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# Context-sensitive grammar
## Formal definition {#formal_definition}
### Formal grammar {#formal_grammar}
Let us notate a formal grammar as $G = (N, \Sigma, P, S)$, with $N$ a set of nonterminal symbols, $\Sigma$ a set of terminal symbols, $P$ a set of production rules, and $S \in N$ the start symbol.
A string $u \in (N \cup \Sigma)^*$ *directly yields*, or *directly derives to*, a string $v \in (N \cup \Sigma)^*$, denoted as $u \Rightarrow v$, if *v* can be obtained from *u* by an application of some production rule in *P*, that is, if $u = \gamma L \delta$ and $v = \gamma R \delta$, where $(L \to R) \in P$ is a production rule, and $\gamma, \delta \in (N \cup \Sigma)^*$ is the unaffected left and right part of the string, respectively. More generally, *u* is said to *yield*, or *derive to*, *v*, denoted as $u \Rightarrow^* v$, if *v* can be obtained from *u* by repeated application of production rules, that is, if $u = u_0 \Rightarrow ... \Rightarrow u_n = v$ for some *n* ≥ 0 and some strings $u_1, ..., u_{n-1} \in (N \cup \Sigma)^*$. In other words, the relation $\Rightarrow^*$ is the reflexive transitive closure of the relation $\Rightarrow$.
The **language** of the grammar *G* is the set of all terminal-symbol strings derivable from its start symbol, formally: $L(G) = \{ w \in \Sigma^* \mid S \Rightarrow^* w \}$. Derivations that do not end in a string composed of terminal symbols only are possible, but do not contribute to *L*(*G*).
### Context-sensitive grammar {#context_sensitive_grammar}
A formal grammar is **context-sensitive** if each rule in *P* is either of the form $S \to \varepsilon$ where $\varepsilon$ is the empty string, or of the form
: α*A*β → αγβ
with *A* ∈ *N*, $\alpha, \beta\in (N \cup \Sigma \setminus\{S\})^*$, and $\gamma\in (N \cup \Sigma \setminus\{S\})^+$.
The name *context-sensitive* is explained by the α and β that form the context of *A* and determine whether *A* can be replaced with γ or not. By contrast, in a context-free grammar, no context is present: the left hand side of every production rule is just a nonterminal.
The string γ is not allowed to be empty. Without this restriction, the resulting grammars become equal in power to unrestricted grammars.
### (Weakly) equivalent definitions {#weakly_equivalent_definitions}
A noncontracting grammar is a grammar in which for any production rule, of the form *u* → *v*, the length of *u* is less than or equal to the length of *v*.
Every context-sensitive grammar is noncontracting, while every noncontracting grammar can be converted into an equivalent context-sensitive grammar; the two classes are weakly equivalent.
Some authors use the term *context-sensitive grammar* to refer to noncontracting grammars in general.
The **left-context**- and **right-context**-sensitive grammars are defined by restricting the rules to just the form α*A* → αγ and to just *A*β → γβ, respectively. The languages generated by these grammars are also the full class of context-sensitive languages. The equivalence was established by Penttonen normal form.
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# Context-sensitive grammar
## Examples
### *a^n^b^n^c^n^*
The following context-sensitive grammar, with start symbol *S*, generates the canonical non-context-free language { *a^n^b^n^c^n^* \| *n* ≥ 1 } :
----------- ----- --------- ------- ----- ----- -----
1\. *S* → *a* *B* *C*
2\. *S* → *a* *S* *B*
3\. *C* *B* → *C* *Z*
4\. *C* *Z* → *W* *Z*
5\. *W* *Z* → *W* *C*
6\. *W* *C* → *B* *C*
7\. *a* *B* → *a* *b*
8\. *b* *B* → *b* *b*
9\. *b* *C* → *b* *c*
10\. *c* *C* → *c* *c*
----------- ----- --------- ------- ----- ----- -----
Rules 1 and 2 allow for blowing-up *S* to *a*^*n*^*BC*(*BC*)^*n*−1^; rules 3 to 6 allow for successively exchanging each *CB* to *BC* (four rules are needed for that since a rule *CB* → *BC* wouldn\'t fit into the scheme α*A*β → αγβ); rules 7--10 allow replacing a non-terminal *B* or *C* with its corresponding terminal *b* or *c*, respectively, provided it is in the right place. A generation chain for *`{{not a typo|aaabbbccc}}`{=mediawiki}* is:
: *S*
: →~2~ `{{not a typo|'''aSBC'''}}`{=mediawiki}
: →~2~ `{{not a typo|''a'''aSBC'''BC''}}`{=mediawiki}
: →~1~ `{{not a typo|''aa'''aBC'''BCBC''}}`{=mediawiki}
: →~3~ `{{not a typo|''aaaB'''CZ'''CBC''}}`{=mediawiki}
: →~4~ `{{not a typo|''aaaB'''WZ'''CBC''}}`{=mediawiki}
: →~5~ `{{not a typo|''aaaB'''WC'''CBC''}}`{=mediawiki}
: →~6~ `{{not a typo|''aaaB'''BC'''CBC''}}`{=mediawiki}
: →~3~ `{{not a typo|''aaaBBC'''CZ'''C''}}`{=mediawiki}
: →~4~ `{{not a typo|''aaaBBC'''WZ'''C''}}`{=mediawiki}
: →~5~ `{{not a typo|''aaaBBC'''WC'''C''}}`{=mediawiki}
: →~6~ `{{not a typo|''aaaBBC'''BC'''C''}}`{=mediawiki}
: →~3~ `{{not a typo|''aaaBB'''CZ'''CC''}}`{=mediawiki}
: →~4~ `{{not a typo|''aaaBB'''WZ'''CC''}}`{=mediawiki}
: →~5~ `{{not a typo|''aaaBB'''WC'''CC''}}`{=mediawiki}
: →~6~ `{{not a typo|''aaaBB'''BC'''CC''}}`{=mediawiki}
: →~7~ `{{not a typo|''aa'''ab'''BBCCC''}}`{=mediawiki}
: →~8~ `{{not a typo|''aaa'''bb'''BCCC''}}`{=mediawiki}
: →~8~ `{{not a typo|''aaab'''bb'''CCC''}}`{=mediawiki}
: →~9~ `{{not a typo|''aaabb'''bc'''CC''}}`{=mediawiki}
: →~10~ `{{not a typo|''aaabbb'''cc'''C''}}`{=mediawiki}
: →~10~ `{{not a typo|''aaabbbc'''cc'''''}}`{=mediawiki}
:
### *a^n^b^n^c^n^d^n^*, etc. {#anbncndn_etc.}
More complicated grammars can be used to parse { *a^n^b^n^c^n^d^n^* \| *n* ≥ 1 }, and other languages with even more letters. Here we show a simpler approach using non-contracting grammars: Start with a kernel of regular productions generating the sentential forms $(ABCD)^{n}abcd$ and then include the non contracting productions $p_{Da} : Da\rightarrow aD$, $p_{Db} : Db\rightarrow bD$, $p_{Dc} : Dc\rightarrow cD$, $p_{Dd} : Dd\rightarrow dd$, $p_{Ca} : Ca\rightarrow aC$, $p_{Cb} : Cb\rightarrow bC$, $p_{Cc} : Cc\rightarrow cc$, $p_{Ba} : Ba\rightarrow aB$, $p_{Bb} : Bb\rightarrow bb$, $p_{Aa} : Aa\rightarrow aa$.
### *a^m^b^n^c^m^d^n^*
A non contracting grammar (for which there is an equivalent CSG) for the language $L_{Cross} = \{ a^mb^nc^{m}d^{n} \mid m \ge 1, n \ge 1 \}$ is defined by
$$p_0 : S \rightarrow RT$$,
$$p_1 : R\rightarrow aRC | aC$$,
$$p_3 : T\rightarrow BTd | Bd$$,
$$p_5 : CB\rightarrow BC$$,
$$p_6 : aB\rightarrow ab$$,
$$p_7 : bB\rightarrow bb$$,
$$p_8 : Cd\rightarrow cd$$, and
$$p_9 : Cc\rightarrow cc$$.
With these definitions, a derivation for $a^3b^2c^3d^2$ is: $S
\Rightarrow_{p_0} RT
\Rightarrow_{p^{2}_{1}p_{2}} a^3C^3T
\Rightarrow_{p_{3}p_{4} } a^3C^3B^2d^2
\Rightarrow_{p^{6}_{5} } a^3B^2C^3d^2
\Rightarrow_{p_{6}p_{7} } a^3b^2C^3d^2
\Rightarrow_{p_{8}p^{2}_{9}} a^3b^2c^3d^2$.
### *a*^2^*i*^^
A noncontracting grammar for the language { *a*^2^*i*^^ \| *i* ≥ 1 } is constructed in Example 9.5 (p. 224) of (Hopcroft, Ullman, 1979):
1. $S\rightarrow [ACaB]$
2. \\begin{cases}
\\ \[Ca\]a\\rightarrow aa\[Ca\] \\\\ \\ \[Ca\]\[aB\]\\rightarrow aa\[CaB\] \\\\ \\ \[ACa\]a\\rightarrow \[Aa\]a\[Ca\] \\\\ \\ \[ACa\]\[aB\]\\rightarrow \[Aa\]a\[CaB\] \\\\ \\ \[ACaB\]\\rightarrow \[Aa\]\[aCB\] \\\\ \\ \[CaB\]\\rightarrow a\[aCB\] \\end{cases}
1. $[aCB]\rightarrow [aDB]$
2. $[aCB]\rightarrow [aE]$
3. \\begin{cases}
\\ a\[Da\]\\rightarrow \[Da\]a \\\\ \\ \[aDB\]\\rightarrow \[DaB\] \\\\ \\ \[Aa\]\[Da\]\\rightarrow \[ADa\]a \\\\ \\ a\[DaB\]\\rightarrow \[Da\]\[aB\] \\\\ \\ \[Aa\]\[DaB\]\\rightarrow \[ADa\]\[aB\] \\end{cases}
1. $[ADa]\rightarrow [ACa]$
2. \\begin{cases}
\\ a\[Ea\]\\rightarrow \[Ea\]a \\\\ \\ \[aE\]\\rightarrow \[Ea\] \\\\ \\ \[Aa\]\[Ea\]\\rightarrow \[AEa\]a \\end{cases}
1. $[AEa]\rightarrow a$
## Kuroda normal form {#kuroda_normal_form}
Every context-sensitive grammar which does not generate the empty string can be transformed into a weakly equivalent one in Kuroda normal form. \"Weakly equivalent\" here means that the two grammars generate the same language. The normal form will not in general be context-sensitive, but will be a noncontracting grammar.
The Kuroda normal form is an actual normal form for non-contracting grammars.
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# Context-sensitive grammar
## Properties and uses {#properties_and_uses}
### Equivalence to linear bounded automaton {#equivalence_to_linear_bounded_automaton}
A formal language can be described by a context-sensitive grammar if and only if it is accepted by some linear bounded automaton (LBA). In some textbooks this result is attributed solely to Landweber and Kuroda. Others call it the Myhill--Landweber--Kuroda theorem. (Myhill introduced the concept of deterministic LBA in 1960. Peter S. Landweber published in 1963 that the language accepted by a deterministic LBA is context sensitive. Kuroda introduced the notion of non-deterministic LBA and the equivalence between LBA and CSGs in 1964.)
it is still an open question whether every context-sensitive language can be accepted by a *deterministic* LBA.
### Closure properties {#closure_properties}
Context-sensitive languages are closed under complement. This 1988 result is known as the Immerman--Szelepcsényi theorem. Moreover, they are closed under union, intersection, concatenation, substitution, inverse homomorphism, and Kleene plus.
Every recursively enumerable language *L* can be written as *h*(*L*) for some context-sensitive language *L* and some string homomorphism *h*.
### Computational problems {#computational_problems}
The decision problem that asks whether a certain string *s* belongs to the language of a given context-sensitive grammar *G*, is PSPACE-complete. Moreover, there are context-sensitive grammars whose languages are PSPACE-complete. In other words, there is a context-sensitive grammar *G* such that deciding whether a certain string *s* belongs to the language of *G* is PSPACE-complete (so *G* is fixed and only *s* is part of the input of the problem).
The emptiness problem for context-sensitive grammars (given a context-sensitive grammar *G*, is *L*(*G*)=∅ ?) is undecidable.
### As model of natural languages {#as_model_of_natural_languages}
Savitch has proven the following theoretical result, on which he bases his criticism of CSGs as basis for natural language: for any recursively enumerable set *R*, there exists a context-sensitive language/grammar *G* which can be used as a sort of proxy to test membership in *R* in the following way: given a string *s*, *s* is in *R* if and only if there exists a positive integer *n* for which *sc^n^* is in G, where *c* is an arbitrary symbol not part of *R*.
It has been shown that nearly all natural languages may in general be characterized by context-sensitive grammars, but the whole class of CSGs seems to be much bigger than natural languages. Worse yet, since the aforementioned decision problem for CSGs is PSPACE-complete, that makes them totally unworkable for practical use, as a polynomial-time algorithm for a PSPACE-complete problem would imply P=NP.
It was proven that some natural languages are not context-free, based on identifying so-called cross-serial dependencies and unbounded scrambling phenomena. However this does not necessarily imply that the class of CSGs is necessary to capture \"context sensitivity\" in the colloquial sense of these terms in natural languages. For example, linear context-free rewriting systems (LCFRSs) are strictly weaker than CSGs but can account for the phenomenon of cross-serial dependencies; one can write a LCFRS grammar for {*a^n^b^n^c^n^d^n^* \| *n* ≥ 1} for example.
Ongoing research on computational linguistics has focused on formulating other classes of languages that are \"mildly context-sensitive\" whose decision problems are feasible, such as tree-adjoining grammars, combinatory categorial grammars, coupled context-free languages, and linear context-free rewriting systems. The languages generated by these formalisms properly lie between the context-free and context-sensitive languages.
More recently, the class PTIME has been identified with range concatenation grammars, which are now considered to be the most expressive of the mild-context sensitive language classes
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# Connected space
In topology and related branches of mathematics, a **connected space** is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces.
A subset of a topological space $X$ is a `{{visible anchor|connected set}}`{=mediawiki} if it is a connected space when viewed as a subspace of $X$.
Some related but stronger conditions are path connected, simply connected, and $n$-connected. Another related notion is locally connected, which neither implies nor follows from connectedness.
## Formal definition {#formal_definition}
A topological space $X$ is said to be `{{visible anchor|disconnected}}`{=mediawiki} if it is the union of two disjoint non-empty open sets. Otherwise, $X$ is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice.
For a topological space $X$ the following conditions are equivalent:
1. $X$ is connected, that is, it cannot be divided into two disjoint non-empty open sets.
2. The only subsets of $X$ which are both open and closed (clopen sets) are $X$ and the empty set.
3. The only subsets of $X$ with empty boundary are $X$ and the empty set.
4. $X$ cannot be written as the union of two non-empty separated sets (sets for which each is disjoint from the other\'s closure).
5. All continuous functions from $X$ to $\{ 0, 1 \}$ are constant, where $\{ 0, 1 \}$ is the two-point space endowed with the discrete topology.
Historically this modern formulation of the notion of connectedness (in terms of no partition of $X$ into two separated sets) first appeared (independently) with N.J. Lennes, Frigyes Riesz, and Felix Hausdorff at the beginning of the 20th century. See `{{harv|Wilder|1978}}`{=mediawiki} for details.
### Connected components {#connected_components}
Given some point $x$ in a topological space $X,$ the union of any collection of connected subsets such that each contains $x$ will once again be a connected subset. The connected component of a point $x$ in $X$ is the union of all connected subsets of $X$ that contain $x;$ it is the unique largest (with respect to $\subseteq$) connected subset of $X$ that contains $x.$ The maximal connected subsets (ordered by inclusion $\subseteq$) of a non-empty topological space are called the connected components of the space. The components of any topological space $X$ form a partition of $X$: they are disjoint, non-empty and their union is the whole space. Every component is a closed subset of the original space. It follows that, in the case where their number is finite, each component is also an open subset. However, if their number is infinite, this might not be the case; for instance, the connected components of the set of the rational numbers are the one-point sets (singletons), which are not open. Proof: Any two distinct rational numbers $q_1<q_2$ are in different components. Take an irrational number $q_1 < r < q_2,$ and then set $A = \{q \in \Q : q < r\}$ and $B = \{q \in \Q : q > r\}.$ Then $(A,B)$ is a separation of $\Q,$ and $q_1 \in A, q_2 \in B$. Thus each component is a one-point set.
Let $\Gamma_x$ be the connected component of $x$ in a topological space $X,$ and $\Gamma_x'$ be the intersection of all clopen sets containing $x$ (called quasi-component of $x$). Then $\Gamma_x \subset \Gamma'_x$ where the equality holds if $X$ is compact Hausdorff or locally connected.
### Disconnected spaces {#disconnected_spaces}
A space in which all components are one-point sets is called `{{visible anchor|totally disconnected}}`{=mediawiki}. Related to this property, a space $X$ is called `{{visible anchor|totally separated}}`{=mediawiki} if, for any two distinct elements $x$ and $y$ of $X$, there exist disjoint open sets $U$ containing $x$ and $V$ containing $y$ such that $X$ is the union of $U$ and $V$. Clearly, any totally separated space is totally disconnected, but the converse does not hold. For example, take two copies of the rational numbers $\Q$, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdorff, and the condition of being totally separated is strictly stronger than the condition of being Hausdorff.
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# Connected space
## Examples
- The closed interval $[0, 2)$ in the standard subspace topology is connected; although it can, for example, be written as the union of $[0, 1)$ and $[1, 2),$ the second set is not open in the chosen topology of $[0, 2).$
- The union of $[0, 1)$ and $(1, 2]$ is disconnected; both of these intervals are open in the standard topological space $[0, 1) \cup (1, 2].$
- $(0, 1) \cup \{ 3 \}$ is disconnected.
- A convex subset of $\R^n$ is connected; it is actually simply connected.
- A Euclidean plane excluding the origin, $(0, 0),$ is connected, but is not simply connected. The three-dimensional Euclidean space without the origin is connected, and even simply connected. In contrast, the one-dimensional Euclidean space without the origin is not connected.
- A Euclidean plane with a straight line removed is not connected since it consists of two half-planes.
- $\R$, the space of real numbers with the usual topology, is connected.
- The Sorgenfrey line is disconnected.
- If even a single point is removed from $\mathbb{R}$, the remainder is disconnected. However, if even a countable infinity of points are removed from $\R^n$, where $n \geq 2,$ the remainder is connected. If $n\geq 3$, then $\R^n$ remains simply connected after removal of countably many points.
- Any topological vector space, e.g. any Hilbert space or Banach space, over a connected field (such as $\R$ or $\Complex$), is simply connected.
- Every discrete topological space with at least two elements is disconnected, in fact such a space is totally disconnected. The simplest example is the discrete two-point space.
