of the boundary value problem given initial data f in L2(Ω) is a function u satisfying
$D(f,g)=(u,g)$
for all g.
For both the Dirichlet and Neumann problem
$D(f,g)=\left(f_{x},g_{x}\right)+\left(f_{y},g_{y}\right).$
For the Dirichlet problem H = H1
0
(Ω)
. In this case
$D(f,g)=(\Delta f,g),\qquad f,g\in H.$
By the trace theorem the solution satisfies u|Ω = 0 in H½(∂Ω).
For the Neumann problem H is taken to be H1(Ω).
Application to Neumann problem
The classical Neumann problem on Ω consists in solving the boundary value problem
${\begin{cases}\Delta u=f,&f,u\in C^{\infty }(\Omega ^{-}),\\\partial _{n}u=0&{\text{on }}\partial \Omega \end{cases}}$
Green's theorem implies that for u, v ∈ C∞(Ω−)
$(\Delta u,v)=(u_{x},v_{x})+(u_{y},v_{y})-(\partial _{n}u,v)_{\partial \Omega }.$
Thus if Δu = 0 in Ω and satisfies the Neumann boundary conditions, ux = uy = 0, and so u is constant in Ω.
Hence the Neumann problem has a unique solution up to adding constants.[17]
Consider the Hermitian form on H1(Ω) defined by
$\displaystyle {D(f,g)=(u_{x},v_{x})+(u_{y},v_{y}).}$
Since H1(Ω) is in duality with H−1
0
(Ω)
, there is a unique element Lu in H−1
0
(Ω)
such that
$\displaystyle {D(u,v)=(Lu,v).}$
The map I + L is an isometry of H1(Ω) onto H−1
0
(Ω)
, so in particular L is bounded.
In fact
$((L+I)u,v)=(u,v)_{(1)}.$
So
$\|(L+I)u\|_{(-1)}=\sup _{\|v\|_{(1)}=1}|((L+I)u,v)|=\sup _{\|v\|_{(1)}=1}|(u,v)_{(1)}|=\|u\|_{(1)}.$
On the other hand, any f in H−1
0
(Ω)
defines a bounded conjugate-linear form on H1(Ω) sending v to ( f, v). By the Riesz–Fischer theorem, there exists u ∈ H1(Ω) such that
$\displaystyle {(f,v)=(u,v)_{(1)}.}$
Hence (L + I)u = f and so L + I is surjective. Define a bounded linear operator T on L2(Ω) by
$T=R_{1}(I+L)^{-1}R_{0},$
where R1 is the map H1(Ω) → L2(Ω), a compact operator, and R0 is the map L2(Ω) → H−1
0
(Ω)
, its adjoint, so also compact.
The operator T has the following properties:
• T is a contraction since it is a composition of contractions
• T is compact, since R0 and R1 are compact by Rellich's theorem
• T is self-adjoint, since if f, g ∈ L2(Ω), they can be written f = (L + I)u, g = (L + I)v with u, v ∈ H1(Ω) so
$(Tf,g)=(u,(I+L)v)=(u,v)_{(1)}=((I+L)u,v)=(f,Tg).$
• T has positive spectrum and kernel (0), for
$(Tf,f)=(u,u)_{(1)}\geq 0,$
and Tf = 0 implies u = 0 and hence f = 0.
• There is a complete orthonormal basis fn of L2(Ω) consisting of eigenfunctions of T. Thus
$Tf_{n}=\mu _{n}f_{n}$
with 0 < μn ≤ 1 and μn decreasing to 0.
• The eigenfunctions all lie in H1(Ω) since the image of T lies in H1(Ω).
• The fn are eigenfunctions of L with
$\displaystyle {Lf_{n}=\lambda _{n}f_{n},\qquad \lambda _{n}=\mu _{n}^{-1}-1.}$
Thus λn are non-negative and increase to ∞.
• The eigenvalue 0 occurs with multiplicity one and corresponds to the constant function. For if u ∈ H1(Ω) satisfies Lu = 0, then
$(u_{x},u_{x})+(u_{y},u_{y})=(Lu,u)=0,$
so u is constant.
Regularity for Neumann problem
Weak solutions are strong solutions
The first main regularity result shows that a weak solution expressed in terms of the operator L and the Dirichlet form D is a strong solution in the classical sense, expressed in terms of the Laplacian Δ and the Neumann boundary conditions. Thus if u = Tf with u ∈ H1(Ω), f ∈ L2(Ω), then u ∈ H2(Ω), satisfies Δu + u = f and ∂nu|∂Ω = 0. Moreover, for some constant C independent of u,
$\|u\|_{(2)}\leq C\|\Delta u\|_{(0)}+C\|u\|_{(1)}.$
Note that
$\|u\|_{(1)}\leq \|Lu\|_{(-1)}+\|u\|_{(0)},$
since
${\begin{aligned}\|u\|_{(1)}^{2}&=|(Lu,u)|+\|u\|_{(0)}^{2}\\&\leq \|Lu\|_{(-1)}\|u\|_{(1)}+\|u\|_{(0)}\|u\|_{(1)}.\end{aligned}}$
Take a decomposition u = ψu + (1 − ψ)u with ψ supported in Ω and 1 − ψ supported in a collar of the boundary.