- On the other hand, a finite set might be connected. For example, the spectrum of a discrete valuation ring consists of two points and is connected. It is an example of a Sierpiński space.
- The Cantor set is totally disconnected; since the set contains uncountably many points, it has uncountably many components.
- If a space $X$ is homotopy equivalent to a connected space, then $X$ is itself connected.
- The topologist\'s sine curve is an example of a set that is connected but is neither path connected nor locally connected.
- The general linear group $\operatorname{GL}(n, \R)$ (that is, the group of $n$-by-$n$ real, invertible matrices) consists of two connected components: the one with matrices of positive determinant and the other of negative determinant. In particular, it is not connected. In contrast, $\operatorname{GL}(n, \Complex)$ is connected. More generally, the set of invertible bounded operators on a complex Hilbert space is connected.
- The spectra of commutative local ring and integral domains are connected. More generally, the following are equivalent
1. The spectrum of a commutative ring $R$ is connected
2. Every finitely generated projective module over $R$ has constant rank.
3. $R$ has no idempotent $\ne 0, 1$ (i.e., $R$ is not a product of two rings in a nontrivial way).
An example of a space that is not connected is a plane with an infinite line deleted from it. Other examples of disconnected spaces (that is, spaces which are not connected) include the plane with an annulus removed, as well as the union of two disjoint closed disks, where all examples of this paragraph bear the subspace topology induced by two-dimensional Euclidean space.
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# Connected space
## Path connectedness {#path_connectedness}
A `{{visible anchor|path-connected space}}`{=mediawiki} is a stronger notion of connectedness, requiring the structure of a path. A path from a point $x$ to a point $y$ in a topological space $X$ is a continuous function $f$ from the unit interval $[0,1]$ to $X$ with $f(0)=x$ and $f(1)=y$. A `{{visible anchor|path-component}}`{=mediawiki} of $X$ is an equivalence class of $X$ under the equivalence relation which makes $x$ equivalent to $y$ if and only if there is a path from $x$ to $y$. The space $X$ is said to be path-connected (or pathwise connected or $\mathbf{0}$-connected) if there is exactly one path-component. For non-empty spaces, this is equivalent to the statement that there is a path joining any two points in $X$. Again, many authors exclude the empty space.
Every path-connected space is connected. The converse is not always true: examples of connected spaces that are not path-connected include the extended long line $L^*$ and the topologist\'s sine curve.
Subsets of the real line $\R$ are connected if and only if they are path-connected; these subsets are the intervals and rays of $\R$. Also, open subsets of $\R^n$ or $\C^n$ are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same for finite topological spaces.
## Arc connectedness {#arc_connectedness}
A space $X$ is said to be arc-connected or arcwise connected if any two topologically distinguishable points can be joined by an arc, which is an embedding $f : [0, 1] \to X$. An arc-component of $X$ is a maximal arc-connected subset of $X$; or equivalently an equivalence class of the equivalence relation of whether two points can be joined by an arc or by a path whose points are topologically indistinguishable.
Every Hausdorff space that is path-connected is also arc-connected; more generally this is true for a $\Delta$-Hausdorff space, which is a space where each image of a path is closed. An example of a space which is path-connected but not arc-connected is given by the line with two origins; its two copies of $0$ can be connected by a path but not by an arc.
Intuition for path-connected spaces does not readily transfer to arc-connected spaces. Let $X$ be the line with two origins. The following are facts whose analogues hold for path-connected spaces, but do not hold for arc-connected spaces:
- Continuous image of arc-connected space may not be arc-connected: for example, a quotient map from an arc-connected space to its quotient with countably many (at least 2) topologically distinguishable points cannot be arc-connected due to too small cardinality.
- Arc-components may not be disjoint. For example, $X$ has two overlapping arc-components.
- Arc-connected product space may not be a product of arc-connected spaces. For example, $X \times \mathbb{R}$ is arc-connected, but $X$ is not.
- Arc-components of a product space may not be products of arc-components of the marginal spaces. For example, $X \times \mathbb{R}$ has a single arc-component, but $X$ has two arc-components.
- If arc-connected subsets have a non-empty intersection, then their union may not be arc-connected. For example, the arc-components of $X$ intersect, but their union is not arc-connected.
## Local connectedness {#local_connectedness}
A topological space is said to be locally connected at a point $x$ if every neighbourhood of $x$ contains a connected open neighbourhood. It is locally connected if it has a base of connected sets. It can be shown that a space $X$ is locally connected if and only if every component of every open set of $X$ is open.
Similarly, a topological space is said to be `{{visible anchor|locally path-connected}}`{=mediawiki} if it has a base of path-connected sets. An open subset of a locally path-connected space is connected if and only if it is path-connected. This generalizes the earlier statement about $\R^n$ and $\C^n$, each of which is locally path-connected. More generally, any topological manifold is locally path-connected. Locally connected does not imply connected, nor does locally path-connected imply path connected. A simple example of a locally connected (and locally path-connected) space that is not connected (or path-connected) is the union of two separated intervals in $\R$, such as $(0,1) \cup (2,3)$.
A classic example of a connected space that is not locally connected is the so-called topologist\'s sine curve, defined as $T = \{(0,0)\} \cup \left\{ \left(x, \sin\left(\tfrac{1}{x}\right)\right) : x \in (0, 1] \right\}$, with the Euclidean topology induced by inclusion in $\R^2$.
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# Connected space
## Set operations {#set_operations}
The intersection of connected sets is not necessarily connected.
The union of connected sets is not necessarily connected, as can be seen by considering $X=(0,1) \cup (1,2)$.
Each ellipse is a connected set, but the union is not connected, since it can be partitioned into two disjoint open sets $U$ and $V$.
This means that, if the union $X$ is disconnected, then the collection $\{X_i\}$ can be partitioned into two sub-collections, such that the unions of the sub-collections are disjoint and open in $X$ (see picture). This implies that in several cases, a union of connected sets `{{em|is}}`{=mediawiki} necessarily connected. In particular:
1. If the common intersection of all sets is not empty ($\bigcap X_i \neq \emptyset$), then obviously they cannot be partitioned to collections with disjoint unions. Hence the union of connected sets with non-empty intersection is connected.
2. If the intersection of each pair of sets is not empty ($\forall i,j: X_i \cap X_j \neq \emptyset$) then again they cannot be partitioned to collections with disjoint unions, so their union must be connected.
3. If the sets can be ordered as a \"linked chain\", i.e. indexed by integer indices and $\forall i: X_i \cap X_{i+1} \neq \emptyset$, then again their union must be connected.
4. If the sets are pairwise-disjoint and the quotient space $X / \{X_i\}$ is connected, then `{{mvar|X}}`{=mediawiki} must be connected. Otherwise, if $U \cup V$ is a separation of `{{mvar|X}}`{=mediawiki} then $q(U) \cup q(V)$ is a separation of the quotient space (since $q(U), q(V)$ are disjoint and open in the quotient space).
The set difference of connected sets is not necessarily connected. However, if $X \supseteq Y$ and their difference $X \setminus Y$ is disconnected (and thus can be written as a union of two open sets $X_1$ and $X_2$), then the union of $Y$ with each such component is connected (i.e. $Y \cup X_{i}$ is connected for all $i$).
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# Connected space
## Theorems
- **Main theorem of connectedness** : Let $X$ and $Y$ be topological spaces and let $f:X\rightarrow Y$ be a continuous function. If $X$ is (path-)connected then the image $f(X)$ is (path-)connected. This result can be considered a generalization of the intermediate value theorem.
- Every path-connected space is connected.
- In a locally path-connected space, every open connected set is path-connected.
- Every locally path-connected space is locally connected.
- A locally path-connected space is path-connected if and only if it is connected.
- The closure of a connected subset is connected. Furthermore, any subset between a connected subset and its closure is connected.
- The connected components are always closed (but in general not open)
- The connected components of a locally connected space are also open.
- The connected components of a space are disjoint unions of the path-connected components (which in general are neither open nor closed).
- Every quotient of a connected (resp. locally connected, path-connected, locally path-connected) space is connected (resp. locally connected, path-connected, locally path-connected).
- Every product of a family of connected (resp. path-connected) spaces is connected (resp. path-connected).
- Every open subset of a locally connected (resp. locally path-connected) space is locally connected (resp. locally path-connected).
- Every manifold is locally path-connected.
- Arc-wise connected space is path connected, but path-wise connected space may not be arc-wise connected
- Continuous image of arc-wise connected set is arc-wise connected.
## Graphs
Graphs have path connected subsets, namely those subsets for which every pair of points has a path of edges joining them. However, it is not always possible to find a topology on the set of points which induces the same connected sets. The 5-cycle graph (and any $n$-cycle with $n>3$ odd) is one such example.
As a consequence, a notion of connectedness can be formulated independently of the topology on a space. To wit, there is a category of connective spaces consisting of sets with collections of connected subsets satisfying connectivity axioms; their morphisms are those functions which map connected sets to connected sets `{{harv|Muscat|Buhagiar|2006}}`{=mediawiki}. Topological spaces and graphs are special cases of connective spaces; indeed, the finite connective spaces are precisely the finite graphs.
However, every graph can be canonically made into a topological space, by treating vertices as points and edges as copies of the unit interval (see topological graph theory#Graphs as topological spaces). Then one can show that the graph is connected (in the graph theoretical sense) if and only if it is connected as a topological space.
## Stronger forms of connectedness {#stronger_forms_of_connectedness}
There are stronger forms of connectedness for topological spaces, for instance:
- If there exist no two disjoint non-empty open sets in a topological space $X$, $X$ must be connected, and thus hyperconnected spaces are also connected.
- Since a simply connected space is, by definition, also required to be path connected, any simply connected space is also connected. If the \"path connectedness\" requirement is dropped from the definition of simple connectivity, a simply connected space does not need to be connected.
- Yet stronger versions of connectivity include the notion of a contractible space. Every contractible space is path connected and thus also connected.
In general, any path connected space must be connected but there exist connected spaces that are not path connected. The deleted comb space furnishes such an example, as does the above-mentioned topologist\'s sine curve
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# Condensation polymer
In polymer chemistry, **condensation polymers** are any kind of polymers whose process of polymerization involves a condensation reaction (i.e. a small molecule, such as water or methanol, is produced as a byproduct). Natural proteins as well as some common plastics such as nylon and PETE are formed in this way. Condensation polymers are formed by polycondensation, when the polymer is formed by condensation reactions between species of all degrees of polymerization, or by condensative chain polymerization, when the polymer is formed by sequential addition of monomers to an active site in a chain reaction. The main alternative forms of polymerization are chain polymerization and polyaddition, both of which give addition polymers.
Condensation polymerization is a form of step-growth polymerization. Linear polymers are produced from bifunctional monomers, i.e. compounds with two reactive end-groups. Common condensation polymers include polyesters, polyamides such as nylon, polyacetals, and proteins.
## Polyamides
One important class of condensation polymers are polyamides. They arise from the reaction of carboxylic acid and an amine. Examples include nylons and proteins. When prepared from amino-carboxylic acids, e.g. amino acids, the stoichiometry of the polymerization includes co-formation of water:
: n H~2~N-X-CO~2~H → \[HN-X-C(O)\]~n~ + (n-1) H~2~O
When prepared from diamines and dicarboxylic acids, e.g. the production of nylon 66, the polymerization produces two molecules of water per repeat unit:
: n H~2~N-X-NH~2~ + n HO~2~C-Y-CO~2~H → \[HN-X-NHC(O)-Y-C(O)\]~n~ + (2n-1) H~2~O
:
## Polyesters
Another important class of condensation polymers are polyesters. They arise from the reaction of a carboxylic acid and an alcohol. An example is polyethyleneterephthalate, the common plastic PET (recycling #1 in the USA):
: n HO-X-OH + n HO~2~C-Y-CO~2~H → \[O-X-O~2~C-Y-C(O)\]~n~ + (2n-1) H~2~O
## Safety and environmental considerations {#safety_and_environmental_considerations}
Condensation polymers tend to be more biodegradable than addition polymers. The peptide or ester bonds between monomers can be hydrolysed, especially in the presence of catalysts or bacterial enzymes
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# Timeline of computing
**Timeline of computing** presents events in the history of computing organized by year and grouped into six topic areas: predictions and concepts, first use and inventions, hardware systems and processors, operating systems, programming languages, and new application areas
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# List of cartoonists
This is a **list of cartoonists**, visual artists who specialize in drawing cartoons. This list includes only notable cartoonists and is not meant to be exhaustive. Note that the word \'cartoon\' only took on its modern sense after its use in Punch magazine in the 1840s - artists working earlier than that are more correctly termed \'caricaturists\',
## Notable cartoonists {#notable_cartoonists}
- Scott Adams, *Dilbert*
- Charles Addams (1938--1988), macabre cartoons featured in *The New Yorker* and elsewhere
- Attila Adorjany
- Sarah Andersen, known for *Sarah\'s Scribbles*
- Barry Appleby
- Dan Piraro
- Sergio Aragonés, known for his contributions to *Mad*
- Graciela Aranis (1908--1996), Chilean painter, cartoonist
- Peter Arno (1904--1968), cartoons featured in *The New Yorker* and elsewhere
- Arotxa (Rodolfo Arotxarena)
- Jim Bamber, cartoonist of *Autosport*, magazine specialising in motor sports
- Edgar Henry Banger
- Carl Barks, inventor of *Duckburg* and many of its characters like *Scrooge McDuck* and *Gladstone Gander*; Fantagraphics Books called him \"the Hans Christian Andersen of comic books.\"
- Sumanta Baruah
- Aminollah Rezaei
- Niko Barun
- Nancy Beiman, \"FurBabies\"
- Darrin Bell, *Candorville* and *Rudy Park*
- Steve Bell, *The Guardian* (UK)
- Stephen Bentley, \"Herb and Jamaal\"
- Jim Benton, known for his cartoons on reddit, GoComics, and Instagram as well as *It\'s Happy Bunny*,*Dear Dumb Diary*,*Franny K Stein*,*Catwad*,*Batman Squad*,*Attack of the Stuff*
- Oscar Berger, *Aesop\'s Foibles (1947)*; active 1920s--1960s
- Mark Beyer, *Amy and Jordan*, *Agony*
- Brumsic Brandon Jr., \"Luther\"; with his daughter Barbara Brandon-Croft, first family of cartoonists (father/daughter) to each be nationally syndicated in the U.S. mainstream press
- Barbara Brandon-Croft, \"Where I\'m Coming From\"; first Black woman cartoonist to be nationally syndicated in the U.S. mainstream press
- Berkeley Breathed, *Bloom County* and *Outland*
- Frédéric-Antonin Breysse
- Ed Brubaker
- Henry Bunbury 18th Century British caricaturist
- Tom Bunk, cartoonist for *Mad*
- Stanley Burnside, *Sideburns*
- Mark Burrier
- John Byrne
- Al Capp, *Li\'l Abner*
- Tom Cheney, staff cartoonist for *The New Yorker*
- Edgar Church
- Chester Commodore, political cartoonist
- George Cruikshank 19th Century British caricaturist
- Isaac Cruikshank 18th Century British caricaturist
- Isaac Robert Cruikshank 19th Century British caricaturist
- Robert Crumb, *Mr. Natural*, *Fritz the Cat*, *Keep on Truckin\'*
- Natalie d\'Arbeloff
- Jack Davis
- Jim Davis, *Garfield*
- Abner Dean
- Arifur Rahman
- Narayan Debnath, Indian cartoonist known for *Handa Bhonda*, *Bantul the Great*, and *Nonte Phonte*
- Richard Decker, *The New Yorker*
- Walt Disney, *Mickey Mouse*, *Donald Duck*
- Ralph Waddell Douglass
- Stan Drake
- George du Maurier, also the author of Trilby
- Robert W. Edgren, American political cartoonist known for his \"Sketches from Death\" from the Spanish--American War
- Will Eisner, *The Spirit*
- Otto Eppers
- Charles Evenden
- Lyonel Feininger, rare fine artist who did strips, *The Kin-der-Kids* and *Wee Willie Winkie\'s World*
- Rod Filbrandt
- David Fletcher
- Ellen Forney
- André François
- André Franquin, *Spirou et Fantasio*, *Gaston Lagaffe*, *Marsupilami*
- Yuliy Abramovich Ganf, Soviet Russian
- Eddie Germano
- Denis Gifford, strips in *Whizzer and Chips*, *Knockout*, *Marvelman*
- Carl Giles
- James Gillray, 18th century British, called \"the father of the political cartoon\".
- John Glashan, *Genius*
- Rube Goldberg, cartoons of complex and convoluted machines doing very simple tasks.
- Larry Gonick, *The Cartoon History of the Universe series, Kokopelli & Company*
- Cleven \"Goodie\" Goudeau, known for his pioneering Afrocentric images on greeting cards
- Jimmy Gownley, *Amelia Rules! series, Simon & Schuster*
- Bud Grace, \"Ernie/Piranha Club\"
- Mel Graff, "The Adventures of Patsy", "Secret Agent X-9"
- Matt Groening, *Life in Hell*, *The Simpsons*, *Futurama*
- Sam Gross, for his *The New Yorker* work, plus many other magazines
- Shekhar Gurera, well known for his quirky cartoons about India\'s political and social trends
- William Haefeli
- Martin Handford, *Where\'s Wally?*
- Steven Harris
- Butch Hartman, *The Fairly OddParents*, *T.U.F.F. Puppy*, *Danny Phantom*, *Bunsen Is a Beast*
- Andrew Kennaway Henderson
- Henfil, Brazilian cartoonist
- Hergé, *The Adventures of Tintin*
- George Herriman, *Krazy Kat*
- Herblock American cartoonist
- Watson Heston
- Stephen Hillenburg (1961--2018), *SpongeBob SquarePants*
- Bill Hinds, \"Tank McNamara\"
- Dick Hodgins, Jr.
- William Hogarth, English pictorial satirist and editorial cartoonist; credited with pioneering western sequential art; work ranged from realistic portraiture to comic strip
- Bill Holbrook, *On the Fastrack*, *Safe Havens*, and *Kevin and Kell*
- Nicole Hollander, *Sylvia*
- John Holmstrom
- Geoff \"Jeff\" Hook, Australian
- George William Houghton, British golf cartoonist
- Jim Hummel
- Edgar Pierre Jacobs, *Blake and Mortimer*
- Al Jaffee, *Mad*
- Kirk Jarvinen
- S. Jithesh, World\'s Fastest Performing Cartoonist
- Herbert Johnson
- Mike Judge, *Beavis and Butt-head*, *King of the Hill*, *The Goode Family*
- Arja Kajermo
- Avi Katz
- Bil Keane, \"Family Circus\"
- Jeff Keane. \"Family Circus\"
- Walt Kelly, *Pogo*
- Rik Kemp
- Molly Kiely
- Wyncie King
- Jeff Kinney, *Diary of a Wimpy Kid*
- Rick Kirkman, \"Baby Blues\"
- Heinrich Kley
- B. Kliban
- John Kricfalusi, *The Ren & Stimpy Show*
- Abril Lamarque
- Gary Larson, *The Far Side*
- Rick Law, *Beyond the Veil*
- R K Laxman, cartoonist for *The Times of India*, India
- Mell Lazarus. \"Momma, Miss Peach\"
- John Leech, 19th-century *Punch* cartoonist
- Jonathan Lemon, *Alley Oop*
- Michael Leunig, Australian
- Arnold Levin
- David Liljemark
- Neil Lonsdale (1907--1989), New Zealand editorial cartoonist
- David Low, New Zealand political cartoonist and caricaturist
- Jay Lynch
- Trey Parker and Matt Stone, *South Park*
- Seth MacFarlane, *Family Guy*, *American Dad!*, *The Cleveland Show*
- Manjul, *India Today*, *The Economic Times* and *Daily News and Analysis*
- Bob Mankoff, *The New Yorker*
- Jack Markow
- Don Martin, \"Mad\"
- Enrico Mazzanti
- Scott McCloud, *Zot!*, *Understanding Comics*
- Aaron McGruder, *The Boondocks*
- Ronald Michaud
- Yevgeniy Migunov
- Mario Miranda, *The Economic Times*, India
- Shigeru Mizuki, *Ge Ge Ge no Kitaro*, master of horror of Japanese manga
- Guillermo Mordillo
- Arthur Moreland
- Lorin Morgan-Richards
- Morris, *Lucky Luke*
- Joe Murray, *Rocko\'s Modern Life* and *Camp Lazlo!*
- Rachel Nabors
- Ogden Nash
- Nigar Nazar, first female cartoonist of the Muslim world, creator of cartoon character \"Gogi\"
- Roy Nelson
- Richard Newton, 18th century British caricaturist
- Mana Neyestani, Iranian cartoonist
- Ajit Ninan, *India Today* and *The Times of India*
- Floyd Norman
- Murray Olderman, sports columnist, author of 14 books, National Cartoonist Society Sports Cartoon Award for 1974 and 1978
- Jack Edward Oliver
- Jackie Ormes, \"Torchy Brown in \'Dixie to Harlem\", \"Candy\", \"Patty-Jo \'n\' Ginger\", \"Torchy in \'Hearbeats\'\"; first Black woman cartoonist to be published nationally in the U.S. (not via syndication)
- Bruce Ozella
- Paul Palnik, American Jewish cartoonist
- Gary Panter
- Virgil Franklin Partch, known as \"VIP;\" leading American gag cartoonist from the 1940s to the 1980s
- Alan Stuart Paterson, New Zealand cartoonist
- Andrea Pazienza
- René Pellos, French cartoonist
- Bob Penuelas, *Wilbur Kookmeyer*
- Camillus Perera
- Bruce Petty
- Peyo, *The Smurfs*, *Steven Strong*, *Johan and Peewit*
- S. D. Phadnis, Indian cartoonist
- Ziraldo Alves Pinto, Brazilian cartoonist
- Hugo Pratt, *Corto Maltese*
- Ken Pyne
- Quino (Joaquín Salvador Lavado), Argentine cartoonist and social satirist, known for *Mafalda*
- Jacki Randall
- Roy Raymonde, 20th Century English cartoonist whose work appeared principally in Punch (magazine) and Playboy
- Bob Rich, American award-winning cartoonist
- Danny Antonucci, Canadian animator famous for the animated show *Ed, Edd n Eddy*
- Pat Ventura, cartoonist who created shorts for *Nickelodeon* and *Cartoon Network*
- Maxwell Atoms, the creator of *The Grim Adventures of Billy & Mandy*
- W. Heath Robinson, British satirist known for drawings of convoluted machines, similar to Rube Goldberg
- Christine Roche
- Artie Romero
- Ed \"Big Daddy\" Roth
- Thomas Rowlandson 18th Century British caricaturist
- Martin Rowson British political cartoonist
- Øystein Runde
- Malik Sajad Indian cartoonist, author of graphic novel *Munnu - A Boy from Kashmir*\'
- Armando Salas
- Gerald Scarfe ( political)
- Jerry Scott, \"Baby Blues, Zits\"
- Ronald Searle, St Trinians, Molesworth, *The Rake\'s Progress*, editorial work
- Elzie Crisler Segar, *Popeye*
- Sempé
- Claude Serre
- James Affleck Shepherd
- Lee Sheppard
- Gilbert Shelton
- Mahmoud Shokraye
- Shel Silverstein
- Posy Simmonds, *The Silent Three of St Botolph\'s*, *Gemma Bovery*
- Siné
- Vishavjit Singh
- Jeff Smith, *Bone*, *RASL*, *Shazam!: The Monster Society of Evil*, *Little Mouse Gets Ready*
- Mauricio de Sousa, *Monica\'s Gang*, *Chuck Billy \'n\' Folks*, *The Cavern Clan*
- Art Spiegelman, author of *Maus*; co-editor of *RAW* magazine
- Dan Spiegle
- George Sprod, *Punch* and other publications
- Ralph Steadman, editorial cartoonist and book illustrator
- Ralph Stein
- Saul Steinberg
- Jay Stephens
- Matt Stone, with Trey Parker, co-creator of *South Park*
- Jakob Martin Strid
- Ed Subitzky, known for his *National Lampoon* work, also *The New York Times*
- Joost Swarte, Dutch comic artist known for his ligne claire or clear line style of drawing
- Betty Swords
- Les Tanner, political cartoonist
- Howard Tayler, pioneered web-cartooning as a profession
- Raina Telgemeier
- Osamu Tezuka, *Astro Boy*, *Phoenix*; known as the \"God\" of Japanese manga who defined modern Japanese cartooning
- Bal Thackeray, formed a political party in India
- Lefred Thouron
- Morrie Turner, credited with the first multicultural syndicated cartoon strip
- Albert Uderzo, *Asterix*
- Jim Unger, Canadian cartoonist: *Herman*
- Willy Vandersteen, *Spike and Suzy*, *De Rode Ridder*
- Joan Vizcarra
- Vicco von Bülow, *Loriot*
- Keith Waite, New Zealand-born English editorial cartoonist
- Mort Walker, *Beetle Bailey*, *Hi and Lois*
- Arthur Watts
- Ben Wicks, Canadian cartoonist and illustrator: *The Outsider*, *Wicks*
- S. Clay Wilson, *Zap Comix*, *Underground Comix*
- Shannon Wright
- Rhie Won-bok
- Bianca Xunise, \"Six Chix\"; first nonbinary cartoonist to be nationally syndicated in the U.S. mainstream press
- Art Young
- José Zabala-Santos
- Zapiro
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# List of cartoonists
## Cartoonists of comic strips {#cartoonists_of_comic_strips}
- Scott Adams, *Dilbert*
- Alex Akerbladh
- Bill Amend, *FoxTrot*
- George Baker, *Sad Sack*
- Tom Batiuk, *Funky Winkerbean*
- Murray Ball, *Footrot Flats*
- Darrin Bell, *Candorville*, *Rudy Park*
- Stephen Bentley, \"Herb and Jamaal\"
- Jerry Bittle
- Boulet, pseudonym of French cartoonist Gilles Roussel
- Brumsic Brandon Jr., \"Luther\"
- Barbara Brandon-Croft, \"Where I\'m Coming From\"
- Berkeley Breathed, *Bloom County* (1980s American social-political), *Outland*, *Opus*
- Dave Breger, *Mister Breger*
- Dik Browne, *Hi and Lois*, *Hägar the Horrible*
- Ernie Bushmiller, *Nancy*
- Milton Caniff, *Terry and the Pirates*, *Steve Canyon*
- Al Capp, *Li\'l Abner*
- Ad Carter, *Just Kids*
- Jok Church, *You Can With Beakman and Jax*
- Francis Cleetus, *It\'s Geek 2 Me*
- Mitch Clem, *Nothing Nice to Say*, *San Antonio Rock City*
- Darby Conley, *Get Fuzzy*
- Joan Cornellà
- Dave Coverly, *Speed Bump*
- Max Crivello
- Alex Raymond, *Flash Gordon*, *Jungle Jim*, *Rip Kirby*
- Stan Cross, *The Potts*
- Stacy Curtis, *Cul de Sac*
- Lyman Dally, *Max Rep*
- Harry Grant Dart
- Lou Darvas
- Jim Davis, *Gnorm Gnat*, *Garfield*, *U.S. Acres*, a *Mr. Potato Head* comic strip
- Reginald Ben Davis
- Derf Backderf (John Backderf)
- Brad Diller
- J. C. Duffy, *The Fusco Brothers*
- Edwina Dumm
- Frank Dunne
- Benita Epstein, *Six Chix*
- Larry Feign, *The World of Lily Wong*
- Norm Feuti, *Retail*
- George Fett, *Sniffy and Norbert*
- Charles Fincher, creator of *Thadeus & Weez* and *The Daily Scribble*
- Bud Fisher, *Mutt and Jeff*
- Ham Fisher, *Joe Palooka*
- Evelyn Flinders, *The Silent Three*
- Harold Rudolf Foster, *Prince Valiant* and *Tarzan*
- J.D. Frazer, *User Friendly*
- David Füleki, *78 Tage auf der Straße des Hasses*
- Paul Gilligan, *Pooch Cafe*
- Erich von Götha de la Rosière
- Chester Gould, *Dick Tracy*
- Bud Grace, \"Ernie/Piranha Club\", \"Babs and Aldo\"
- Mel Graff, "The Adventures of Patsy", "Secret Agent X-9"
- Bill Griffith, *Zippy the Pinhead*
- Milt Gross
- Cathy Guisewite, *Cathy*
- Nicholas Gurewitch, *Perry Bible Fellowship*
- Alex Hallatt
- Johnny Hart, *B.C.*, *The Wizard of Id*
- Bill Hinds, *Tank McNamara*, *Cleats*, *Buzz Beamer*
- Bill Holman, *Smokey Stover*
- Daniel Hulet
- Billy Ireland
- Tatsuya Ishida, *Sinfest*
- Tove and Lars Jansson, *The Moomins*
- Ferd Johnson, *Moon Mullins*
- Kerry G. Johnson, *Harambee Hills*, caricaturist and children\'s book illustrator
- Russell Johnson, *Mister Oswald*
- Lynn Johnston, *For Better or For Worse*
- Eric Jolliffe, *Andy*
- Bil Keane, *Family Circus*
- Jeff Keane, *Family Circus*
- Walt Kelly, *Pogo*
- James Kemsley, *Ginger Meggs*
- Hank Ketcham, *Dennis the Menace*
- Kazu Kibuishi, *Copper*
- Frank King, *Gasoline Alley*
- Rick Kirkman, \"Baby Blues\"
- Keith Knight, *The K Kronicles*
- Charles Kuhn, *Grandma*
- Fred Lasswell, *Barney Google*
- Mell Lazarus, \"Momma, Miss Peach\"
- Virginio Livraghi
- Les Lumsdon, \"Basil\", \"Nipper\", \"Caspar\"
- Edgar Martin
- Clifford McBride, *Napoleon*
- Winsor McCay, *Little Nemo*
- Patrick McDonnell, *Mutts*
- Brian McFadden, *Big Fat Whale*
- Aaron McGruder, creator of the controversial strip *The Boondocks*
- George McManus, *Bringing Up Father*
- Caesar Meadows
- Dale Messick, *Brenda Starr*
- Tim Molloy
- Bill Murray, *Sonny Boy*
- Fred Negro, *Pub Strip*
- Chris Onstad, *Achewood*
- Jackie Ormes, \"Torchy Brown in \'Dixie to Harlem\'\", \"Torchy in \'Heartbeats\'\"
- Phil Ortiz
- Frode Øverli, *Pondus*
- Nina Paley, *Nina\'s Adventures*, *Fluff*, *The Hots*
- Brant Parker, *The Wizard of Id*
- Stephan Pastis, *Pearls Before Swine*
- Charles Peattie and Russell Taylor, *Alex*
- Mike Peters, *Mother Goose & Grimm*
- Keats Petree
- Stan Pitt, *Larry Flynn, Detective*
- Vic Pratt
- Dariush Ramezani
- John Rivas, *Bonzzo*
- Valentina Romeo, *Jonathan Steele*, *Dylan Dog*, *Morgan Lost*, *Nathan Never*
- Leigh Rubin, *Rubes*
- Warren Sattler, *Grubby*, *Billy the Kid* and *Yang*, as well as contributing artist for *Barnaby* daily, *The Jackson Twins*, *Bringing Up Father* and *Hi and Lois*
- Charles M. Schulz, *Peanuts*, *Young Pillars*
- Jerry Scott, \"Baby Blues, Zits, Nancy\"
- Caroll Spinney, *Harvey*
- Lee W. Stanley, *The Old Home Town*
- Cliff Sterrett, *Polly and Her Pals*
- Kris Straub, *Starslip Crisis*, *Checkerboard Nightmare*
- Henry Matthew Talintyre
- Harold Tamblyn-Watts
- Russell Taylor and Charles Peattie, *Alex*
- Richard Thompson, *Cul de Sac*
- Jim Toomey, *Sherman\'s Lagoon*
- Harry J. Tuthill, *The Bungle Family*
- Gustave Verbeek, *The Upside Downs*, *The Terrors of the Tiny Tads*
- Mort Walker, *Beetle Bailey*, *Hi and Lois*
- Bill Watterson, *Calvin and Hobbes*
- Bob Weber, *Moose & Molly*
- Monty Wedd, *Ned Kelly*
- Alex Williams, *Queen\'s Counsel*
- Tom Wilson, *Ziggy*
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# List of cartoonists
## Cartoonists of single-panel cartoons {#cartoonists_of_single_panel_cartoons}
- Charles Addams
- Gene Ahern
- Glen Baxter
- Belsky
- Jim Benton
- Rupert Besley
- Charles Boyce, *Compu-Toon*
- Barry Bradfield
- Sheree Bradford-Lea
- Bo Brown
- Ivan Brunetti
- John Callahan
- Irwin Caplan
- Patrick Chappatte (Chappatte)
- Roz Chast
- Chumy Chúmez
- Mariza Dias Costa
- Wilbur Dawbarn
- Chon Day
- Donelan
- Denise Dorrance
- Nick Downes
- Mort Drucker
- Vladimir Flórez
- Stanley Arthur Franklin
- Carl Giles (Giles), *Daily Express*
- Ted Goff
- Bud Grace
- Sam Gross
- Dick Guindon
- William Haefeli
- Jessica Hagy
- Baron Halpenny
- Sidney Harris
- William Haselden
- Bill Hoest
- Judy Horacek
- Stan Hunt
- Hank Ketcham
- Ted Key
- John F. Knott, creator of Old Man Texas, Dallas Morning News, 1905-1957
- Clyde Lamb
- Gary Larson
- Mel Lazarus
- Robert Leighton
- George Lichty
- Mike Lynch
- Lorin Morgan-Richards
- Fred Neher
- John Norment
- Don Orehek
- Jackie Ormes, \"Candy\", \"Patty-Jo \'n\' Ginger\"
- W. B. Park
- Virgil Partch
- Dave Pascal
- Mad Peck
- Matt Percival
- Martin Perscheid
- Josefina Tanganelli Plana
- Gardner Rea
- John Reiner
- Dan Reynolds
- Mischa Richter
- Victoria Roberts
- Burr Shafer
- Vahan Shirvanian
- Chris Slane
- Grant Snider
- Dan Steffan
- James Thurber
- Jerry Van Amerongen
- H. T. Webster
- Gluyas Williams
- J. R. Williams, *Out Our Way*
- Gahan Wilson
- George Wolfe
- Kevin Woodcock
- Bianca Xunise
- Bill Yates
- ZAK, pseudonym of Belgian cartoonist Jacques Moeraert
- Zero
## Cartoonists of comic books {#cartoonists_of_comic_books}
- Carlo Ambrosini
- Jack Herbert
- Sergio Aragonés, *Mad*; creator of *Groo the Wanderer*
- Daniel A. Baker
- Ken Battefield
- Jim Benton, *Catwad*,*Batman Squad*,*Attack of the Stuff*
- Bill Benulis, *War is Hell*
- Steve Bialik
- François Bourgeon, *Le Cycle de Cyann*
- Anna Brandoli
- Reg Bunn
- Ben Caldwell, creator of the Dare Detectives
- Aldo Capitanio
- Onofrio Catacchio
- Domitille Collardey
- Carlo Cossio, *Dick Fulmine*
- Jason Craig
- Hugleikur Dagsson
- Dame Darcy, creator of *Meat Cake*
- Patryck de Froidmont
- Gianni De Luca, *Commissario Spada*
- Dan DeCarlo, *Archie*, *Josie and the Pussycats*, *Sabrina, the Teenage Witch*
- Kim Deitch creator of *Waldo the Cat* and comic novels
- Vince Deporter, DC Comics; Nickelodeon, Spirou (Belgium)
- Julie Doucet, creator of *Dirty Plotte*, *My New York Diary*
- Will Elder, *Mad*, *Little Annie Fanny* in *Playboy*
- Steve Fiorilla, mini-comics
- Andy Fish
- Brad W. Foster, creator of *Mechthings* mini-comics, *The Mechthings*, *Adventures of Olivia* mini-comics
- Chandra Free
- Vernon Grant, creator of *The Love Rangers*
- Dick Hafer
- Marc Hansen, creator of Ralph Snart
- Los Bros Hernandez, creators of *Love and Rockets*
- Don Hillsman II
- Yvonne Hutton
- Al Jaffee, *Mad*, *Snappy Answers to Stupid Questions*
- Robyn E
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# Chemical reaction
A **chemical reaction** is a process that leads to the chemical transformation of one set of chemical substances to another. When chemical reactions occur, the atoms are rearranged and the reaction is accompanied by an energy change as new products are generated. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breaking of chemical bonds between atoms, with no change to the nuclei (no change to the elements present), and can often be described by a chemical equation. Nuclear chemistry is a sub-discipline of chemistry that involves the chemical reactions of unstable and radioactive elements where both electronic and nuclear changes can occur.
The substance (or substances) initially involved in a chemical reaction are called reactants or reagents. Chemical reactions are usually characterized by a chemical change, and they yield one or more products, which usually have properties different from the reactants. Reactions often consist of a sequence of individual sub-steps, the so-called elementary reactions, and the information on the precise course of action is part of the reaction mechanism. Chemical reactions are described with chemical equations, which symbolically present the starting materials, end products, and sometimes intermediate products and reaction conditions.
Chemical reactions happen at a characteristic reaction rate at a given temperature and chemical concentration. Some reactions produce heat and are called exothermic reactions, while others may require heat to enable the reaction to occur, which are called endothermic reactions. Typically, reaction rates increase with increasing temperature because there is more thermal energy available to reach the activation energy necessary for breaking bonds between atoms.
A reaction may be classified as redox in which oxidation and reduction occur or non-redox in which there is no oxidation and reduction occurring. Most simple redox reactions may be classified as a combination, decomposition, or single displacement reaction.