The operator L is characterized by
$(Lf,g)=\left(f_{x},g_{x}\right)+\left(f_{y},g_{y}\right)=(\Delta f,g)_{\Omega }-\left(\partial _{n}f,g\right)_{\partial \Omega },\qquad f,g\in C^{\infty }(\Omega ^{-}).$
Then
$([L,\psi ]f,g)=([\Delta ,\psi ]f,g),$
so that
$\displaystyle {[L,\psi ]=-[L,1-\psi ]=\Delta \psi +2\psi _{x}\partial _{x}+2\psi _{y}\partial _{y}.}$
The function v = ψu and w = (1 − ψ)u are treated separately, v being essentially subject to usual elliptic regularity considerations for interior points while w requires special treatment near the boundary using difference quotients. Once the strong properties are established in terms of ∆ and the Neumann boundary conditions, the "bootstrap" regularity results can be proved exactly as for the Dirichlet problem.
Interior estimates
The function v = ψu lies in H1
0
(Ω1)
where Ω1 is a region with closure in Ω. If f ∈ C∞
c
(Ω)
and g ∈ C∞(Ω−)
$(Lf,g)=(\Delta f,g)_{\Omega }.$
By continuity the same holds with f replaced by v and hence Lv = ∆v. So
$\Delta v=Lv=L(\psi u)=\psi Lu+[L,\psi ]u=\psi (f-u)+[\Delta ,\psi ]u.$
Hence regarding v as an element of H1(T2), ∆v ∈ L2(T2). Hence v ∈ H2(T2). Since v = φv for φ ∈ C∞
c
(Ω)
, we have v ∈ H2
0
(Ω)
. Moreover,
$\|v\|_{(2)}^{2}=\|\Delta v\|^{2}+2\|v\|_{(1)}^{2},$
so that
$\|v\|_{(2)}\leq C\left(\|\Delta (v)\|+\|v\|_{(1)}\right).$
Boundary estimates
The function w = (1 − ψ)u is supported in a collar contained in a tubular neighbourhood of the boundary. The difference quotients δhw can be formed for the flow St and lie in H1(Ω), so the first inequality is applicable:
${\begin{aligned}\|\delta _{h}w\|_{(1)}&\leq \|L\delta _{h}w\|_{(-1)}+\|\delta _{h}w\|_{(0)}\\&\leq \|[L,\delta _{h}]w\|_{(-1)}+\|\delta _{h}Lw\|_{(-1)}+\|\delta _{h}w\|_{(0)}\\&\leq \|[L,\delta _{h}]w\|_{(-1)}+C\|Lw\|_{(0)}+C\|w\|_{(1)}.\end{aligned}}$
The commutators [L, δh] are uniformly bounded as operators from H1(Ω) to H−1
0
(Ω)
. This is equivalent to checking the inequality
$\left|\left(\left[L,\delta _{h}\right]g,h\right)\right|\leq A\|g\|_{(1)}\|h\|_{(1)},$
for g, h smooth functions on a collar. This can be checked directly on an annulus, using invariance of Sobolev spaces under dffeomorphisms and the fact that for the annulus the commutator of δh with a differential operator is obtained by applying the difference operator to the coefficients after having applied Rh to the function:[18]
$\left[\delta ^{h},\sum a_{\alpha }\partial ^{\alpha }\right]=\left(\delta ^{h}(a_{\alpha })\circ R_{h}\right)\partial ^{\alpha }.$
Hence the difference quotients δhw are uniformly bounded, and therefore Yw ∈ H1(Ω) with
$\|Yw\|_{(1)}\leq C\|Lw\|_{(0)}+C^{\prime }\|w\|_{(1)}.$
Hence Vw ∈ H1(Ω) and Vw satisfies a similar inequality to Yw:
$\|Vw\|_{(1)}\leq C^{\prime \prime }\left(\|Lw\|_{(0)}+\|w\|_{(1)}\right).$
Let W be the orthogonal vector field. As for the Dirichlet problem, to show that w ∈ H2(Ω), it suffices to show that Ww ∈ H1(Ω).
To check this, it is enough to show that VWw, W 2u ∈ L2(Ω). As before
${\begin{aligned}A&=\Delta -V^{2}-W^{2}\\B&=[V,W]\end{aligned}}$
are vector fields. On the other hand, (Lw, φ) = (∆w, φ) for φ ∈ C∞
c
(Ω)
, so that Lw and ∆w define the same distribution on Ω. Hence
${\begin{aligned}\left(W^{2}w,\varphi \right)&=\left(Lw-V^{2}w-Au,\varphi \right),\\(VWw,\varphi )&=(WVw+Bw,\varphi ).\end{aligned}}$
Since the terms on the right hand side are pairings with functions in L2(Ω), the regularity criterion shows that Ww ∈ H2(Ω). Hence Lw = ∆w since both terms lie in L2(Ω) and have the same inner products with φ's.
Moreover, the inequalities for Vw show that
${\begin{aligned}\|Ww\|_{(1)}&\leq C\left(\|VWw\|_{(0)}+\left\|W^{2}w\right\|_{(0)}\right)\\&\leq C\left\|\left(\Delta -V^{2}-A\right)w\right\|_{(0)}+C\|(WV+B)w\|_{(0)}\\&\leq C_{1}\|Lw\|_{(0)}+C_{1}\|w\|_{(1)}.\end{aligned}}$
Hence
${\begin{aligned}\|w\|_{(2)}&\leq C\left(\|Vw\|_{(1)}+\|Ww\|_{(1)}\right)\\&\leq C^{\prime }\|\Delta w\|_{(0)}+C^{\prime }\|w\|_{(1)}.\end{aligned}}$
It follows that u = v + w ∈ H2(Ω). Moreover,
${\begin{aligned}\|u\|_{(2)}&\leq C\left(\|\Delta v\|+\|\Delta w\|+\|v\|_{(1)}+\|w\|_{(1)}\right)\\&\leq C^{\prime }\left(\|\psi \Delta u\|+\|(1-\psi )\Delta u\|+2\|[\Delta ,\psi ]u\|+\|u\|_{(1)}\right)\\&\leq C^{\prime \prime }\left(\|\Delta u\|+\|u\|_{(1)}\right).\end{aligned}}$
Neumann boundary conditions
Since u ∈ H2(Ω), Green's theorem is applicable by continuity. Thus for v ∈ H1(Ω),
${\begin{aligned}(f,v)&=(Lu,v)+(u,v)\\&=(u_{x},v_{x})+(u_{y},v_{y})+(u,v)\\&=((\Delta +I)u,v)+(\partial _{n}u,v)_{\partial \Omega }\\&=(f,v)+(\partial _{n}u,v)_{\partial \Omega }.\end{aligned}}$
Hence the Neumann boundary conditions are satisfied:
$\partial _{n}u|_{\partial \Omega }=0,$
where the left hand side is regarded as an element of H½(∂Ω) and hence L2(∂Ω).