Different chemical reactions are used during chemical synthesis in order to obtain the desired product. In biochemistry, a consecutive series of chemical reactions (where the product of one reaction is the reactant of the next reaction) form metabolic pathways. These reactions are often catalyzed by protein enzymes. Enzymes increase the rates of biochemical reactions, so that metabolic syntheses and decompositions impossible under ordinary conditions can occur at the temperature and concentrations present within a cell.
The general concept of a chemical reaction has been extended to reactions between entities smaller than atoms, including nuclear reactions, radioactive decays and reactions between elementary particles, as described by quantum field theory.
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# Chemical reaction
## History
Chemical reactions such as combustion in fire, fermentation and the reduction of ores to metals were known since antiquity. Initial theories of transformation of materials were developed by Greek philosophers, such as the Four-Element Theory of Empedocles stating that any substance is composed of the four basic elements -- fire, water, air and earth. In the Middle Ages, chemical transformations were studied by alchemists. They attempted, in particular, to convert lead into gold, for which purpose they used reactions of lead and lead-copper alloys with sulfur.
The artificial production of chemical substances already was a central goal for medieval alchemists. Examples include the synthesis of ammonium chloride from organic substances as described in the works (c. 850--950) attributed to Jābir ibn Ḥayyān, or the production of mineral acids such as sulfuric and nitric acids by later alchemists, starting from c. 1300. The production of mineral acids involved the heating of sulfate and nitrate minerals such as copper sulfate, alum and saltpeter. In the 17th century, Johann Rudolph Glauber produced hydrochloric acid and sodium sulfate by reacting sulfuric acid and sodium chloride. With the development of the lead chamber process in 1746 and the Leblanc process, allowing large-scale production of sulfuric acid and sodium carbonate, respectively, chemical reactions became implemented into the industry. Further optimization of sulfuric acid technology resulted in the contact process in the 1880s, and the Haber process was developed in 1909--1910 for ammonia synthesis.
From the 16th century, researchers including Jan Baptist van Helmont, Robert Boyle, and Isaac Newton tried to establish theories of experimentally observed chemical transformations. The phlogiston theory was proposed in 1667 by Johann Joachim Becher. It postulated the existence of a fire-like element called \"phlogiston\", which was contained within combustible bodies and released during combustion. This proved to be false in 1785 by Antoine Lavoisier who found the correct explanation of the combustion as a reaction with oxygen from the air.
Joseph Louis Gay-Lussac recognized in 1808 that gases always react in a certain relationship with each other. Based on this idea and the atomic theory of John Dalton, Joseph Proust had developed the law of definite proportions, which later resulted in the concepts of stoichiometry and chemical equations.
Regarding the organic chemistry, it was long believed that compounds obtained from living organisms were too complex to be obtained synthetically. According to the concept of vitalism, organic matter was endowed with a \"vital force\" and distinguished from inorganic materials. This separation was ended however by the synthesis of urea from inorganic precursors by Friedrich Wöhler in 1828. Other chemists who brought major contributions to organic chemistry include Alexander William Williamson with his synthesis of ethers and Christopher Kelk Ingold, who, among many discoveries, established the mechanisms of substitution reactions.
## Characteristics
The general characteristics of chemical reactions are:
- Evolution of a gas
- Formation of a precipitate
- Change in temperature
- Change in state
## Equations
Chemical equations are used to graphically illustrate chemical reactions. They consist of chemical or structural formulas of the reactants on the left and those of the products on the right. They are separated by an arrow (→) which indicates the direction and type of the reaction; the arrow is read as the word \"yields\". The tip of the arrow points in the direction in which the reaction proceeds. A double arrow (`{{eqm}}`{=mediawiki}) pointing in opposite directions is used for equilibrium reactions. Equations should be balanced according to the stoichiometry, the number of atoms of each species should be the same on both sides of the equation. This is achieved by scaling the number of involved molecules (A, B, C and D in a schematic example below) by the appropriate integers *a, b, c* and *d*.
:
More elaborate reactions are represented by reaction schemes, which in addition to starting materials and products show important intermediates or transition states. Also, some relatively minor additions to the reaction can be indicated above the reaction arrow; examples of such additions are water, heat, illumination, a catalyst, etc. Similarly, some minor products can be placed below the arrow, often with a minus sign. Retrosynthetic analysis can be applied to design a complex synthesis reaction. Here the analysis starts from the products, for example by splitting selected chemical bonds, to arrive at plausible initial reagents. A special arrow (⇒) is used in retro reactions.
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# Chemical reaction
## Elementary reactions {#elementary_reactions}
The elementary reaction is the smallest division into which a chemical reaction can be decomposed, it has no intermediate products. Most experimentally observed reactions are built up from many elementary reactions that occur in parallel or sequentially. The actual sequence of the individual elementary reactions is known as reaction mechanism. An elementary reaction involves a few molecules, usually one or two, because of the low probability for several molecules to meet at a certain time.
The most important elementary reactions are unimolecular and bimolecular reactions. Only one molecule is involved in a unimolecular reaction; it is transformed by isomerization or a dissociation into one or more other molecules. Such reactions require the addition of energy in the form of heat or light. A typical example of a unimolecular reaction is the cis--trans isomerization, in which the cis-form of a compound converts to the trans-form or vice versa.
In a typical dissociation reaction, a bond in a molecule splits (**ruptures**) resulting in two molecular fragments. The splitting can be homolytic or heterolytic. In the first case, the bond is divided so that each product retains an electron and becomes a neutral radical. In the second case, both electrons of the chemical bond remain with one of the products, resulting in charged ions. Dissociation plays an important role in triggering chain reactions, such as hydrogen--oxygen or polymerization reactions.
: AB -\> A + B
: Dissociation of a molecule AB into fragments A and B
For bimolecular reactions, two molecules collide and react with each other. Their merger is called chemical synthesis or an addition reaction.
: A + B -\> AB
Another possibility is that only a portion of one molecule is transferred to the other molecule. This type of reaction occurs, for example, in redox and acid-base reactions. In redox reactions, the transferred particle is an electron, whereas in acid-base reactions it is a proton. This type of reaction is also called metathesis.
: HA + B -\> A + HB
for example
: NaCl + AgNO3 -\> NaNO3 + AgCl(v)
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# Chemical reaction
## Chemical equilibrium {#chemical_equilibrium}
Most chemical reactions are reversible; that is, they can and do run in both directions. The forward and reverse reactions are competing with each other and differ in reaction rates. These rates depend on the concentration and therefore change with the time of the reaction: the reverse rate gradually increases and becomes equal to the rate of the forward reaction, establishing the so-called chemical equilibrium. The time to reach equilibrium depends on parameters such as temperature, pressure, and the materials involved, and is determined by the minimum free energy. In equilibrium, the Gibbs free energy of reaction must be zero. The pressure dependence can be explained with the Le Chatelier\'s principle. For example, an increase in pressure due to decreasing volume causes the reaction to shift to the side with fewer moles of gas.
The reaction yield stabilizes at equilibrium but can be increased by removing the product from the reaction mixture or changed by increasing the temperature or pressure. A change in the concentrations of the reactants does not affect the equilibrium constant but does affect the equilibrium position.
## Thermodynamics
Chemical reactions are determined by the laws of thermodynamics. Reactions can proceed by themselves if they are exergonic, that is if they release free energy. The associated free energy change of the reaction is composed of the changes of two different thermodynamic quantities, enthalpy and entropy:
:; $\Delta G = \Delta H - T \cdot \Delta S$.
:
: `{{mvar|G}}`{=mediawiki}: free energy, `{{mvar|H}}`{=mediawiki}: enthalpy, `{{mvar|T}}`{=mediawiki}: temperature, `{{mvar|S}}`{=mediawiki}: entropy, `{{math|Δ}}`{=mediawiki}: difference (change between original and product)
Reactions can be exothermic, where Δ*H* is negative and energy is released. Typical examples of exothermic reactions are combustion, precipitation and crystallization, in which ordered solids are formed from disordered gaseous or liquid phases. In contrast, in endothermic reactions, heat is consumed from the environment. This can occur by increasing the entropy of the system, often through the formation of gaseous or dissolved reaction products, which have higher entropy. Since the entropy term in the free-energy change increases with temperature, many endothermic reactions preferably take place at high temperatures. On the contrary, many exothermic reactions such as crystallization occur preferably at lower temperatures. A change in temperature can sometimes reverse the sign of the enthalpy of a reaction, as for the carbon monoxide reduction of molybdenum dioxide:
: 2CO(g) + MoO2(s) -\> 2CO2(g) + Mo(s); $\Delta H^o = +21.86 \ \text{kJ at 298 K}$
This reaction to form carbon dioxide and molybdenum is endothermic at low temperatures, becoming less so with increasing temperature. Δ*H*° is zero at `{{val|1855|ul=K}}`{=mediawiki}, and the reaction becomes exothermic above that temperature.
Changes in temperature can also reverse the direction tendency of a reaction. For example, the water gas shift reaction
: CO(g) + H2O({v}) \<=\> CO2(g) + H2(g)
is favored by low temperatures, but its reverse is favored by high temperatures. The shift in reaction direction tendency occurs at `{{val|1100|u=K}}`{=mediawiki}.
Reactions can also be characterized by their internal energy change, which takes into account changes in the entropy, volume and chemical potentials. The latter depends, among other things, on the activities of the involved substances.
:; ${d}U = T\cdot {d}S - p\cdot {d}V + \mu\cdot {d}n$
:
: `{{mvar|U}}`{=mediawiki}: internal energy, `{{mvar|S}}`{=mediawiki}: entropy, `{{mvar|p}}`{=mediawiki}: pressure, `{{mvar|μ}}`{=mediawiki}: chemical potential, `{{mvar|n}}`{=mediawiki}: number of molecules, `{{mvar|d}}`{=mediawiki}: small change sign
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# Chemical reaction
## Kinetics
The speed at which reactions take place is studied by reaction kinetics. The rate depends on various parameters, such as:
- Reactant concentrations, which usually make the reaction happen at a faster rate if raised through increased collisions per unit of time. Some reactions, however, have rates that are *independent* of reactant concentrations, due to a limited number of catalytic sites. These are called zero order reactions.
- Surface area available for contact between the reactants, in particular solid ones in heterogeneous systems. Larger surface areas lead to higher reaction rates.
- Pressure -- increasing the pressure decreases the volume between molecules and therefore increases the frequency of collisions between the molecules.
- Activation energy, which is defined as the amount of energy required to make the reaction start and carry on spontaneously. Higher activation energy implies that the reactants need more energy to start than a reaction with lower activation energy.
- Temperature, which hastens reactions if raised, since higher temperature increases the energy of the molecules, creating more collisions per unit of time,
- The presence or absence of a catalyst. Catalysts are substances that make weak bonds with reactants or intermediates and change the pathway (mechanism) of a reaction which in turn increases the speed of a reaction by lowering the activation energy needed for the reaction to take place. A catalyst is not destroyed or changed during a reaction, so it can be used again.
- For some reactions, the presence of electromagnetic radiation, most notably ultraviolet light, is needed to promote the breaking of bonds to start the reaction. This is particularly true for reactions involving radicals.
Several theories allow calculating the reaction rates at the molecular level. This field is referred to as reaction dynamics. The rate *v* of a first-order reaction, which could be the disintegration of a substance A, is given by:
: $v= -\frac {d[\ce{A}]}{dt}= k \cdot [\ce{A}].$
Its integration yields:
: $\ce{[A]}(t) = \ce{[A]}_{0} \cdot e^{-k\cdot t}.$
Here *k* is the first-order rate constant, having dimension 1/time, \[A\](*t*) is the concentration at a time *t* and \[A\]~0~ is the initial concentration. The rate of a first-order reaction depends only on the concentration and the properties of the involved substance, and the reaction itself can be described with a characteristic half-life. More than one time constant is needed when describing reactions of higher order. The temperature dependence of the rate constant usually follows the Arrhenius equation:
$$k = k_0 e^{{-E_a}/{k_{B}T}}$$
where *E*~a~ is the activation energy and *k*~B~ is the Boltzmann constant. One of the simplest models of reaction rate is the collision theory. More realistic models are tailored to a specific problem and include the transition state theory, the calculation of the potential energy surface, the Marcus theory and the Rice--Ramsperger--Kassel--Marcus (RRKM) theory.
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# Chemical reaction
## Reaction types {#reaction_types}
### Four basic types {#four_basic_types}
#### Synthesis
In a synthesis reaction, two or more simple substances combine to form a more complex substance. These reactions are in the general form: A + B-\>AB
Two or more reactants yielding one product is another way to identify a synthesis reaction. One example of a synthesis reaction is the combination of iron and sulfur to form iron(II) sulfide: 8Fe + S8-\>8FeS
Another example is simple hydrogen gas combined with simple oxygen gas to produce a more complex substance, such as water.
#### Decomposition
A decomposition reaction is when a more complex substance breaks down into its more simple parts. It is thus the opposite of a synthesis reaction and can be written as AB-\>A + B
One example of a decomposition reaction is the electrolysis of water to make oxygen and hydrogen gas: 2H2O-\>2H2 + O2
#### Single displacement {#single_displacement}
In a single displacement reaction, a single uncombined element replaces another in a compound; in other words, one element trades places with another element in a compound. These reactions come in the general form of: A + BC-\>AC + B
One example of a single displacement reaction is when magnesium replaces hydrogen in water to make solid magnesium hydroxide and hydrogen gas: Mg + 2H2O-\>Mg(OH)2 (v) + H2 (\^)
#### Double displacement {#double_displacement}
In a double displacement reaction, the anions and cations of two compounds switch places and form two entirely different compounds. These reactions are in the general form: AB + CD-\>AD + CB
For example, when barium chloride (BaCl~2~) and magnesium sulfate (MgSO~4~) react, the SO~4~^2−^ anion switches places with the 2Cl^−^ anion, giving the compounds BaSO~4~ and MgCl~2~.
Another example of a double displacement reaction is the reaction of lead(II) nitrate with potassium iodide to form lead(II) iodide and potassium nitrate: Pb(NO3)2 + 2KI-\>PbI2(v) + 2KNO3
### Forward and backward reactions {#forward_and_backward_reactions}
According to Le Chatelier\'s Principle, reactions may proceed in the forward or reverse direction until they end or reach equilibrium.
#### Forward reactions {#forward_reactions}
Reactions that proceed in the forward direction (from left to right) to approach equilibrium are often called spontaneous reactions, that is, $\Delta G$ is negative, which means that if they occur at constant temperature and pressure, they decrease the Gibbs free energy of the reaction. They require less energy to proceed in the forward direction. Reactions are usually written as forward reactions in the direction in which they are spontaneous. Examples:
- Reaction of hydrogen and oxygen to form water.
: \+ `{{chem|O|2}}`{=mediawiki} `{{eqm}}`{=mediawiki} `{{chem|2H|2|O}}`{=mediawiki}
- Dissociation of acetic acid in water into acetate ions and hydronium ions.
: \+ `{{chem|H|2|O}}`{=mediawiki} `{{eqm}}`{=mediawiki} `{{chem|CH|3|COO|-}}`{=mediawiki} + `{{chem|H|3|O|+}}`{=mediawiki}
#### Backward reactions {#backward_reactions}
Reactions that proceed in the backward direction to approach equilibrium are often called non-spontaneous reactions, that is, $\Delta G$ is positive, which means that if they occur at constant temperature and pressure, they increase the Gibbs free energy of the reaction. They require input of energy to proceed in the forward direction. Examples include:
- Charging a normal DC battery (consisting of electrolytic cells) from an external electrical power source
- Photosynthesis driven by absorption of electromagnetic radiation usually in the form of sunlight
: \+ `{{underset| water |H<sub>2</sub>O}}`{=mediawiki} + `{{underset|light energy|photons}}`{=mediawiki} → `{{underset|carbohydrate|[CH<sub>2</sub>O]}}`{=mediawiki} + `{{underset| oxygen |O<sub>2</sub>}}`{=mediawiki}
### Combustion
In a combustion reaction, an element or compound reacts with an oxidant, usually oxygen, often producing energy in the form of heat or light. Combustion reactions frequently involve a hydrocarbon. For instance, the combustion of 1 mole (114 g) of octane in oxygen C8H18(l) + 25/2 O2(g)-\>8CO2 + 9H2O(l)
releases 5500 kJ. A combustion reaction can also result from carbon, magnesium or sulfur reacting with oxygen. 2Mg(s) + O2-\>2MgO(s) S(s) + O2(g)-\>SO2(g)
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# Chemical reaction
## Reaction types {#reaction_types}
### Oxidation and reduction {#oxidation_and_reduction}
Redox reactions can be understood in terms of the transfer of electrons from one involved species (reducing agent) to another (oxidizing agent). In this process, the former species is *oxidized* and the latter is *reduced*. Though sufficient for many purposes, these descriptions are not precisely correct. Oxidation is better defined as an increase in oxidation state of atoms and reduction as a decrease in oxidation state. In practice, the transfer of electrons will always change the oxidation state, but there are many reactions that are classed as \"redox\" even though no electron transfer occurs (such as those involving covalent bonds).
In the following redox reaction, hazardous sodium metal reacts with toxic chlorine gas to form the ionic compound sodium chloride, or common table salt: 2Na(s) + Cl2(g)-\>2NaCl(s)
In the reaction, sodium metal goes from an oxidation state of 0 (a pure element) to +1: in other words, the sodium lost one electron and is said to have been oxidized. On the other hand, the chlorine gas goes from an oxidation of 0 (also a pure element) to −1: the chlorine gains one electron and is said to have been reduced. Because the chlorine is the one reduced, it is considered the electron acceptor, or in other words, induces oxidation in the sodium -- thus the chlorine gas is considered the oxidizing agent. Conversely, the sodium is oxidized or is the electron donor, and thus induces a reduction in the other species and is considered the *reducing agent*.
Which of the involved reactants would be a reducing or oxidizing agent can be predicted from the electronegativity of their elements. Elements with low electronegativities, such as most metals, easily donate electrons and oxidize -- they are reducing agents. On the contrary, many oxides or ions with high oxidation numbers of their non-oxygen atoms, such as `{{chem|link=hydrogen peroxide|H|2|O|2}}`{=mediawiki}, `{{chem|link=permanganate|MnO|4|-}}`{=mediawiki}, `{{chem|link=chromium trioxide|CrO|3}}`{=mediawiki}, `{{chem|link=dichromate|Cr|2|O|7|2-}}`{=mediawiki}, or `{{chem|link=Osmium(VIII) oxide|OsO|4}}`{=mediawiki}, can gain one or two extra electrons and are strong oxidizing agents.
For some main-group elements the number of electrons donated or accepted in a redox reaction can be predicted from the electron configuration of the reactant element. Elements try to reach the low-energy noble gas configuration, and therefore alkali metals and halogens will donate and accept one electron, respectively. Noble gases themselves are chemically inactive.
The overall redox reaction can be balanced by combining the oxidation and reduction half-reactions multiplied by coefficients such that the number of electrons lost in the oxidation equals the number of electrons gained in the reduction.
An important class of redox reactions are the electrolytic electrochemical reactions, where electrons from the power supply at the negative electrode are used as the reducing agent and electron withdrawal at the positive electrode as the oxidizing agent. These reactions are particularly important for the production of chemical elements, such as chlorine or aluminium. The reverse process, in which electrons are released in redox reactions and chemical energy is converted to electrical energy, is possible and used in batteries.
### Complexation
In complexation reactions, several ligands react with a metal atom to form a coordination complex. This is achieved by providing lone pairs of the ligand into empty orbitals of the metal atom and forming dipolar bonds. The ligands are Lewis bases, they can be both ions and neutral molecules, such as carbon monoxide, ammonia or water. The number of ligands that react with a central metal atom can be found using the 18-electron rule, saying that the valence shells of a transition metal will collectively accommodate 18 electrons, whereas the symmetry of the resulting complex can be predicted with the crystal field theory and ligand field theory. Complexation reactions also include ligand exchange, in which one or more ligands are replaced by another, and redox processes which change the oxidation state of the central metal atom.
### Acid--base reactions {#acidbase_reactions}
In the Brønsted--Lowry acid--base theory, an acid--base reaction involves a transfer of protons (H^+^) from one species (the acid) to another (the base). When a proton is removed from an acid, the resulting species is termed that acid\'s conjugate base. When the proton is accepted by a base, the resulting species is termed that base\'s conjugate acid. In other words, acids act as proton donors and bases act as proton acceptors according to the following equation: \\underset{acid}{HA} + \\underset{base}{B} \<=\> \\underset{conjugated\\ base}{A\^-} + \\underset{conjugated\\ acid}{HB+}
The reverse reaction is possible, and thus the acid/base and conjugated base/acid are always in equilibrium. The equilibrium is determined by the acid and base dissociation constants (*K*~a~ and *K*~b~) of the involved substances. A special case of the acid-base reaction is the neutralization where an acid and a base, taken at the exact same amounts, form a neutral salt.
Acid-base reactions can have different definitions depending on the acid-base concept employed. Some of the most common are:
- Arrhenius definition: Acids dissociate in water releasing H~3~O^+^ ions; bases dissociate in water releasing OH^−^ ions.
- Brønsted--Lowry definition: Acids are proton (H^+^) donors, bases are proton acceptors; this includes the Arrhenius definition.
- Lewis definition: Acids are electron-pair acceptors, and bases are electron-pair donors; this includes the Brønsted-Lowry definition.
### Precipitation
Precipitation is the formation of a solid in a solution or inside another solid during a chemical reaction. It usually takes place when the concentration of dissolved ions exceeds the solubility limit and forms an insoluble salt. This process can be assisted by adding a precipitating agent or by the removal of the solvent. Rapid precipitation results in an amorphous or microcrystalline residue and a slow process can yield single crystals. The latter can also be obtained by recrystallization from microcrystalline salts.
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# Chemical reaction
## Reaction types {#reaction_types}
### Solid-state reactions {#solid_state_reactions}
Reactions can take place between two solids. However, because of the relatively small diffusion rates in solids, the corresponding chemical reactions are very slow in comparison to liquid and gas phase reactions. They are accelerated by increasing the reaction temperature and finely dividing the reactant to increase the contacting surface area.
### Reactions at the solid/gas interface {#reactions_at_the_solidgas_interface}
The reaction can take place at the solid\|gas interface, surfaces at very low pressure such as ultra-high vacuum. Via scanning tunneling microscopy, it is possible to observe reactions at the solid\|gas interface in real space, if the time scale of the reaction is in the correct range. Reactions at the solid\|gas interface are in some cases related to catalysis.
### Photochemical reactions {#photochemical_reactions}
In photochemical reactions, atoms and molecules absorb energy (photons) of the illumination light and convert it into an excited state. They can then release this energy by breaking chemical bonds, thereby producing radicals. Photochemical reactions include hydrogen--oxygen reactions, radical polymerization, chain reactions and rearrangement reactions.
Many important processes involve photochemistry. The premier example is photosynthesis, in which most plants use solar energy to convert carbon dioxide and water into glucose, disposing of oxygen as a side-product. Humans rely on photochemistry for the formation of vitamin D, and vision is initiated by a photochemical reaction of rhodopsin. In fireflies, an enzyme in the abdomen catalyzes a reaction that results in bioluminescence. Many significant photochemical reactions, such as ozone formation, occur in the Earth atmosphere and constitute atmospheric chemistry.
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# Chemical reaction
## Catalysis
In catalysis, the reaction does not proceed directly, but through a reaction with a third substance known as catalyst. Although the catalyst takes part in the reaction, forming weak bonds with reactants or intermediates, it is returned to its original state by the end of the reaction and so is not consumed. However, it can be inhibited, deactivated or destroyed by secondary processes. Catalysts can be used in a different phase (heterogeneous) or in the same phase (homogeneous) as the reactants. In heterogeneous catalysis, typical secondary processes include coking where the catalyst becomes covered by polymeric side products. Additionally, heterogeneous catalysts can dissolve into the solution in a solid-liquid system or evaporate in a solid--gas system. Catalysts can only speed up the reaction -- chemicals that slow down the reaction are called inhibitors. Substances that increase the activity of catalysts are called promoters, and substances that deactivate catalysts are called catalytic poisons. With a catalyst, a reaction that is kinetically inhibited by high activation energy can take place in the circumvention of this activation energy.
Heterogeneous catalysts are usually solids, powdered in order to maximize their surface area. Of particular importance in heterogeneous catalysis are the platinum group metals and other transition metals, which are used in hydrogenations, catalytic reforming and in the synthesis of commodity chemicals such as nitric acid and ammonia. Acids are an example of a homogeneous catalyst, they increase the nucleophilicity of carbonyls, allowing a reaction that would not otherwise proceed with electrophiles. The advantage of homogeneous catalysts is the ease of mixing them with the reactants, but they may also be difficult to separate from the products. Therefore, heterogeneous catalysts are preferred in many industrial processes.
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# Chemical reaction
## Reactions in organic chemistry {#reactions_in_organic_chemistry}
In organic chemistry, in addition to oxidation, reduction or acid-base reactions, a number of other reactions can take place which involves covalent bonds between carbon atoms or carbon and heteroatoms (such as oxygen, nitrogen, halogens, etc.). Many specific reactions in organic chemistry are name reactions designated after their discoverers.
One of the most industrially important reactions is the cracking of heavy hydrocarbons at oil refineries to create smaller, simpler molecules. This process is used to manufacture gasoline. Specific types of organic reactions may be grouped by their reaction mechanisms (particularly substitution, addition and elimination) or by the types of products they produce (for example, methylation, polymerisation and halogenation).
### Substitution
In a substitution reaction, a functional group in a particular chemical compound is replaced by another group. These reactions can be distinguished by the type of substituting species into a nucleophilic, electrophilic or radical substitution. `{{multiple image | direction = vertical | image1 = SN1 reaction mechanism.png|width1 = 300|image2 = SN2 reaction mechanism.png|width2 = 300| caption1 = S<sub>N</sub>1 mechanism| caption2 = S<sub>N</sub>2 mechanism}}`{=mediawiki}
In the first type, a nucleophile, an atom or molecule with an excess of electrons and thus a negative charge or partial charge, replaces another atom or part of the \"substrate\" molecule. The electron pair from the nucleophile attacks the substrate forming a new bond, while the leaving group departs with an electron pair. The nucleophile may be electrically neutral or negatively charged, whereas the substrate is typically neutral or positively charged. Examples of nucleophiles are hydroxide ion, alkoxides, amines and halides. This type of reaction is found mainly in aliphatic hydrocarbons, and rarely in aromatic hydrocarbon. The latter have high electron density and enter nucleophilic aromatic substitution only with very strong electron withdrawing groups. Nucleophilic substitution can take place by two different mechanisms, S~N~1 and S~N~2. In their names, S stands for substitution, N for nucleophilic, and the number represents the kinetic order of the reaction, unimolecular or bimolecular. `{{multiple image | direction = vertical
| align = right
| width = 120
| image1= Walden-inversion-3D-balls.png
|caption1=The three steps of an [[SN2 reaction|S<sub>N</sub>2 reaction]]. The nucleophile is green and the leaving group is red
|image2=SN2-Walden-before-and-after-horizontal-3D-balls.png
|caption2=S<sub>N</sub>2 reaction causes stereo inversion (Walden inversion)
}}`{=mediawiki}
The S~N~1 reaction proceeds in two steps. First, the leaving group is eliminated creating a carbocation. This is followed by a rapid reaction with the nucleophile.
In the S~N~2 mechanisms, the nucleophile forms a transition state with the attacked molecule, and only then the leaving group is cleaved. These two mechanisms differ in the stereochemistry of the products. S~N~1 leads to the non-stereospecific addition and does not result in a chiral center, but rather in a set of geometric isomers (*cis/trans*). In contrast, a reversal (Walden inversion) of the previously existing stereochemistry is observed in the S~N~2 mechanism.
Electrophilic substitution is the counterpart of the nucleophilic substitution in that the attacking atom or molecule, an electrophile, has low electron density and thus a positive charge. Typical electrophiles are the carbon atom of carbonyl groups, carbocations or sulfur or nitronium cations. This reaction takes place almost exclusively in aromatic hydrocarbons, where it is called electrophilic aromatic substitution. The electrophile attack results in the so-called σ-complex, a transition state in which the aromatic system is abolished. Then, the leaving group, usually a proton, is split off and the aromaticity is restored. An alternative to aromatic substitution is electrophilic aliphatic substitution. It is similar to the nucleophilic aliphatic substitution and also has two major types, S~E~1 and S~E~2.
In the third type of substitution reaction, radical substitution, the attacking particle is a radical. This process usually takes the form of a chain reaction, for example in the reaction of alkanes with halogens. In the first step, light or heat disintegrates the halogen-containing molecules producing radicals. Then the reaction proceeds as an avalanche until two radicals meet and recombine.
:;X. + R-H -\> X-H + R.
:;R. + X2 -\> R-X + X.
:
: Reactions during the chain reaction of radical substitution
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# Chemical reaction
## Reactions in organic chemistry {#reactions_in_organic_chemistry}
### Addition and elimination {#addition_and_elimination}
The addition and its counterpart, the elimination, are reactions that change the number of substituents on the carbon atom, and form or cleave multiple bonds. Double and triple bonds can be produced by eliminating a suitable leaving group. Similar to the nucleophilic substitution, there are several possible reaction mechanisms that are named after the respective reaction order. In the E1 mechanism, the leaving group is ejected first, forming a carbocation. The next step, the formation of the double bond, takes place with the elimination of a proton (deprotonation). The leaving order is reversed in the E1cb mechanism, that is the proton is split off first. This mechanism requires the participation of a base. Because of the similar conditions, both reactions in the E1 or E1cb elimination always compete with the S~N~1 substitution.
The E2 mechanism also requires a base, but there the attack of the base and the elimination of the leaving group proceed simultaneously and produce no ionic intermediate. In contrast to the E1 eliminations, different stereochemical configurations are possible for the reaction product in the E2 mechanism, because the attack of the base preferentially occurs in the anti-position with respect to the leaving group. Because of the similar conditions and reagents, the E2 elimination is always in competition with the S~N~2-substitution.
The counterpart of elimination is an addition where double or triple bonds are converted into single bonds. Similar to substitution reactions, there are several types of additions distinguished by the type of the attacking particle. For example, in the electrophilic addition of hydrogen bromide, an electrophile (proton) attacks the double bond forming a carbocation, which then reacts with the nucleophile (bromine). The carbocation can be formed on either side of the double bond depending on the groups attached to its ends, and the preferred configuration can be predicted with the Markovnikov\'s rule. This rule states that \"In the heterolytic addition of a polar molecule to an alkene or alkyne, the more electronegative (nucleophilic) atom (or part) of the polar molecule becomes attached to the carbon atom bearing the smaller number of hydrogen atoms.\"
If the addition of a functional group takes place at the less substituted carbon atom of the double bond, then the electrophilic substitution with acids is not possible. In this case, one has to use the hydroboration--oxidation reaction, wherein the first step, the boron atom acts as electrophile and adds to the less substituted carbon atom. In the second step, the nucleophilic hydroperoxide or halogen anion attacks the boron atom.
While the addition to the electron-rich alkenes and alkynes is mainly electrophilic, the nucleophilic addition plays an important role in the carbon-heteroatom multiple bonds, and especially its most important representative, the carbonyl group. This process is often associated with elimination so that after the reaction the carbonyl group is present again. It is, therefore, called an addition-elimination reaction and may occur in carboxylic acid derivatives such as chlorides, esters or anhydrides. This reaction is often catalyzed by acids or bases, where the acids increase the electrophilicity of the carbonyl group by binding to the oxygen atom, whereas the bases enhance the nucleophilicity of the attacking nucleophile.
Nucleophilic addition of a carbanion or another nucleophile to the double bond of an alpha, beta-unsaturated carbonyl compound can proceed via the Michael reaction, which belongs to the larger class of conjugate additions. This is one of the most useful methods for the mild formation of C--C bonds.
Some additions which can not be executed with nucleophiles and electrophiles can be succeeded with free radicals. As with the free-radical substitution, the radical addition proceeds as a chain reaction, and such reactions are the basis of the free-radical polymerization.
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# Chemical reaction
## Reactions in organic chemistry {#reactions_in_organic_chemistry}
### Other organic reaction mechanisms {#other_organic_reaction_mechanisms}
`{{multiple image | direction = vertical
| align = right
| width = 220
| image1= Diels Alder Mechanismus.svg
|caption1=Mechanism of a Diels-Alder reaction
| image2= Diels Alder Orbitale.svg
|caption2=Orbital overlap in a Diels-Alder reaction}}`{=mediawiki}
In a rearrangement reaction, the carbon skeleton of a molecule is rearranged to give a structural isomer of the original molecule. These include hydride shift reactions such as the Wagner-Meerwein rearrangement, where a hydrogen, alkyl or aryl group migrates from one carbon to a neighboring carbon. Most rearrangements are associated with the breaking and formation of new carbon-carbon bonds. Other examples are sigmatropic reaction such as the Cope rearrangement.
Cyclic rearrangements include cycloadditions and, more generally, pericyclic reactions, wherein two or more double bond-containing molecules form a cyclic molecule. An important example of cycloaddition reaction is the Diels--Alder reaction (the so-called \[4+2\] cycloaddition) between a conjugated diene and a substituted alkene to form a substituted cyclohexene system.
Whether a certain cycloaddition would proceed depends on the electronic orbitals of the participating species, as only orbitals with the same sign of wave function will overlap and interact constructively to form new bonds. Cycloaddition is usually assisted by light or heat. These perturbations result in a different arrangement of electrons in the excited state of the involved molecules and therefore in different effects. For example, the \[4+2\] Diels-Alder reactions can be assisted by heat whereas the \[2+2\] cycloaddition is selectively induced by light. Because of the orbital character, the potential for developing stereoisomeric products upon cycloaddition is limited, as described by the Woodward--Hoffmann rules.
## Biochemical reactions {#biochemical_reactions}
Biochemical reactions are mainly controlled by complex proteins called enzymes, which are usually specialized to catalyze only a single, specific reaction. The reaction takes place in the active site, a small part of the enzyme which is usually found in a cleft or pocket lined by amino acid residues, and the rest of the enzyme is used mainly for stabilization. The catalytic action of enzymes relies on several mechanisms including the molecular shape (\"induced fit\"), bond strain, proximity and orientation of molecules relative to the enzyme, proton donation or withdrawal (acid/base catalysis), electrostatic interactions and many others.
The biochemical reactions that occur in living organisms are collectively known as metabolism. Among the most important of its mechanisms is the anabolism, in which different DNA and enzyme-controlled processes result in the production of large molecules such as proteins and carbohydrates from smaller units. Bioenergetics studies the sources of energy for such reactions. Important energy sources are glucose and oxygen, which can be produced by plants via photosynthesis or assimilated from food and air, respectively. All organisms use this energy to produce adenosine triphosphate (ATP), which can then be used to energize other reactions. Decomposition of organic material by fungi, bacteria and other micro-organisms is also within the scope of biochemistry.
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# Chemical reaction
## Applications
Chemical reactions are central to chemical engineering, where they are used for the synthesis of new compounds from natural raw materials such as petroleum, mineral ores, and oxygen in air. It is essential to make the reaction as efficient as possible, maximizing the yield and minimizing the number of reagents, energy inputs and waste. Catalysts are especially helpful for reducing the energy required for the reaction and increasing its reaction rate.
Some specific reactions have their niche applications. For example, the thermite reaction is used to generate light and heat in pyrotechnics and welding. Although it is less controllable than the more conventional oxy-fuel welding, arc welding and flash welding, it requires much less equipment and is still used to mend rails, especially in remote areas.
## Monitoring
Mechanisms of monitoring chemical reactions depend strongly on the reaction rate. Relatively slow processes can be analyzed in situ for the concentrations and identities of the individual ingredients. Important tools of real-time analysis are the measurement of pH and analysis of optical absorption (color) and emission spectra. A less accessible but rather efficient method is the introduction of a radioactive isotope into the reaction and monitoring how it changes over time and where it moves to; this method is often used to analyze the redistribution of substances in the human body. Faster reactions are usually studied with ultrafast laser spectroscopy where utilization of femtosecond lasers allows short-lived transition states to be monitored at a time scaled down to a few femtoseconds
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# Casiquiare canal
The **Casiquiare river** or **canal** (`{{IPA|es|kasiˈkjaɾe}}`{=mediawiki}) is a natural distributary of the upper Orinoco flowing southward into the Rio Negro, in Venezuela, South America. As such, it forms a unique natural canal between the Orinoco and Amazon river systems. It is the world\'s largest river of the kind that links two major river systems, a so-called bifurcation. The area forms a water divide, more dramatically at regional flood stage.
## Etymology
The name *Casiquiare*, first used in that form by Manuel Román, likely derives from the Ye\'kuana language name of the river, *Kashishiwadi*.
## Discovery
The first European to describe it was Spanish Jesuit missionary and explorer Cristóbal Diatristán de Acuña in 1639.
In 1744 a Jesuit priest named Manuel Román, while ascending the Orinoco River in the region of La Esmeralda, met some Portuguese slave-traders from the settlements on the Rio Negro. The Portuguese insisted they were not in Spanish territory but on a tributary of the Amazon; they invited Román back with them to prove their claim. He accompanied them on their return, by way of the Casiquiare canal, and afterwards retraced his route to the Orinoco. Along the way, he made first contact with the Ye\'kuana people, whom he enlisted to help in his journey. Charles Marie de La Condamine, seven months later, was able to give to the *Académie française* an account of Father Román\'s voyage, and thus confirm the existence of this waterway, first reported by Father Acuña in 1639.