Regularity of strong solutions
The main result here states that if u ∈ Hk+1 (k ≥ 1), ∆u ∈ Hk and ∂nu|∂Ω = 0, then u ∈ Hk+2 and
$\|u\|_{(k+2)}\leq C\|\Delta u\|_{(k)}+C\|u\|_{(k+1)},$
for some constant independent of u.
Like the corresponding result for the Dirichlet problem, this is proved by induction on k ≥ 1. For k = 1, u is also a weak solution of the Neumann problem so satisfies the estimate above for k = 0. The Neumann boundary condition can be written
$Zu|_{\partial \Omega }=0.$
Since Z commutes with the vector field Y corresponding to the period flow St, the inductive method of proof used for the Dirichlet problem works equally well in this case: for the difference quotients δh preserve the boundary condition when expressed in terms of Z.[19]
Smoothness of eigenfunctions
It follows by induction from the regularity theorem for the Neumann problem that the eigenfunctions of D in H1(Ω) lie in C∞(Ω−). Moreover, any solution of Du = f with f in C∞(Ω−) and u in H1(Ω) must have u in C∞(Ω−). In both cases by the vanishing properties, the normal derivatives of the eigenfunctions and u vanish on ∂Ω.
Solving the associated Neumann problem
The method above can be used to solve the associated Neumann boundary value problem:
${\begin{cases}\Delta f|_{\Omega }=0\\\partial _{n}f|_{\partial \Omega }=g&g\in C^{\infty }(\partial \Omega )\end{cases}}$
By Borel's lemma g is the restriction of a function G ∈ C∞(Ω−). Let F be a smooth function such that ∂nF = G near the boundary. Let u be the solution of ∆u = −∆F with ∂nu = 0. Then f = u + F solves the boundary value problem.[20]
Notes
1. Bers, John & Schechter 1979, pp. 192–193
2. Chazarain & Piriou 1982
3. Folland 1995, p. 226
4. Folland 1995
5. See:
• Agmon 2010
• Folland 1995, pp. 219–223
• Chazarain & Piriou 1982, p. 94
6. Taylor 2011
7. Folland 1995, p. 84
8. Taylor 2011, pp. 323–325
9. Chazarain & Piriou 1982
10. Taylor 2011, p. 275
11. Renardy & Rogers 2004, pp. 214–218
12. Hörmander 1990, pp. 240–241
13. Renardy & Rogers 2004
14. Chazarain & Piriou 1982
15. Renardy & Rogers 2004
16. Folland 1995, pp. 231–248
17. Taylor 2011
18. Folland 1995, pp. 255–260
19. Taylor 2011, p. 348
20. Folland 1995, p. 85
References
• John, Fritz (1982), Partial differential equations, Applied Mathematical Sciences, vol. 1 (4th ed.), Springer-Verlag, ISBN 0-387-90609-6
• Bers, Lipman; John, Fritz; Schechter, Martin (1979), Partial differential equations, with supplements by Lars Gȧrding and A. N. Milgram, Lectures in Applied Mathematics, vol. 3A, American Mathematical Society, ISBN 0-8218-0049-3
• Agmon, Shmuel (2010), Lectures on Elliptic Boundary Value Problems, American Mathematical Society, ISBN 978-0-8218-4910-1
• Stein, Elias M. (1970), Singular Integrals and Differentiability Properties of Functions, Princeton University Press
• Greene, Robert E.; Krantz, Steven G. (2006), Function theory of one complex variable, Graduate Studies in Mathematics, vol. 40 (3rd ed.), American Mathematical Society, ISBN 0-8218-3962-4
• Taylor, Michael E. (2011), Partial differential equations I. Basic theory, Applied Mathematical Sciences, vol. 115 (2nd ed.), Springer, ISBN 978-1-4419-7054-1
• Zimmer, Robert J. (1990), Essential results of functional analysis, Chicago Lectures in Mathematics, University of Chicago Press, ISBN 0-226-98337-4
• Folland, Gerald B. (1995), Introduction to partial differential equations (2nd ed.), Princeton University Press, ISBN 0-691-04361-2
• Chazarain, Jacques; Piriou, Alain (1982), Introduction to the Theory of Linear Partial Differential Equations, Studies in Mathematics and Its Applications, vol. 14, Elsevier, ISBN 0-444-86452-0
• Bell, Steven R. (1992), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, ISBN 0-8493-8270-X
• Warner, Frank W. (1983), Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics, vol. 94, Springer, ISBN 0-387-90894-3
• Griffiths, Phillip; Harris, Joseph (1994), Principles of Algebraic Geometry, Wiley Interscience, ISBN 0-471-05059-8
• Courant, R. (1950), Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces, Interscience
• Schiffer, M.; Hawley, N. S. (1962), "Connections and conformal mapping", Acta Math., 107 (3–4): 175–274, doi:10.1007/bf02545790
• Hörmander, Lars (1990), The analysis of linear partial differential operators, I. Distribution theory and Fourier analysis (2nd ed.), Springer-Verlag, ISBN 3-540-52343-X
• Renardy, Michael; Rogers, Robert C. (2004), An Introduction to Partial Differential Equations, Texts in Applied Mathematics, vol. 13 (2nd ed.), Springer, ISBN 0-387-00444-0
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Wikipedia
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Order-5 dodecahedral honeycomb
In hyperbolic geometry, the order-5 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {5,3,5}, it has five dodecahedral cells around each edge, and each vertex is surrounded by twenty dodecahedra. Its vertex figure is an icosahedron.