Little credence was given to Román\'s statement until it was verified, in 1756, by the Spanish Boundary-line Commission of José Yturriaga and Solano. In 1800 German scientist Alexander von Humboldt and French botanist Aimé Bonpland explored the river. `{{citation needed span|During a 1924–25 expedition, [[Alexander H. Rice Jr.]] of [[Harvard University]] traveled up the Orinoco, traversed the Casiquiare canal, and descended the Rio Negro to the Amazon at Manaus. It was the first expedition to use aerial photography and [[shortwave radio]] for mapping of the region.|reason=Rice's WP article indicates he explored the Casiquiare Canal in 1919, and the use of aerial photography and shortwave radio was on a later expedition elsewhere.|date=November 2024}}`{=mediawiki} In 1968 the Casiquiare was navigated by an SRN6 hovercraft during a The Geographical Journal expedition.
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# Casiquiare canal
## Geography
The origin of the Casiquiare, at the River Orinoco, is 9 mi below the mission of La Esmeralda at 3 8 18.5 N 65 52 42.5 W region:VE-X_type:landmark display=inline, and about 123 m above sea level. Its mouth at the Rio Negro, an affluent of the Amazon River, is near the town of San Carlos and is 91 m above sea level.
The general course is south-west, and its length, including windings, is about 200 mi. Its width, at its bifurcation with the Orinoco, is approximately 300 ft, with a current towards the Rio Negro of 0.75 mph. However, as it gains in volume from the very numerous tributary streams, large and small, that it receives en route, its velocity increases, and in the wet season reaches 5 mph, even 8 mph in certain stretches. It broadens considerably as it approaches its mouth, where it is about 1750 ft wide. The volume of water the Casiquiare captures from the Orinoco is small in comparison to what it accumulates in its course. Nevertheless, the geological processes are ongoing, and evidence points to a slow and gradual increase in the size of Casiquiare. It is likely that stream capture is in progress, i.e. what currently is the uppermost Orinoco basin, including Cunucunuma River, eventually will be entirely diverted by the Casiquiare into the Amazon basin.
In flood time, it is said to have a second connection with the Rio Negro by a branch, which it throws off to the westward, called the Itinivini, which leaves it at a point about 50 mi above its mouth. In the dry season, it has shallows, and is obstructed by sandbanks, a few rapids and granite rocks. Its shores are densely wooded, and the soil more fertile than that along the Rio Negro. The general slope of the plains through which the canal runs is south-west, but those of the Rio Negro slope south-east.
The Casiquiare is not a sluggish canal on a flat tableland, but a great, rapid river which, if its upper waters had not found contact with the Orinoco, perhaps by cutting back, would belong entirely to the Negro branch of the Amazon.
To the west of the Casiquiare, there is a much shorter and easier portage between the Orinoco and Amazon basins, called the isthmus of Pimichin, which is reached by ascending the Temi branch of the Atabapo River, an affluent of the Orinoco. Although the Temi is somewhat obstructed, it is believed that it could easily be made navigable for small craft. The isthmus is 10 mi across, with undulating ground, nowhere over 50 ft high, with swamps and marshes. In the early 20th century, it was much used for the transit of large canoes, which were hauled across it from the Temi River and reached the Rio Negro by a little stream called the Pimichin.
## Hydrographic divide {#hydrographic_divide}
The Casiquiare canal -- Orinoco River hydrographic divide is a representation of the hydrographic water divide that delineates the separation between the Orinoco Basin and the Amazon Basin. (The Orinoco Basin flows west--north--northeast into the Caribbean; the Amazon Basin flows east into the western Atlantic in the extreme northeast of Brazil.)
Essentially the river divide is a west-flowing, upriver section of Venezuela\'s Orinoco River with an outflow to the south into the Amazon Basin. This named outflow is the Casiquiare canal, which, as it heads downstream (southerly), picks up speed and also accumulates water volume.
The greatest manifestation of the divide is during floods. During flood stage, the Casiquiare\'s main outflow point into the Rio Negro is supplemented by an overflow that is a second, and more minor, entry river bifurcation into the Rio Negro and upstream from its major, common low-water entry confluence with the Rio Negro. At flood, the river becomes an area flow source, far more than a narrow confined river.
The Casiquiare canal connects the upper Orinoco, 9 mi below the mission of Esmeraldas, with the Rio Negro affluent of the Amazon River near the town of San Carlos.
The simplest description (besides the entire area-floodplain) of the water divide is a \"south-bank Orinoco River strip\" at the exit point of the Orinoco, also the origin of the Casiquiare canal. However, during the Orinoco\'s flood stage, that single, simply defined \"origin of the canal\" is turned into a region, and an entire strip along the southern bank of the Orinoco River
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# Cuboctahedron
A **cuboctahedron** is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e., an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral. Its dual polyhedron is the rhombic dodecahedron.
## Construction
The cuboctahedron can be constructed in many ways:
- Its construction can be started by attaching two regular triangular cupolas base-to-base. This is similar to one of the Johnson solids, triangular orthobicupola. The difference is that the triangular orthobicupola is constructed with one of the cupolas twisted so that similar polygonal faces are adjacent, whereas the cuboctahedron is not. As a result, the cuboctahedron may also called the *triangular gyrobicupola*.
- Its construction can be started from a cube or a regular octahedron, marking the midpoints of their edges, and cutting off all the vertices at those points. This process is known as rectification, making the cuboctahedron being named the *rectified cube* and *rectified octahedron*.
- An alternative construction is by cutting off all vertices (truncation) of a regular tetrahedron and beveling the edges. This process is termed cantellation, lending the cuboctahedron an alternate name of *cantellated tetrahedron*.
From all of these constructions, the cuboctahedron has 14 faces: 8 equilateral triangles and 6 squares. It also has 24 edges and 12 vertices.
The Cartesian coordinates for the vertices of a cuboctahedron with edge length $\sqrt{2}$ centered at the origin are: $(\pm 1, \pm 1, 0), \qquad (\pm 1, 0, \pm 1), \qquad (0, \pm 1, \pm 1).$
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# Cuboctahedron
## Properties
### Measurement and other metric properties {#measurement_and_other_metric_properties}
The surface area of a cuboctahedron $A$ can be determined by summing all the area of its polygonal faces. The volume of a cuboctahedron $V$ can be determined by slicing it off into two regular triangular cupolas, summing up their volume. Given that the edge length $a$, its surface area and volume are: $\begin{align}
A &= \left(6+2\sqrt{3}\right)a^2 &&\approx 9.464a^2 \\
V &= \frac{5 \sqrt{2}}{3} a^3 &&\approx 2.357a^3.
\end{align}$
The dihedral angle of a cuboctahedron can be calculated with the angle of triangular cupolas. The dihedral angle of a triangular cupola between square-to-triangle is approximately 125°, that between square-to-hexagon is 54.7°, and that between triangle-to-hexagon is 70.5°. Therefore, the dihedral angle of a cuboctahedron between square-to-triangle, on the edge where the base of two triangular cupolas are attached is 54.7° + 70.5° approximately 125°. Therefore, the dihedral angle of a cuboctahedron between square-to-triangle is approximately 125°.
Buckminster Fuller found that the cuboctahedron is the only polyhedron in which the distance between its center to the vertex is the same as the length of its edges. In other words, it has the same length vectors in three-dimensional space, known as *vector equilibrium*. The rigid struts and the flexible vertices of a cuboctahedron may also be transformed progressively into a regular icosahedron, regular octahedron, regular tetrahedron. Fuller named this the *jitterbug transformation*.
A cuboctahedron has the Rupert property, meaning there is a polyhedron of the same or larger size that can pass through its hole.
### Symmetry and classification {#symmetry_and_classification}
The cuboctahedron is an Archimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex. The cuboctahedron has two symmetries, resulting from the constructions as has mentioned above: the same symmetry as the regular octahedron or cube, the octahedral symmetry $\mathrm{O}_\mathrm{h}$, and the same symmetry as the regular tetrahedron, tetrahedral symmetry $\mathrm{T}_\mathrm{d}$. The polygonal faces that meet for every vertex are two equilateral triangles and two squares, and the vertex figure of a cuboctahedron is 3.4.3.4. The dual of a cuboctahedron is rhombic dodecahedron.
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# Cuboctahedron
## Properties
### Radial equilateral symmetry {#radial_equilateral_symmetry}
In a cuboctahedron, the long radius (center to vertex) is the same as the edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Its center is like the apical vertex of a canonical pyramid: one edge length away from *all* the other vertices. (In the case of the cuboctahedron, the center is in fact the apex of 6 square and 8 triangular pyramids). This radial equilateral symmetry is a property of only a few uniform polytopes, including the two-dimensional hexagon, the three-dimensional cuboctahedron, and the four-dimensional 24-cell and 8-cell (tesseract). *Radially equilateral* polytopes are those that can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge. Therefore, all the interior elements which meet at the center of these polytopes have equilateral triangle inward faces, as in the dissection of the cuboctahedron into 6 square pyramids and 8 tetrahedra.
Each of these radially equilateral polytopes also occurs as cells of a characteristic space-filling tessellation: the tiling of regular hexagons, the rectified cubic honeycomb (of alternating cuboctahedra and octahedra), the 24-cell honeycomb and the tesseractic honeycomb, respectively. Each tessellation has a dual tessellation; the cell centers in a tessellation are cell vertices in its dual tessellation. The densest known regular sphere-packing in two, three and four dimensions uses the cell centers of one of these tessellations as sphere centers.
Because it is radially equilateral, the cuboctahedron\'s center is one edge length distant from the 12 vertices.
## Configuration matrix {#configuration_matrix}
The cuboctahedron can be represented as a configuration matrix with elements grouped by symmetry transitivity classes. A configuration matrix is a matrix in which the rows and columns correspond to the elements of a polyhedron as in the vertices, edges, and faces. The diagonal of a matrix denotes the number of each element that appears in a polyhedron, whereas the non-diagonal of a matrix denotes the number of the column\'s elements that occur in or at the row\'s element.
The cuboctahedron has 1 transitivity class of 12 vertices, 1 class of 24 edges, and 2 classes of faces: 8 triangular and 6 square; each element in a matrix\'s diagonal. The 24 edges can be seen in 4 central hexagons.
With octahedral symmetry (orbifold 432), the squares have the 4-fold symmetry, triangles the 3-fold symmetry, and vertices the 2-fold symmetry. With tetrahedral symmetry (orbifold 332) the 24 vertices split into 2 edge classes, and the 8 triangles split into 2 face classes. The square symmetry is reduced to 2-fold.
+---------------------------+----------------------------------------------+
| Octahedral symmetry (432) | |
+===========================+==============================================+
| | -------------- ------ ------ ------ ------ |
| | \(432\) v~1~ e~1~ f~1~ f~2~ |
| | v~1\ (Z~2~)~ 12 \|4 \|2 \|2 |
| | e~1~ \|2 24 \|1 \|1 |
| | f~1\ (Z~3~)~ \|3 \|3 8 \* |
| | f~2\ (Z~4~)~ \|4 \|4 \* 6 |
| | -------------- ------ ------ ------ ------ |
| | |
| | : Configuration |
+---------------------------+----------------------------------------------+
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# Cuboctahedron
## Graph
The skeleton of a cuboctahedron may be represented as the graph, one of the Archimedean graph. It has 12 vertices and 24 edges. It is quartic graph, which is four vertices connecting each vertex.
The graph of a cuboctahedron may be constructed as the line graph of the cubical graph, making it becomes the locally linear graph.
The 24 edges can be partitioned into 2 sets isomorphic to tetrahedral symmetry. The edges can also be partitioned into 4 hexagonal cycles, representing centrosymmetry, with only opposite vertices and edges in the same transitivity class.
+------------------------------+------------------------+----------------------+
| Octahedral (48 automorphism) | | Tetrahedral (24 aut) |
+==============================+========================+======================+
| | ------ ------ ------ | |
| | \\ v~1~ e~1~ | |
| | v~1~ 12 \|4 | |
| | e~1~ \|2 24 | |
| | ------ ------ ------ | |
| | | |
| | : Configuration | |
+------------------------------+------------------------+----------------------+
## Related polyhedra and honeycomb {#related_polyhedra_and_honeycomb}
The cuboctahedron shares its skeleton with the two nonconvex uniform polyhedra, the cubohemioctahedron and octahemioctahedron. These polyhedrons are constructed from the skeleton of a cuboctahedron in which the four hexagonal planes bisect its diagonal, intersecting its interior. Adding six squares or eight equilateral triangles results in the cubohemicotahedron or octahemioctahedron, respectively.
The cuboctahedron 2-covers the tetrahemihexahedron, which accordingly has the same abstract vertex figure (two triangles and two squares: $3 \cdot 4 \cdot 3 \cdot 4$) and half the vertices, edges, and faces. (The actual vertex figure of the tetrahemihexahedron is $3 \cdot 4 \cdot \frac{3}{2} \cdot 4$, with the $\frac{a}{2}$ factor due to the cross.)
The cuboctahedron can be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point. This dissection is expressed in the tetrahedral-octahedral honeycomb where pairs of square pyramids are combined into octahedra.
## Appearance
The cuboctahedron was probably known to Plato: Heron\'s *Definitiones* quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares
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# Christian Goldbach
**Christian Goldbach** (`{{IPAc-en|ˈ|ɡ|oʊ|l|d|b|ɑː|k}}`{=mediawiki} `{{respell|GOHLD|bahk}}`{=mediawiki}, `{{IPA|de|ˈkʁɪsti̯a(ː)n ˈɡɔltbax|lang}}`{=mediawiki}; 18 March 1690 -- 20 November 1764) was a Prussian mathematician connected with some important research mainly in number theory; he also studied law and took an interest in and a role in the Russian court. After traveling around Europe in his early life, he landed in Russia in 1725 as a professor at the newly founded Saint Petersburg Academy of Sciences. Goldbach jointly led the academy in 1737. However, he relinquished duties in the academy in 1742 and worked in the Russian Ministry of Foreign Affairs until his death in 1764. He is remembered today for Goldbach\'s conjecture and the Goldbach--Euler Theorem. He had a close friendship with famous mathematician Leonhard Euler, serving as inspiration for Euler\'s mathematical pursuits.
## Biography
### Early life {#early_life}
Born in the Duchy of Prussia\'s capital Königsberg, part of Brandenburg-Prussia, Goldbach was the son of a pastor. He studied at the Royal Albertus University. After finishing his studies he went on long educational trips from 1710 to 1724 through Europe, visiting other German states, England, the Netherlands, Italy, and France, meeting with many famous mathematicians, such as Gottfried Leibniz, Leonhard Euler, and Nicholas I Bernoulli. These acquaintances started Goldbach\'s interest in mathematics. He briefly attended Oxford University in 1713 and, while he was there, Goldbach studied mathematics with John Wallis and Isaac Newton. Also, Goldbach\'s travels fostered his interest in philology, archaeology, metaphysics, ballistics, and medicine. Between 1717 and 1724, Goldbach published his first few papers which, while minor, credited his mathematical ability. Back in Königsberg, he became acquainted with Georg Bernhard Bilfinger and Jakob Hermann.
### Saint Petersburg Academy of Sciences {#saint_petersburg_academy_of_sciences}
Goldbach followed Bilfinger and Hermann to the newly opened St. Petersburg Academy of Sciences in 1725. Christian Wolff had invited and had written recommendations for all the Germans who traveled to Saint Petersburg for the academy except Goldbach. Goldbach wrote to the president-designate of the academy, petitioning for a position in the academy, using his past publications and knowledge in medicine and law as qualifications. Goldbach was then hired to a five-year contract as a professor of mathematics and historian of the academy. As historian of the academy, he recorded each academy meeting from the opening of the school in 1725 until January 1728. Goldbach worked with famous mathematicians like Leonhard Euler, Daniel Bernoulli, Johann Bernoulli, and Jean le Rond d\'Alembert. Goldbach also played a part in Euler\'s decision to academically pursue mathematics instead of medicine, cementing mathematics as the premier research field of the academy in the 1730s.
### Russian government work {#russian_government_work}
In 1728, when Peter II became Tsar of Russia, Goldbach became Peter II and Anna\'s, Peter II\'s cousin, tutor. Peter II moved the Russian court from St. Petersburg to Moscow in 1729, so Goldbach followed him to Moscow. Goldbach started a correspondence with Euler in 1729, in which some of Goldbach\'s most important mathematics contributions can be found. Upon Peter II\'s death in 1730, Goldbach stopped teaching but continued to assist Empress Anna. In 1732, Goldbach returned to the St. Petersburg Academy of Sciences and stayed in the Russian government when Anna moved the court back to St. Petersburg. Upon return to the academy, Goldbach was named corresponding secretary. With Goldbach\'s return, his friend Euler continued his teaching and research at the academy as well. Then, in 1737, Goldbach and J.D. Schumacher took over the administration of the academy. Also, Goldbach took on duty in Russian court under Empress Anna. He managed to retain his influence in court after the death of Anna and the rule of Empress Elizabeth. In 1742 he entered the Russian Ministry of Foreign Affairs, stepping away from the academy once more. Goldbach was gifted land and increased salary for his good work and rise in the Russian government. In 1760, Goldbach created new guidelines for the education of the royal children which would remain in place for 100 years. He died on 20 November 1764, aged 74, in Moscow.
Christian Goldbach was multilingual -- he wrote a diary in German and Latin, his letters were written in German, Latin, French, and Italian and for official documents he used Russian, German and Latin.
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# Christian Goldbach
## Contributions
Goldbach is most noted for his correspondence with Leibniz, Euler, and Bernoulli, especially in his 1742 letter to Euler stating his Goldbach\'s conjecture. He also studied and proved some theorems on perfect powers, such as the Goldbach--Euler theorem, and made several notable contributions to analysis. He also proved a result concerning Fermat numbers that is called Goldbach\'s theorem.
### Impact on Euler {#impact_on_euler}
It is Goldbach and Euler\'s correspondence that contains some of Goldbach\'s most important contributions to mathematics, specifically number theory. Goldbach and Euler\'s friendship survived Goldbach\'s move to Moscow in 1728 and communication ensued. Their correspondence spanned 196 letters over 35 years written in Latin, German, and French. These letters spanned a wide range of topics, including various mathematics topics. Goldbach was the leading influence on Euler\'s interest and work in number theory. Most of the letters discuss Euler\'s research in number theory as well as differential calculus. Until the late 1750s, Euler\'s correspondence on his number theory research was almost exclusively with Goldbach. Goldbach\'s earlier mathematical work and ideas in letters to Euler directly influenced some of Euler\'s work. In 1729, Euler solved two problems pertaining to sequences which had stumped Goldbach. Ensuingly, Euler outlined the solutions to Goldbach. Also, in 1729 Goldbach closely approximated the Basel problem, which prompted Euler\'s interest and concurring breakthrough solution. Goldbach, through his letters, kept Euler focused on number theory in the 1730s by discussing Fermat\'s conjecture with Euler. Euler subsequently offered a proof to the conjecture, crediting Goldbach with introducing him to the subfield. Euler proceeded to write 560 writings, published posthumously in four volumes of Opera omnia, with Goldbach\'s influence guiding some of the writings. Goldbach\'s famous conjecture and his writings with Euler prove him to be one of a handful of mathematicians who understood complex number theory in light of Fermat\'s revolutionary ideas on the topic
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# Convex set
In geometry, a set of points is **convex** if it contains every line segment between two points in the set. For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex.