Order-5 dodecahedral honeycomb
Perspective projection view
from center of Poincaré disk model
TypeHyperbolic regular honeycomb
Uniform hyperbolic honeycomb
Schläfli symbol{5,3,5}
t0{5,3,5}
Coxeter-Dynkin diagram
Cells{5,3} (regular dodecahedron)
Faces{5} (pentagon)
Edge figure{5} (pentagon)
Vertex figure
icosahedron
DualSelf-dual
Coxeter groupK3, [5,3,5]
PropertiesRegular
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.
Description
The dihedral angle of a Euclidean regular dodecahedron is ~116.6°, so no more than three of them can fit around an edge in Euclidean 3-space. In hyperbolic space, however, the dihedral angle is smaller than it is in Euclidean space, and depends on the size of the figure; the smallest possible dihedral angle is 60°, for an ideal hyperbolic regular dodecahedron with infinitely long edges. The dodecahedra in this dodecahedral honeycomb are sized so that all of their dihedral angles are exactly 72°.
Images
Related polytopes and honeycombs
There are four regular compact honeycombs in 3D hyperbolic space:
Four regular compact honeycombs in H3
{5,3,4}
{4,3,5}
{3,5,3}
{5,3,5}
There is another honeycomb in hyperbolic 3-space called the order-4 dodecahedral honeycomb, {5,3,4}, which has only four dodecahedra per edge. These honeycombs are also related to the 120-cell which can be considered as a honeycomb in positively curved space (the surface of a 4-dimensional sphere), with three dodecahedra on each edge, {5,3,3}. Lastly the dodecahedral ditope, {5,3,2} exists on a 3-sphere, with 2 hemispherical cells.
There are nine uniform honeycombs in the [5,3,5] Coxeter group family, including this regular form. Also the bitruncated form, t1,2{5,3,5}, , of this honeycomb has all truncated icosahedron cells.
[5,3,5] family honeycombs
{5,3,5}
r{5,3,5}
t{5,3,5}
rr{5,3,5}
t0,3{5,3,5}
2t{5,3,5}
tr{5,3,5}
t0,1,3{5,3,5}
t0,1,2,3{5,3,5}
The Seifert–Weber space is a compact manifold that can be formed as a quotient space of the order-5 dodecahedral honeycomb.
This honeycomb is a part of a sequence of polychora and honeycombs with icosahedron vertex figures:
{p,3,5} polytopes
Space S3 H3
Form Finite Compact Paracompact Noncompact
Name {3,3,5}
{4,3,5}
{5,3,5}
{6,3,5}
{7,3,5}
{8,3,5}
... {∞,3,5}
Image
Cells
{3,3}
{4,3}
{5,3}
{6,3}
{7,3}
{8,3}
{∞,3}
This honeycomb is a part of a sequence of regular polytopes and honeycombs with dodecahedral cells:
{5,3,p} polytopes
Space S3 H3
Form Finite Compact Paracompact Noncompact
Name {5,3,3} {5,3,4} {5,3,5} {5,3,6} {5,3,7} {5,3,8} ... {5,3,∞}
Image
Vertex
figure
{3,3}
{3,4}
{3,5}
{3,6}
{3,7}
{3,8}
{3,∞}
{p,3,p} regular honeycombs
Space S3 Euclidean E3 H3
Form Finite Affine Compact Paracompact Noncompact
Name {3,3,3} {4,3,4} {5,3,5} {6,3,6} {7,3,7} {8,3,8} ...{∞,3,∞}
Image
Cells
{3,3}
{4,3}
{5,3}
{6,3}
{7,3}
{8,3}
{∞,3}
Vertex
figure
{3,3}
{3,4}
{3,5}
{3,6}
{3,7}
{3,8}
{3,∞}
Rectified order-5 dodecahedral honeycomb
Rectified order-5 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolr{5,3,5}
t1{5,3,5}
Coxeter diagram
Cellsr{5,3}
{3,5}
Facestriangle {3}
pentagon {5}
Vertex figure
pentagonal prism
Coxeter group${\overline {K}}_{3}$, [5,3,5]
PropertiesVertex-transitive, edge-transitive
The rectified order-5 dodecahedral honeycomb, , has alternating icosahedron and icosidodecahedron cells, with a pentagonal prism vertex figure.