The boundary of a convex set in the plane is always a convex curve. The intersection of all the convex sets that contain a given subset `{{mvar|A}}`{=mediawiki} of Euclidean space is called the convex hull of `{{mvar|A}}`{=mediawiki}. It is the smallest convex set containing `{{mvar|A}}`{=mediawiki}.
A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets. The branch of mathematics devoted to the study of properties of convex sets and convex functions is called convex analysis.
Spaces in which convex sets are defined include the Euclidean spaces, the affine spaces over the real numbers, and certain non-Euclidean geometries.
## Definitions
Let `{{mvar|S}}`{=mediawiki} be a vector space or an affine space over the real numbers, or, more generally, over some ordered field (this includes Euclidean spaces, which are affine spaces). A subset `{{mvar|C}}`{=mediawiki} of `{{mvar|S}}`{=mediawiki} is **convex** if, for all `{{mvar|x}}`{=mediawiki} and `{{mvar|y}}`{=mediawiki} in `{{mvar|C}}`{=mediawiki}, the line segment connecting `{{mvar|x}}`{=mediawiki} and `{{mvar|y}}`{=mediawiki} is included in `{{mvar|C}}`{=mediawiki}.
This means that the affine combination `{{math|(1 − ''t'')''x'' + ''ty''}}`{=mediawiki} belongs to `{{mvar|C}}`{=mediawiki} for all `{{mvar|x,y}}`{=mediawiki} in `{{mvar|C}}`{=mediawiki} and `{{mvar|t}}`{=mediawiki} in the interval `{{math|[0, 1]}}`{=mediawiki}. This implies that convexity is invariant under affine transformations. Further, it implies that a convex set in a real or complex topological vector space is path-connected (and therefore also connected).
A set `{{mvar|C}}`{=mediawiki} is **`{{visible anchor|strictly convex}}`{=mediawiki}** if every point on the line segment connecting `{{mvar|x}}`{=mediawiki} and `{{mvar|y}}`{=mediawiki} other than the endpoints is inside the topological interior of `{{mvar|C}}`{=mediawiki}. A closed convex subset is strictly convex if and only if every one of its boundary points is an extreme point.
A set `{{mvar|C}}`{=mediawiki} is **absolutely convex** if it is convex and balanced.
### Examples
The convex subsets of `{{math|'''R'''}}`{=mediawiki} (the set of real numbers) are the intervals and the points of `{{math|'''R'''}}`{=mediawiki}. Some examples of convex subsets of the Euclidean plane are solid regular polygons, solid triangles, and intersections of solid triangles. Some examples of convex subsets of a Euclidean 3-dimensional space are the Archimedean solids and the Platonic solids. The Kepler-Poinsot polyhedra are examples of non-convex sets.
### Non-convex set {#non_convex_set}
A set that is not convex is called a *non-convex set*. A polygon that is not a convex polygon is sometimes called a concave polygon, and some sources more generally use the term *concave set* to mean a non-convex set, but most authorities prohibit this usage.
The complement of a convex set, such as the epigraph of a concave function, is sometimes called a *reverse convex set*, especially in the context of mathematical optimization.
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# Convex set
## Properties
Given `{{mvar|r}}`{=mediawiki} points `{{math|''u''<sub>1</sub>, ..., ''u<sub>r</sub>''}}`{=mediawiki} in a convex set `{{mvar|S}}`{=mediawiki}, and `{{mvar|r}}`{=mediawiki} nonnegative numbers `{{math|''λ''<sub>1</sub>, ..., ''λ<sub>r</sub>''}}`{=mediawiki} such that `{{math|''λ''<sub>1</sub> + ... + ''λ<sub>r</sub>'' {{=}}`{=mediawiki} 1}}, the affine combination $\sum_{k=1}^r\lambda_k u_k$ belongs to `{{mvar|S}}`{=mediawiki}. As the definition of a convex set is the case `{{math|1=''r'' = 2}}`{=mediawiki}, this property characterizes convex sets.
Such an affine combination is called a convex combination of `{{math|''u''<sub>1</sub>, ..., ''u<sub>r</sub>''}}`{=mediawiki}. The **convex hull** of a subset `{{mvar|S}}`{=mediawiki} of a real vector space is defined as the intersection of all convex sets that contain `{{mvar|S}}`{=mediawiki}. More concretely, the convex hull is the set of all convex combinations of points in `{{mvar|S}}`{=mediawiki}. In particular, this is a convex set.
A *(bounded) convex polytope* is the convex hull of a finite subset of some Euclidean space `{{math|'''R'''<sup>''n''</sup>}}`{=mediawiki}.
### Intersections and unions {#intersections_and_unions}
The collection of convex subsets of a vector space, an affine space, or a Euclidean space has the following properties:
1. The empty set and the whole space are convex.
2. The intersection of any collection of convex sets is convex.
3. The *union* of a collection of convex sets is convex if those sets form a chain (a totally ordered set) under inclusion. For this property, the restriction to chains is important, as the union of two convex sets need not be convex.
### Closed convex sets {#closed_convex_sets}
Closed convex sets are convex sets that contain all their limit points. They can be characterised as the intersections of *closed half-spaces* (sets of points in space that lie on and to one side of a hyperplane).
From what has just been said, it is clear that such intersections are convex, and they will also be closed sets. To prove the converse, i.e., every closed convex set may be represented as such intersection, one needs the supporting hyperplane theorem in the form that for a given closed convex set `{{mvar|C}}`{=mediawiki} and point `{{mvar|P}}`{=mediawiki} outside it, there is a closed half-space `{{mvar|H}}`{=mediawiki} that contains `{{mvar|C}}`{=mediawiki} and not `{{mvar|P}}`{=mediawiki}. The supporting hyperplane theorem is a special case of the Hahn--Banach theorem of functional analysis.
### Face of a convex set {#face_of_a_convex_set}
A **face** of a convex set $C$ is a convex subset $F$ of $C$ such that whenever a point $p$ in $F$ lies strictly between two points $x$ and $y$ in $C$, both $x$ and $y$ must be in $F$. Equivalently, for any $x,y\in C$ and any real number $0<t<1$ such that $(1-t)x+ty$ is in $F$, $x$ and $y$ must be in $F$. According to this definition, $C$ itself and the empty set are faces of $C$; these are sometimes called the *trivial faces* of $C$. An **extreme point** of $C$ is a point that is a face of $C$.
Let $C$ be a convex set in $\R^n$ that is compact (or equivalently, closed and bounded). Then $C$ is the convex hull of its extreme points. More generally, each compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points (the Krein--Milman theorem).
For example:
- A triangle in the plane (including the region inside) is a compact convex set. Its nontrivial faces are the three vertices and the three edges. (So the only extreme points are the three vertices.)
- The only nontrivial faces of the closed unit disk $\{ (x,y) \in \R^2: x^2+y^2 \leq 1 \}$ are its extreme points, namely the points on the unit circle $S^1 = \{ (x,y) \in \R^2: x^2+y^2=1 \}$.
### Convex sets and rectangles {#convex_sets_and_rectangles}
Let `{{mvar|C}}`{=mediawiki} be a convex body in the plane (a convex set whose interior is non-empty). We can inscribe a rectangle *r* in `{{mvar|C}}`{=mediawiki} such that a homothetic copy *R* of *r* is circumscribed about `{{mvar|C}}`{=mediawiki}. The positive homothety ratio is at most 2 and: $\tfrac{1}{2} \cdot\operatorname{Area}(R) \leq \operatorname{Area}(C) \leq 2\cdot \operatorname{Area}(r)$\
### Blaschke-Santaló diagrams {#blaschke_santaló_diagrams}
The set $\mathcal{K}^2$ of all planar convex bodies can be parameterized in terms of the convex body diameter *D*, its inradius *r* (the biggest circle contained in the convex body) and its circumradius *R* (the smallest circle containing the convex body). In fact, this set can be described by the set of inequalities given by $2r \le D \le 2R$ $R \le \frac{\sqrt{3}}{3} D$ $r + R \le D$ $D^2 \sqrt{4R^2-D^2} \le 2R (2R + \sqrt{4R^2 -D^2})$ and can be visualized as the image of the function *g* that maps a convex body to the `{{math|'''R'''<sup>2</sup>}}`{=mediawiki} point given by (*r*/*R*, *D*/2*R*). The image of this function is known a (*r*, *D*, *R*) Blachke-Santaló diagram. Alternatively, the set $\mathcal{K}^2$ can also be parametrized by its width (the smallest distance between any two different parallel support hyperplanes), perimeter and area.
### Other properties {#other_properties}
Let *X* be a topological vector space and $C \subseteq X$ be convex.
- $\operatorname{Cl} C$ and $\operatorname{Int} C$ are both convex (i.e. the closure and interior of convex sets are convex).
- If $a \in \operatorname{Int} C$ and $b \in \operatorname{Cl} C$ then $[a, b[ \, \subseteq \operatorname{Int} C$ (where $[a, b[ \, := \left\{ (1 - r) a + r b : 0 \leq r < 1 \right\}$).
- If $\operatorname{Int} C \neq \emptyset$ then:
- $\operatorname{cl} \left( \operatorname{Int} C \right) = \operatorname{Cl} C$, and
- $\operatorname{Int} C = \operatorname{Int} \left( \operatorname{Cl} C \right) = C^i$, where $C^{i}$ is the algebraic interior of *C*.
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# Convex set
## Convex hulls and Minkowski sums {#convex_hulls_and_minkowski_sums}
### Convex hulls {#convex_hulls}
Every subset `{{mvar|A}}`{=mediawiki} of the vector space is contained within a smallest convex set (called the *convex hull* of `{{mvar|A}}`{=mediawiki}), namely the intersection of all convex sets containing `{{mvar|A}}`{=mediawiki}. The convex-hull operator Conv() has the characteristic properties of a closure operator:
- *extensive*: `{{math|''S'' ⊆ Conv(''S'')}}`{=mediawiki},
- *non-decreasing*: `{{math|''S'' ⊆ ''T''}}`{=mediawiki} implies that `{{math|Conv(''S'') ⊆ Conv(''T'')}}`{=mediawiki}, and
- *idempotent*: `{{math|Conv(Conv(''S'')) {{=}}`{=mediawiki} Conv(*S*)}}.
The convex-hull operation is needed for the set of convex sets to form a lattice, in which the \"*join*\" operation is the convex hull of the union of two convex sets $\operatorname{Conv}(S)\vee\operatorname{Conv}(T) = \operatorname{Conv}(S\cup T) = \operatorname{Conv}\bigl(\operatorname{Conv}(S)\cup\operatorname{Conv}(T)\bigr).$ The intersection of any collection of convex sets is itself convex, so the convex subsets of a (real or complex) vector space form a complete lattice.
### Minkowski addition {#minkowski_addition}
In a real vector-space, the *Minkowski sum* of two (non-empty) sets, `{{math|''S''<sub>1</sub>}}`{=mediawiki} and `{{math|''S''<sub>2</sub>}}`{=mediawiki}, is defined to be the set `{{math|''S''<sub>1</sub> + ''S''<sub>2</sub>}}`{=mediawiki} formed by the addition of vectors element-wise from the summand-sets $S_1+S_2=\{x_1+x_2: x_1\in S_1, x_2\in S_2\}.$ More generally, the *Minkowski sum* of a finite family of (non-empty) sets `{{math|''S<sub>n</sub>''}}`{=mediawiki} is the set formed by element-wise addition of vectors $\sum_n S_n = \left \{ \sum_n x_n : x_n \in S_n \right \}.$
For Minkowski addition, the *zero set* `{{math|{0} }}`{=mediawiki} containing only the zero vector `{{math|0}}`{=mediawiki} has special importance: For every non-empty subset S of a vector space $S+\{0\}=S;$ in algebraic terminology, `{{math|{0} }}`{=mediawiki} is the identity element of Minkowski addition (on the collection of non-empty sets).
### Convex hulls of Minkowski sums {#convex_hulls_of_minkowski_sums}
Minkowski addition behaves well with respect to the operation of taking convex hulls, as shown by the following proposition:
Let `{{math|''S''<sub>1</sub>, ''S''<sub>2</sub>}}`{=mediawiki} be subsets of a real vector-space, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls $\operatorname{Conv}(S_1+S_2)=\operatorname{Conv}(S_1)+\operatorname{Conv}(S_2).$
This result holds more generally for each finite collection of non-empty sets: $\text{Conv}\left ( \sum_n S_n \right ) = \sum_n \text{Conv} \left (S_n \right).$
In mathematical terminology, the operations of Minkowski summation and of forming convex hulls are commuting operations.
### Minkowski sums of convex sets {#minkowski_sums_of_convex_sets}
The Minkowski sum of two compact convex sets is compact. The sum of a compact convex set and a closed convex set is closed.
The following famous theorem, proved by Dieudonné in 1966, gives a sufficient condition for the difference of two closed convex subsets to be closed. It uses the concept of a **recession cone** of a non-empty convex subset *S*, defined as: $\operatorname{rec} S = \left\{ x \in X \, : \, x + S \subseteq S \right\},$ where this set is a convex cone containing $0 \in X$ and satisfying $S + \operatorname{rec} S = S$. Note that if *S* is closed and convex then $\operatorname{rec} S$ is closed and for all $s_0 \in S$, $\operatorname{rec} S = \bigcap_{t > 0} t (S - s_0).$
**Theorem** (Dieudonné). Let *A* and *B* be non-empty, closed, and convex subsets of a locally convex topological vector space such that $\operatorname{rec} A \cap \operatorname{rec} B$ is a linear subspace. If *A* or *B* is locally compact then *A* − *B* is closed.
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# Convex set
## Generalizations and extensions for convexity {#generalizations_and_extensions_for_convexity}
The notion of convexity in the Euclidean space may be generalized by modifying the definition in some or other aspects. The common name \"generalized convexity\" is used, because the resulting objects retain certain properties of convex sets.
### Star-convex (star-shaped) sets {#star_convex_star_shaped_sets}
Let `{{mvar|C}}`{=mediawiki} be a set in a real or complex vector space. `{{mvar|C}}`{=mediawiki} is **star convex (star-shaped)** if there exists an `{{math|''x''<sub>0</sub>}}`{=mediawiki} in `{{mvar|C}}`{=mediawiki} such that the line segment from `{{math|''x''<sub>0</sub>}}`{=mediawiki} to any point `{{mvar|y}}`{=mediawiki} in `{{mvar|C}}`{=mediawiki} is contained in `{{mvar|C}}`{=mediawiki}. Hence a non-empty convex set is always star-convex but a star-convex set is not always convex.
### Orthogonal convexity {#orthogonal_convexity}
An example of generalized convexity is **orthogonal convexity**.
A set `{{mvar|S}}`{=mediawiki} in the Euclidean space is called **orthogonally convex** or **ortho-convex**, if any segment parallel to any of the coordinate axes connecting two points of `{{mvar|S}}`{=mediawiki} lies totally within `{{mvar|S}}`{=mediawiki}. It is easy to prove that an intersection of any collection of orthoconvex sets is orthoconvex. Some other properties of convex sets are valid as well.
### Non-Euclidean geometry {#non_euclidean_geometry}
The definition of a convex set and a convex hull extends naturally to geometries which are not Euclidean by defining a geodesically convex set to be one that contains the geodesics joining any two points in the set.
### Order topology {#order_topology}
Convexity can be extended for a totally ordered set `{{mvar|X}}`{=mediawiki} endowed with the order topology.
Let `{{math|''Y'' ⊆ ''X''}}`{=mediawiki}. The subspace `{{mvar|Y}}`{=mediawiki} is a convex set if for each pair of points `{{math|''a'', ''b''}}`{=mediawiki} in `{{mvar|Y}}`{=mediawiki} such that `{{math|''a'' ≤ ''b''}}`{=mediawiki}, the interval `{{math|[''a'', ''b''] {{=}}`{=mediawiki} {*x* ∈ *X* {{!}} *a* ≤ *x* ≤ *b*} }} is contained in `{{mvar|Y}}`{=mediawiki}. That is, `{{mvar|Y}}`{=mediawiki} is convex if and only if for all `{{math|''a'', ''b''}}`{=mediawiki} in `{{mvar|Y}}`{=mediawiki}, `{{math|''a'' ≤ ''b''}}`{=mediawiki} implies `{{math|[''a'', ''b''] ⊆ ''Y''}}`{=mediawiki}.
A convex set is `{{em|not}}`{=mediawiki} connected in general: a counter-example is given by the subspace {1,2,3} in `{{math|'''Z'''}}`{=mediawiki}, which is both convex and not connected.
### Convexity spaces {#convexity_spaces}
The notion of convexity may be generalised to other objects, if certain properties of convexity are selected as axioms.
Given a set `{{mvar|X}}`{=mediawiki}, a **convexity** over `{{mvar|X}}`{=mediawiki} is a collection `{{math|''𝒞''}}`{=mediawiki} of subsets of `{{mvar|X}}`{=mediawiki} satisfying the following axioms:
1. The empty set and `{{mvar|X}}`{=mediawiki} are in `{{math|''𝒞''}}`{=mediawiki}.
2. The intersection of any collection from `{{math|''𝒞''}}`{=mediawiki} is in `{{math|''𝒞''}}`{=mediawiki}.
3. The union of a chain (with respect to the inclusion relation) of elements of `{{math|''𝒞''}}`{=mediawiki} is in `{{math|''𝒞''}}`{=mediawiki}.
The elements of `{{math|''𝒞''}}`{=mediawiki} are called convex sets and the pair `{{math|(''X'', ''𝒞'')}}`{=mediawiki} is called a **convexity space**. For the ordinary convexity, the first two axioms hold, and the third one is trivial.
For an alternative definition of abstract convexity, more suited to discrete geometry, see the *convex geometries* associated with antimatroids.
### Convex spaces {#convex_spaces}
Convexity can be generalised as an abstract algebraic structure: a space is convex if it is possible to take convex combinations of points
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# Chupacabra
The **chupacabra** or ***chupacabras*** (`{{IPA|es|tʃupaˈkaβɾas}}`{=mediawiki}, literally \'goat-sucker\', from *chupa*, \'sucks\', and *cabras*, \'goats\') is a legendary creature, or cryptid, in the folklore of parts of the Americas. The name comes from the animal\'s purported vampirism `{{Ndash}}`{=mediawiki} the chupacabra is said to attack and drink the blood of livestock, including goats.