Related tilings and honeycomb
There are four rectified compact regular honeycombs:
Four rectified regular compact honeycombs in H3
Image
Symbols r{5,3,4}
r{4,3,5}
r{3,5,3}
r{5,3,5}
Vertex
figure
r{p,3,5}
Space S3 H3
Form Finite Compact Paracompact Noncompact
Name r{3,3,5}
r{4,3,5}
r{5,3,5}
r{6,3,5}
r{7,3,5}
... r{∞,3,5}
Image
Cells
{3,5}
r{3,3}
r{4,3}
r{5,3}
r{6,3}
r{7,3}
r{∞,3}
Truncated order-5 dodecahedral honeycomb
Truncated order-5 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt{5,3,5}
t0,1{5,3,5}
Coxeter diagram
Cellst{5,3}
{3,5}
Facestriangle {3}
decagon {10}
Vertex figure
pentagonal pyramid
Coxeter group${\overline {K}}_{3}$, [5,3,5]
PropertiesVertex-transitive
The truncated order-5 dodecahedral honeycomb, , has icosahedron and truncated dodecahedron cells, with a pentagonal pyramid vertex figure.
Related honeycombs
Four truncated regular compact honeycombs in H3
Image
Symbols t{5,3,4}
t{4,3,5}
t{3,5,3}
t{5,3,5}
Vertex
figure
Bitruncated order-5 dodecahedral honeycomb
Bitruncated order-5 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbol2t{5,3,5}
t1,2{5,3,5}
Coxeter diagram
Cellst{3,5}
Facespentagon {5}
hexagon {6}
Vertex figure
tetragonal disphenoid
Coxeter group$2\times {\overline {K}}_{3}$, [[5,3,5]]
PropertiesVertex-transitive, edge-transitive, cell-transitive
The bitruncated order-5 dodecahedral honeycomb, , has truncated icosahedron cells, with a tetragonal disphenoid vertex figure.
Related honeycombs
Three bitruncated compact honeycombs in H3
Image
Symbols 2t{4,3,5}
2t{3,5,3}
2t{5,3,5}
Vertex
figure
Cantellated order-5 dodecahedral honeycomb
Cantellated order-5 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolrr{5,3,5}
t0,2{5,3,5}
Coxeter diagram
Cellsrr{5,3}
r{3,5}
{}x{5}
Facestriangle {3}
square {4}
pentagon {5}
Vertex figure
wedge
Coxeter group${\overline {K}}_{3}$, [5,3,5]
PropertiesVertex-transitive
The cantellated order-5 dodecahedral honeycomb, , has rhombicosidodecahedron, icosidodecahedron, and pentagonal prism cells, with a wedge vertex figure.
Related honeycombs
Four cantellated regular compact honeycombs in H3
Image
Symbols rr{5,3,4}
rr{4,3,5}
rr{3,5,3}
rr{5,3,5}
Vertex
figure
Cantitruncated order-5 dodecahedral honeycomb
Cantitruncated order-5 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symboltr{5,3,5}
t0,1,2{5,3,5}
Coxeter diagram
Cellstr{5,3}
t{3,5}
{}x{5}
Facessquare {4}
pentagon {5}
hexagon {6}
decagon {10}
Vertex figure
mirrored sphenoid
Coxeter group${\overline {K}}_{3}$, [5,3,5]
PropertiesVertex-transitive
The cantitruncated order-5 dodecahedral honeycomb, , has truncated icosidodecahedron, truncated icosahedron, and pentagonal prism cells, with a mirrored sphenoid vertex figure.
Related honeycombs
Four cantitruncated regular compact honeycombs in H3
Image
Symbols tr{5,3,4}
tr{4,3,5}
tr{3,5,3}
tr{5,3,5}
Vertex
figure
Runcinated order-5 dodecahedral honeycomb
Runcinated order-5 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt0,3{5,3,5}
Coxeter diagram
Cells{5,3}
{}x{5}
Facessquare {4}
pentagon {5}
Vertex figure
triangular antiprism
Coxeter group$2\times {\overline {K}}_{3}$, [[5,3,5]]
PropertiesVertex-transitive, edge-transitive
The runcinated order-5 dodecahedral honeycomb, , has dodecahedron and pentagonal prism cells, with a triangular antiprism vertex figure.
Related honeycombs
Three runcinated regular compact honeycombs in H3
Image
Symbols t0,3{4,3,5}
t0,3{3,5,3}
t0,3{5,3,5}
Vertex
figure
Runcitruncated order-5 dodecahedral honeycomb
Runcitruncated order-5 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt0,1,3{5,3,5}
Coxeter diagram
Cellst{5,3}
rr{5,3}
{}x{5}
{}x{10}
Facestriangle {3}
square {4}
pentagon {5}
decagon {10}
Vertex figure
isosceles-trapezoidal pyramid
Coxeter group${\overline {K}}_{3}$, [5,3,5]
PropertiesVertex-transitive
The runcitruncated order-5 dodecahedral honeycomb, , has truncated dodecahedron, rhombicosidodecahedron, pentagonal prism, and decagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.
The runcicantellated order-5 dodecahedral honeycomb is equivalent to the runcitruncated order-5 dodecahedral honeycomb.
Related honeycombs
Four runcitruncated regular compact honeycombs in H3
Image
Symbols t0,1,3{5,3,4}
t0,1,3{4,3,5}
t0,1,3{3,5,3}
t0,1,3{5,3,5}
Vertex
figure
Omnitruncated order-5 dodecahedral honeycomb
Omnitruncated order-5 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt0,1,2,3{5,3,5}
Coxeter diagram
Cellstr{5,3}
{}x{10}
Facessquare {4}
hexagon {6}
decagon {10}
Vertex figure
phyllic disphenoid
Coxeter group$2\times {\overline {K}}_{3}$, [[5,3,5]]
PropertiesVertex-transitive
The omnitruncated order-5 dodecahedral honeycomb, , has truncated icosidodecahedron and decagonal prism cells, with a phyllic disphenoid vertex figure.
Related honeycombs
Three omnitruncated regular compact honeycombs in H3
Image
Symbols t0,1,2,3{4,3,5}
t0,1,2,3{3,5,3}
t0,1,2,3{5,3,5}
Vertex
figure
See also
• Convex uniform honeycombs in hyperbolic space
• Regular tessellations of hyperbolic 3-space
• 57-cell - An abstract regular polychoron which shared the {5,3,5} symbol.
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)
• 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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Wikipedia
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Small cubicuboctahedron
In geometry, the small cubicuboctahedron is a uniform star polyhedron, indexed as U13. It has 20 faces (8 triangles, 6 squares, and 6 octagons), 48 edges, and 24 vertices.[1] Its vertex figure is a crossed quadrilateral.
Small cubicuboctahedron
TypeUniform star polyhedron
ElementsF = 20, E = 48
V = 24 (χ = −4)
Faces by sides8{3}+6{4}+6{8}
Coxeter diagram
Wythoff symbol3/2 4 | 4
3 4/3 | 4
Symmetry groupOh, [4,3], *432
Index referencesU13, C38, W69
Dual polyhedronSmall hexacronic icositetrahedron
Vertex figure
4.8.3/2.8
Bowers acronymSocco
The small cubicuboctahedron is a faceting of the rhombicuboctahedron. Its square faces and its octagonal faces are parallel to those of a cube, while its triangular faces are parallel to those of an octahedron: hence the name cubicuboctahedron. The small suffix serves to distinguish it from the great cubicuboctahedron, which also has faces in the aforementioned directions.[2]
Related polyhedra
It shares its vertex arrangement with the stellated truncated hexahedron. It additionally shares its edge arrangement with the rhombicuboctahedron (having the triangular faces and 6 square faces in common), and with the small rhombihexahedron (having the octagonal faces in common).
Rhombicuboctahedron
Small cubicuboctahedron
Small rhombihexahedron
Stellated truncated hexahedron
Related tilings
As the Euler characteristic suggests, the small cubicuboctahedron is a toroidal polyhedron of genus 3 (topologically it is a surface of genus 3), and thus can be interpreted as a (polyhedral) immersion of a genus 3 polyhedral surface, in the complement of its 24 vertices, into 3-space. (A neighborhood of any vertex is topologically a cone on a figure-8, which cannot occur in an immersion. Note that the Richter reference overlooks this fact.) The underlying polyhedron (ignoring self-intersections) defines a uniform tiling of this surface, and so the small cubicuboctahedron is a uniform polyhedron. In the language of abstract polytopes, the small cubicuboctahedron is a faithful realization of this abstract toroidal polyhedron, meaning that it is a nondegenerate polyhedron and that they have the same symmetry group. In fact, every automorphism of the abstract genus 3 surface with this tiling is realized by an isometry of Euclidean space.
Higher genus surfaces (genus 2 or greater) admit a metric of negative constant curvature (by the uniformization theorem), and the universal cover of the resulting Riemann surface is the hyperbolic plane. The corresponding tiling of the hyperbolic plane has vertex figure 3.8.4.8 (triangle, octagon, square, octagon). If the surface is given the appropriate metric of curvature = −1, the covering map is a local isometry and thus the abstract vertex figure is the same. This tiling may be denoted by the Wythoff symbol 3 4 | 4, and is depicted on the right.
Alternatively and more subtly, by chopping up each square face into 2 triangles and each octagonal face into 6 triangles, the small cubicuboctahedron can be interpreted as a non-regular coloring of the combinatorially regular (not just uniform) tiling of the genus 3 surface by 56 equilateral triangles, meeting at 24 vertices, each with degree 7.[3] This regular tiling is significant as it is a tiling of the Klein quartic, the genus 3 surface with the most symmetric metric (automorphisms of this tiling equal isometries of the surface), and the orientation-preseserving automorphism group of this surface is isomorphic to the projective special linear group PSL(2,7), equivalently GL(3,2) (the order 168 group of all orientation-preserving isometries). Note that the small cubicuboctahedron is not a realization of this abstract polyhedron, as it only has 24 orientation-preserving symmetries (not every abstract automorphism is realized by a Euclidean isometry) – the isometries of the small cubicuboctahedron preserve not only the triangular tiling, but also the coloring, and hence are a proper subgroup of the full isometry group.
The corresponding tiling of the hyperbolic plane (the universal covering) is the order-7 triangular tiling. The automorphism group of the Klein quartic can be augmented (by a symmetry which is not realized by a symmetry of the polyhedron, namely "exchanging the two endpoints of the edges that bisect the squares and octahedra) to yield the Mathieu group M24.[4]
See also
• Compound of five small cubicuboctahedra
• List of uniform polyhedra
References
1. Maeder, Roman. "13: small cubicuboctahedron". MathConsult.{{cite web}}: CS1 maint: url-status (link)
2. Webb, Robert. "Small Cubicuboctahedron". Stella: Polyhedron Navigator.{{cite web}}: CS1 maint: url-status (link)
3. (Richter) Note each face in the polyhedron consist of multiple faces in the tiling, hence the description as a "coloring" – two triangular faces constitute a square face and so forth, as per this explanatory image.
4. (Richter)
• Richter, David A., How to Make the Mathieu Group M24, retrieved 2010-04-15
External links
• Eric W. Weisstein, Small cubicuboctahedron (Uniform polyhedron) at MathWorld.
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Wikipedia
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Sociable number
In mathematics, sociable numbers are numbers whose aliquot sums form a periodic sequence. They are generalizations of the concepts of perfect numbers and amicable numbers. The first two sociable sequences, or sociable chains, were discovered and named by the Belgian mathematician Paul Poulet in 1918.[1] In a sociable sequence, each number is the sum of the proper divisors of the preceding number, i.e., the sum excludes the preceding number itself. For the sequence to be sociable, the sequence must be cyclic and return to its starting point.
The period of the sequence, or order of the set of sociable numbers, is the number of numbers in this cycle.
If the period of the sequence is 1, the number is a sociable number of order 1, or a perfect number—for example, the proper divisors of 6 are 1, 2, and 3, whose sum is again 6. A pair of amicable numbers is a set of sociable numbers of order 2. There are no known sociable numbers of order 3, and searches for them have been made up to $5\times 10^{7}$ as of 1970.[2]
It is an open question whether all numbers end up at either a sociable number or at a prime (and hence 1), or, equivalently, whether there exist numbers whose aliquot sequence never terminates, and hence grows without bound.
Example
As an example, the number 1,264,460 is a sociable number whose cyclic aliquot sequence has a period of 4:
The sum of the proper divisors of $1264460$ ($=2^{2}\cdot 5\cdot 17\cdot 3719$) is
1 + 2 + 4 + 5 + 10 + 17 + 20 + 34 + 68 + 85 + 170 + 340 + 3719 + 7438 + 14876 + 18595 + 37190 + 63223 + 74380 + 126446 + 252892 + 316115 + 632230 = 1547860,
the sum of the proper divisors of $1547860$ ($=2^{2}\cdot 5\cdot 193\cdot 401$) is
1 + 2 + 4 + 5 + 10 + 20 + 193 + 386 + 401 + 772 + 802 + 965 + 1604 + 1930 + 2005 + 3860 + 4010 + 8020 + 77393 + 154786 + 309572 + 386965 + 773930 = 1727636,
the sum of the proper divisors of $1727636$ ($=2^{2}\cdot 521\cdot 829$) is
1 + 2 + 4 + 521 + 829 + 1042 + 1658 + 2084 + 3316 + 431909 + 863818 = 1305184, and
the sum of the proper divisors of $1305184$ ($=2^{5}\cdot 40787$) is
1 + 2 + 4 + 8 + 16 + 32 + 40787 + 81574 + 163148 + 326296 + 652592 = 1264460.
List of known sociable numbers
The following categorizes all known sociable numbers as of July 2018 by the length of the corresponding aliquot sequence:
Sequence
length
Number of known
sequences
lowest number
in sequence[3]
1
(Perfect number)
51 6
2
(Amicable number)
1225736919[4] 220
4 5398 1,264,460
5 1 12,496
6 5 21,548,919,483
8 4 1,095,447,416
9 1 805,984,760
28 1 14,316
It is conjectured that if n is congruent to 3 modulo 4 then there is no such sequence with length n.
The 5-cycle sequence is: 12496, 14288, 15472, 14536, 14264
The only known 28-cycle is: 14316, 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946, 48976, 45946, 22976, 22744, 19916, 17716 (sequence A072890 in the OEIS). It was discovered by Ben Orlin.
These two sequences provide the only sociable numbers below 1 million (other than the perfect and amicable numbers).
Searching for sociable numbers
The aliquot sequence can be represented as a directed graph, $G_{n,s}$, for a given integer $n$, where $s(k)$ denotes the sum of the proper divisors of $k$.[5] Cycles in $G_{n,s}$ represent sociable numbers within the interval $[1,n]$. Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs.
Conjecture of the sum of sociable number cycles
It is conjectured that as the number of sociable number cycles with length greater than 2 approaches infinity, the proportion of the sums of the sociable number cycles divisible by 10 approaches 1 (sequence A292217 in the OEIS).
References
1. P. Poulet, #4865, L'Intermédiaire des Mathématiciens 25 (1918), pp. 100–101. (The full text can be found at ProofWiki: Catalan-Dickson Conjecture.)
2. Bratley, Paul; Lunnon, Fred; McKay, John (1970). "Amicable numbers and their distribution". Mathematics of Computation. 24 (110): 431–432. doi:10.1090/S0025-5718-1970-0271005-8. ISSN 0025-5718.
3. https://oeis.org/A003416 cross referenced with https://oeis.org/A052470
4. Sergei Chernykh Amicable pairs list
5. Rocha, Rodrigo Caetano; Thatte, Bhalchandra (2015), Distributed cycle detection in large-scale sparse graphs, Simpósio Brasileiro de Pesquisa Operacional (SBPO), doi:10.13140/RG.2.1.1233.8640
• H. Cohen, On amicable and sociable numbers, Math. Comp. 24 (1970), pp. 423–429
External links
• A list of known sociable numbers
• Extensive tables of perfect, amicable and sociable numbers
• Weisstein, Eric W. "Sociable numbers". MathWorld.
• A003416 (smallest sociable number from each cycle) and A122726 (all sociable numbers) in OEIS
Divisibility-based sets of integers
Overview
• Integer factorization
• Divisor
• Unitary divisor
• Divisor function
• Prime factor
• Fundamental theorem of arithmetic
Factorization forms
• Prime
• Composite
• Semiprime
• Pronic
• Sphenic
• Square-free
• Powerful
• Perfect power
• Achilles
• Smooth
• Regular
• Rough
• Unusual
Constrained divisor sums
• Perfect
• Almost perfect
• Quasiperfect
• Multiply perfect
• Hemiperfect
• Hyperperfect
• Superperfect
• Unitary perfect
• Semiperfect
• Practical
• Erdős–Nicolas
With many divisors
• Abundant
• Primitive abundant
• Highly abundant
• Superabundant
• Colossally abundant
• Highly composite
• Superior highly composite
• Weird
Aliquot sequence-related
• Untouchable
• Amicable (Triple)
• Sociable
• Betrothed
Base-dependent
• Equidigital
• Extravagant
• Frugal
• Harshad
• Polydivisible
• Smith
Other sets
• Arithmetic
• Deficient
• Friendly
• Solitary
• Sublime
• Harmonic divisor
• Descartes
• Refactorable
• Superperfect
Classes of natural numbers
Powers and related numbers
• Achilles
• Power of 2
• Power of 3
• Power of 10
• Square
• Cube
• Fourth power
• Fifth power
• Sixth power
• Seventh power
• Eighth power
• Perfect power
• Powerful
• Prime power
Of the form a × 2b ± 1
• Cullen
• Double Mersenne
• Fermat
• Mersenne
• Proth
• Thabit
• Woodall
Other polynomial numbers
• Hilbert
• Idoneal
• Leyland
• Loeschian
• Lucky numbers of Euler
Recursively defined numbers
• Fibonacci
• Jacobsthal
• Leonardo
• Lucas
• Padovan
• Pell
• Perrin
Possessing a specific set of other numbers
• Amenable
• Congruent
• Knödel
• Riesel
• Sierpiński
Expressible via specific sums
• Nonhypotenuse
• Polite
• Practical
• Primary pseudoperfect
• Ulam
• Wolstenholme
Figurate numbers
2-dimensional
centered
• Centered triangular
• Centered square
• Centered pentagonal
• Centered hexagonal
• Centered heptagonal
• Centered octagonal
• Centered nonagonal
• Centered decagonal
• Star
non-centered
• Triangular
• Square
• Square triangular
• Pentagonal
• Hexagonal
• Heptagonal
• Octagonal
• Nonagonal
• Decagonal
• Dodecagonal
3-dimensional
centered
• Centered tetrahedral
• Centered cube
• Centered octahedral
• Centered dodecahedral
• Centered icosahedral
non-centered
• Tetrahedral
• Cubic
• Octahedral
• Dodecahedral
• Icosahedral
• Stella octangula
pyramidal
• Square pyramidal
4-dimensional
non-centered
• Pentatope
• Squared triangular
• Tesseractic
Combinatorial numbers
• Bell
• Cake
• Catalan
• Dedekind
• Delannoy
• Euler
• Eulerian
• Fuss–Catalan
• Lah
• Lazy caterer's sequence
• Lobb
• Motzkin
• Narayana
• Ordered Bell
• Schröder
• Schröder–Hipparchus
• Stirling first
• Stirling second
• Telephone number
• Wedderburn–Etherington
Primes
• Wieferich
• Wall–Sun–Sun
• Wolstenholme prime
• Wilson
Pseudoprimes
• Carmichael number
• Catalan pseudoprime
• Elliptic pseudoprime
• Euler pseudoprime
• Euler–Jacobi pseudoprime
• Fermat pseudoprime
• Frobenius pseudoprime
• Lucas pseudoprime
• Lucas–Carmichael number
• Somer–Lucas pseudoprime
• Strong pseudoprime
Arithmetic functions and dynamics
Divisor functions
• Abundant
• Almost perfect
• Arithmetic
• Betrothed
• Colossally abundant
• Deficient
• Descartes
• Hemiperfect
• Highly abundant
• Highly composite
• Hyperperfect
• Multiply perfect
• Perfect
• Practical
• Primitive abundant
• Quasiperfect
• Refactorable
• Semiperfect
• Sublime
• Superabundant
• Superior highly composite
• Superperfect
Prime omega functions
• Almost prime
• Semiprime
Euler's totient function
• Highly cototient
• Highly totient
• Noncototient
• Nontotient
• Perfect totient
• Sparsely totient
Aliquot sequences
• Amicable
• Perfect
• Sociable
• Untouchable
Primorial
• Euclid
• Fortunate
Other prime factor or divisor related numbers
• Blum
• Cyclic
• Erdős–Nicolas
• Erdős–Woods
• Friendly
• Giuga
• Harmonic divisor
• Jordan–Pólya
• Lucas–Carmichael
• Pronic
• Regular
• Rough
• Smooth
• Sphenic
• Størmer
• Super-Poulet
• Zeisel
Numeral system-dependent numbers
Arithmetic functions
and dynamics
• Persistence
• Additive
• Multiplicative
Digit sum
• Digit sum
• Digital root
• Self
• Sum-product
Digit product
• Multiplicative digital root
• Sum-product
Coding-related
• Meertens
Other
• Dudeney
• Factorion
• Kaprekar
• Kaprekar's constant
• Keith
• Lychrel
• Narcissistic
• Perfect digit-to-digit invariant
• Perfect digital invariant
• Happy
P-adic numbers-related
• Automorphic
• Trimorphic
Digit-composition related
• Palindromic
• Pandigital
• Repdigit
• Repunit
• Self-descriptive
• Smarandache–Wellin
• Undulating
Digit-permutation related
• Cyclic
• Digit-reassembly
• Parasitic
• Primeval
• Transposable
Divisor-related
• Equidigital
• Extravagant
• Frugal
• Harshad
• Polydivisible
• Smith
• Vampire
Other
• Friedman
Binary numbers
• Evil
• Odious
• Pernicious
Generated via a sieve
• Lucky
• Prime
Sorting related
• Pancake number
• Sorting number
Natural language related
• Aronson's sequence
• Ban
Graphemics related
• Strobogrammatic
• Mathematics portal
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