Physical descriptions of the creature vary. In Puerto Rico and in Hispanic America it is generally described as a heavy creature, reptilian and alien-like, roughly the size of a small bear, and with a row of spines reaching from the neck to the base of the tail, while in the Southwestern United States it is depicted as more dog-like.
Initial sightings and accompanying descriptions first occurred in Puerto Rico in 1995. The creature has since been reported as far north as Maine, as far south as Chile, and even outside the Americas in countries like Russia and the Philippines. All of the reports are anecdotal and have been disregarded as uncorroborated or lacking evidence. Sightings in northern Mexico and the Southern United States have been verified as canids afflicted by mange.
## Name
can be literally translated as \'goat-sucker\', from *chupar* (\'to suck\') and *cabras* (\'goats\'). It is known as both *chupacabras* and *chupacabra* throughout the Americas, with the former being the original name, and the latter a regularization. The name is attributed to Puerto Rican comedian Silverio Pérez, who coined the label in 1995 while commenting on the attacks as a San Juan radio deejay.
## History
In 1975, a series of livestock killings in the small town of Moca, Puerto Rico were attributed to *el vampiro de Moca* (\'the vampire of Moca\'). Initially, it was suspected that the killings were committed by a Satanic cult; later more killings were reported around the island, and many farms reported loss of animal life. Each of the animals was reported to have had its body bled dry through a series of small circular incisions.
The first reported attack eventually attributed to the actual chupacabras occurred in March 1995. Eight sheep were discovered dead in Puerto Rico, each with three puncture wounds in the chest area and reportedly completely drained of blood. A few months later, in August, an eyewitness named Madelyne Tolentino reported seeing the creature in the Puerto Rican town of Canóvanas, where as many as 150 farm animals and pets were reportedly killed.
Puerto Rican comedian and entrepreneur Silverio Pérez is credited with coining the term *chupacabras* soon after the first incidents were reported in the press. Shortly after the first reported incidents in Puerto Rico, other animal deaths were reported in other countries, such as Argentina, Bolivia, Brazil, Chile, Colombia, Dominican Republic, El Salvador, Honduras, Mexico, Nicaragua, Panama, Peru, and the United States.
In 2019 a video recorded by *Mundo Ovni* showed the results of a supposed attack on chickens in the Seburuquillo sector of Lares, Puerto Rico.
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# Chupacabra
## Reputed origin {#reputed_origin}
A five-year investigation by Benjamin Radford, documented in his 2011 book *Tracking the Chupacabra*, concluded that the description given by the original eyewitness in Puerto Rico, Madelyne Tolentino, was based on the creature Sil in the 1995 science-fiction horror film *Species*. The alien creature Sil is nearly identical to Tolentino\'s chupacabra eyewitness account and she had seen the movie before her report: \"It was a creature that looked like the chupacabra, with spines on its back and all\... The resemblance to the chupacabra was really impressive\", Tolentino reported. Radford revealed that Tolentino \"believed that the creatures and events she saw in *Species* were happening in reality in Puerto Rico at the time\", and therefore concludes that \"the most important chupacabra description cannot be trusted\". This, Radford believes, seriously undermines the credibility of the chupacabra as a real animal.
The reports of blood-sucking by the chupacabra were never confirmed by a necropsy, the only way to conclude that the animal was drained of blood. Dr. David Morales, a Puerto Rican veterinarian with the Department of Agriculture, analyzed 300 reported victims of the chupacabra and found that they had not been bled dry.
Radford divided the chupacabra reports into two categories: the reports from Puerto Rico and Latin America, where animals were attacked and it is supposed their blood was extracted; and the reports in the United States of mammals, mostly dogs and coyotes with mange, that people call \"chupacabra\" due to their unusual appearance.
In 2010, University of Michigan biologist Barry O\'Connor concluded that all the chupacabra reports in the United States were simply coyotes infected with the parasite *Sarcoptes scabiei*, whose symptoms would explain most of the features of the chupacabra: they would be left with little fur, thickened skin, and a rank odor. O\'Connor theorized that the attacks on goats occurred \"because these animals are greatly weakened, \[so\] they\'re going to have a hard time hunting. So they may be forced into attacking livestock because it\'s easier than running down a rabbit or a deer.\" Both dogs and coyotes can kill and not consume the prey, either because they are inexperienced, or due to injury or difficulty in killing the prey. The prey can survive the attack and die afterwards from internal bleeding or circulatory shock. The presence of two holes in the neck, corresponding with the canine teeth, are to be expected since this is the only way that most land carnivores have to catch their prey. There are reports of stray Mexican hairless dogs being mistaken for chupacabras.
## Appearance
The most common description of the chupacabra is that of a reptile-like creature, said to have leathery or scaly greenish-gray skin and sharp spines or quills running down its back. It is said to be approximately 3 to high, and stands and hops in a fashion similar to that of a kangaroo. This description was the chief one given to the few Puerto Rican reports in 1995 that claimed to have sighted the creature, with similar reports in parts of Chile and Argentina following.
Another common description of the chupacabra is of a strange breed of wild dog. This form is mostly hairless and has a pronounced spinal ridge, unusually pronounced eye sockets, fangs, and claws. This description started to appear in the early 2000s from reports trailing north from the Yucatán Peninsula, northern Mexico, and then into the United States; becoming the predominant description since. Unlike conventional predators, the chupacabra is said to drain all of the animal\'s blood (and sometimes organs) usually through three holes in the shape of a downwards-pointing triangle, but sometimes through only one or two holes.
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# Chupacabra
## Plausibility of existence {#plausibility_of_existence}
The chupacabra panic first started in late 1995, Puerto Rico: farmers were mass reporting the mysterious killings of various livestock. In these reports, the farmers recalled two puncture wounds on the animal carcasses. Chupacabra killings were soon associated with a seemingly untouched animal carcass other than puncture wounds which were said to be used to suck the blood out of the victim. Reports of such killings began to spread around and eventually out of the country, reaching areas such as Mexico, Brazil, Chile, and the Southern area of the United States.
Most notably, these areas experience frequent, and extreme dry seasons; in the cases of the Puerto Rican reports of 1995 and the Mexican reports of 1996, both countries were currently experiencing or dealing with the aftermath of severe droughts. Investigations carried out in both countries at this time noted a certain dramatic violence in these killings. These environmental conditions could provide a simple explanation for the livestock killings: wild predators losing their usual prey to the drought, therefore being forced to hunt the livestock of farmers for sustenance. Thus, the same theory can be applied to many of the other \'chupacabra\' attacks: that the dry weather had created a more competitive environment for native predators, leading them to prey on livestock to survive. Such an idea can also explain the increased violence in the killings; hungry and desperate predators are driven to hunt livestock to avoid starvation, causing an increase in both the number of livestock killings, and the viciousness of each one.
Evidence of such is provided in page 179 of Benjamin Radford\'s book, *Tracking the Chupacabra: The Vampire Beast in Fact, Fiction, and Folklore.* Radford\'s chart highlights ten significant reports of chupacabra attacks, seven of which had a carcass recovered and examined; these autopsies concluded the causes of death as various animal attacks, as displayed though the animal DNA found on the carcasses. Radford provides further evidence in pages 161-162 of his book, displaying animals who are proven to have fallen victim to regular coyote attacks; thus, explaining that it is not unusual for an animal carcass to be left uneaten while only displaying puncture wounds and/or minimal signs of attack.
The plausibility of the chupacabra\'s existence is also discredited by the varying descriptions of the creature. Depending on the reported sighting, the creature is described with thick skin or fur, wings or no wings, a long tail or no tail, is bat-like, dog-like, or even alien-like. Evidently, the chupacabra has a wide variety of descriptions; to the point where it is hard to believe that all the sightings are of the same creature. A very likely explanation for this phenomenon is that individuals who had heard of the newly popular chupacabra had the creature\'s name fresh in their mind before they happened to see a strange looking animal. They then resort to make sense of their encounter by labelling it as the recently \'discovered\' monster, instead of a more realistic explanation. For example, some scientists hypothesize that what many believe to be a chupacabra is a wild or domestic dog affected by mange, a disease causing a thick buildup of skin and hair loss.
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# Chupacabra
## Related legends {#related_legends}
The \"Ozark Howler\", a large bear-like animal, is the subject of a similar legend.
The Peuchens of Chile also share similarities in their supposed habits, but instead of being dog-like they are described as winged snakes. This legend may have originated from the vampire bat, an animal endemic to the region.
In the Philippines the Sigbin shares many of the chupacabra\'s descriptions.
In 2018 there were reports of suspected chupacabras in Manipur, India. Many domestic animals and poultry were killed in a manner similar to other chupacabra attacks, and several people reported that they had seen creatures. Forensic experts opined that street dogs were responsible for mass killing of domestic animals and poultry after studying the remnants of a corpse.
## Media
- A chupacabra is referred to in the 2009 novel *Drive Your Plow Over the Bones of the Dead*.
- The debut album by Imani Coppola is titled *Chupacabra*.
- In *Indigenous* (2014), the chupacabra is the main antagonist.
- The myth of the chupacabra is mocked in a 2012 episode of the cartoon series *South Park*, titled \"Jewpacabra\", in which antisemitic main character Eric Cartman claims to have seen a Jewish Chupacabra that kills children on Easter.
- The chupacabra was included as one of several vinyl figurines in Cryptozoic Entertainment\'s Cryptkins blind box toy line in 2018. A redesigned series of figurines, including an updated chupacabra, was released in August 2020.
- The search for a chupacabra was featured in the 1997 *The X-Files* episode \"El Mundo Gira\".
- \"Chupacabra\" was the title of the midseason finale of season 4 of the supernatural drama television series *Grimm*, in December 2014.
- *Teen Titans Academy*, a DC Comics book, has a bat-like metahuman called Chupacabra, whose alter ego is Diego Pérez, named in honour of George Pérez (the artist that initially illustrated the Teen Titans).
- A 1999 episode of *Futurama* features a monster called \"El Chupanibre\".
- In the *Jackie Chan Adventures* episode \"The Curse of El Chupacabra\", Jackie Chan\'s friend El Toro gets scratched and infected by a Chupacabra, causing him to transform into another Chupacabra every night, much like a werewolf.
- In season 3 of *Workaholics* called \"To Kill a Chupacabraj\", Blake finds what he believes to be the deceased corpse of the Rancho Chupacabra in the pool, though it turns out to be the neighbor\'s dog.
- In the Netflix original series *The Imperfects*, the character of Juan Ruiz transforms into a chupacabra whenever anyone he cares about is in danger.
- The 2016 film *La leyenda del Chupacabras* features the titular Chupacabra initially as an antagonist before revealing the creature is merely trying to rescue its family.
- The Brazilian Chupa-Cu legend created in 2017 takes its cues from the chupacabra.
- A \"Chupakabura\" plays the role of a tourism mascot for the fictional town of Manoyama in P.A. Works\' 2017 anime *Sakura Quest*. The spelling and pronunciation relates to a retired mascot called \"Kabura Kid\", whose name was a pun alluding to the Japanese word for turnips.
- The 2023 film *Chupa* is about a chupacabra that is saved from scientists who want to capture it to prove it is real and exploit it for medicine.
- The 2010-2011 *Super Sentai* series *Tensou Sentai Goseiger*\'s main antagonist Brajira of the Messiah assumes the guise Buredoran of the Chupacabra when working with the Yuumajuu, the villain faction of the second arc that is based on cryptids.
- The Ukrainian news program TSN used to broadcast fake news about the Chupacabra when no interesting news were there to broadcast.
- In a short titled \"Mission: Chupacabras\" from *Helluva Boss*, a Mexican goat-farmer mistakes Blitzo for a chupacabra and tries to sell him.
- *Chupacabra vs. The Alamo*, a 2013 made-for-TV movie.
- *Guns of El Chupacabra*, a 1997 martial arts based monster film
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# Fire (classical element)
**Fire** is one of the four classical elements along with earth, water and air in ancient Greek philosophy and science. Fire is considered to be both hot and dry and, according to Plato, is associated with the tetrahedron.
## Greek and Roman tradition {#greek_and_roman_tradition}
Fire is one of the four classical elements in ancient Greek philosophy and science. It was commonly associated with the qualities of energy, assertiveness, and passion. In one Greek myth, Prometheus stole *fire* from the gods to protect the otherwise helpless humans, but was punished for this charity.
Fire was one of many *archai* proposed by the pre-Socratics, most of whom sought to reduce the cosmos, or its creation, to a single substance. Heraclitus `{{nowrap|(c. 535 BCE –}}`{=mediawiki} `{{nowrap|c. 475 BCE)}}`{=mediawiki} considered *fire* to be the most fundamental of all elements. He believed fire gave rise to the other three elements: \"All things are an interchange for fire, and fire for all things, just like goods for gold and gold for goods.\" He had a reputation for obscure philosophical principles and for speaking in riddles. He described how fire gave rise to the other elements as the: \"upward-downward path\", (*ὁδὸς ἄνω κάτω*), a \"hidden harmony\" or series of transformations he called the \"turnings of fire\", (*πυρὸς τροπαὶ*), first into *sea*, and half that *sea* into *earth*, and half that *earth* into rarefied *air*. This is a concept that anticipates both the four classical elements of Empedocles and Aristotle\'s transmutation of the four elements into one another.
> This world, which is the same for all, no one of gods or men has made. But it always was and will be: an ever-living fire, with measures of it kindling, and measures going out.
Heraclitus regarded the soul as being a mixture of fire and water, with fire being the more noble part and water the ignoble aspect. He believed the goal of the soul is to be rid of water and become pure fire: the dry soul is the best and it is worldly pleasures that make the soul \"moist\". He was known as the \"weeping philosopher\" and died of hydropsy, a swelling due to abnormal accumulation of fluid beneath the skin.
However, Empedocles of Akragas `{{nowrap|(c. 495 –}}`{=mediawiki} `{{nowrap|c. 435 BCE)}}`{=mediawiki}, is best known for having selected all elements as his *archai* and by the time of Plato `{{nowrap|(427–}}`{=mediawiki}`{{nowrap|347 BCE)}}`{=mediawiki}, the four Empedoclian elements were well established. In the *Timaeus*, Plato\'s major cosmological dialogue, the Platonic solid he associated with fire was the tetrahedron which is formed from four triangles and contains the least volume with the greatest surface area. This also makes fire the element with the smallest number of sides, and Plato regarded it as appropriate for the heat of fire, which he felt is sharp and stabbing, (like one of the points of a tetrahedron).
Plato\'s student Aristotle `{{nowrap|(384–}}`{=mediawiki}`{{nowrap|322 BCE)}}`{=mediawiki} did not maintain his former teacher\'s geometric view of the elements, but rather preferred a somewhat more naturalistic explanation for the elements based on their traditional qualities. Fire the hot and dry element, like the other elements, was an abstract principle and not identical with the normal solids, liquids and combustion phenomena we experience:
> What we commonly call fire. It is not really fire, for fire is an excess of heat and a sort of ebullition; but in reality, of what we call air, the part surrounding the earth is moist and warm, because it contains both vapour and a dry exhalation from the earth.
According to Aristotle, the four elements rise or fall toward their natural place in concentric layers surrounding the center of the Earth and form the terrestrial or sublunary spheres.
In ancient Greek medicine, each of the four humours became associated with an element. Yellow bile was the humor identified with fire, since both were hot and dry. Other things associated with fire and yellow bile in ancient and medieval medicine included the season of summer, since it increased the qualities of heat and aridity; the choleric temperament (of a person dominated by the yellow bile humour); the masculine; and the eastern point of the compass.
In alchemy the chemical element of sulfur was often associated with fire and its alchemical symbol and its symbol was an upward-pointing triangle. In alchemic tradition, metals are incubated by fire in the womb of the Earth and alchemists only accelerate their development.
## Indian tradition {#indian_tradition}
Agni is a Hindu and Vedic deity. The word *agni* is Sanskrit for fire (noun), cognate with Latin *ignis* (the root of English *ignite*), Russian *огонь* (fire), pronounced *agon*. Agni has three forms: fire, lightning and the sun.
Agni is one of the most important of the Vedic gods. He is the god of fire and the accepter of sacrifices. The sacrifices made to Agni go to the deities because Agni is a messenger from and to the other gods. He is ever-young, because the fire is re-lit every day, yet he is also immortal. In Indian tradition fire is also linked to Surya or the Sun and Mangala or Mars, and with the south-east direction.
Teukāya ekendriya is a name used in Jain tradition which refers to Jīvas said to be reincarnated as fire.
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# Fire (classical element)
## Ceremonial magic {#ceremonial_magic}
Fire and the other Greek classical elements were incorporated into the Golden Dawn system. Philosophus (4=7) is the elemental grade attributed to fire; this grade is also attributed to the Qabalistic Sephirah Netzach and the planet Venus. The elemental weapon of fire is the Wand. Each of the elements has several associated spiritual beings. The archangel of fire is Michael, the angel is Aral, the ruler is Seraph, the king is Djin, and the fire elementals (following Paracelsus) are called salamanders. Fire is considered to be active; it is represented by the symbol for Leo and it is referred to the lower right point of the pentacle in the Supreme Invoking Ritual of the Pentacle. Many of these associations have since spread throughout the occult community.
## Tarot
Fire in tarot symbolizes conversion or passion. Many references to fire in tarot are related to the usage of fire in the practice of alchemy, in which the application of fire is a prime method of conversion, and everything that touches fire is changed, often beyond recognition. The symbol of fire was a cue pointing towards transformation, the chemical variant being the symbol delta, which is also the classical symbol for fire. Conversion symbolized can be good, for example, refining raw crudities to gold, as seen in The Devil. Conversion can also be bad, as in The Tower, symbolizing a downfall due to anger. Fire is associated with the suit of rods/wands, and as such, represents passion from inspiration. As an element, fire has mixed symbolism because it represents energy, which can be helpful when controlled, but volatile if left unchecked.
## Modern witchcraft {#modern_witchcraft}
Fire is one of the five elements that appear in most Wiccan traditions influenced by the Golden Dawn system of magic, and Aleister Crowley\'s mysticism, which was in turn inspired by the Golden Dawn.
## Freemasonry
In freemasonry, fire is present, for example, during the ceremony of winter solstice, a symbol also of renaissance and energy. Freemasonry takes the ancient symbolic meaning of fire and recognizes its double nature: creation, light, on the one hand, and destruction and purification, on the other
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# Air (classical element)
**Air** or **Wind** is one of the four classical elements along with water, earth and fire in ancient Greek philosophy and in Western alchemy.
## Greek and Roman tradition {#greek_and_roman_tradition}
According to Plato, it is associated with the octahedron; air is considered to be both hot and wet. The ancient Greeks used two words for air: *aer* meant the dim lower atmosphere, and *aether* meant the bright upper atmosphere above the clouds. Plato, for instance writes that \"So it is with air: there is the brightest variety which we call *aether*, the muddiest which we call mist and darkness, and other kinds for which we have no name\....\" Among the early Greek Pre-Socratic philosophers, Anaximenes (mid-6th century BCE) named air as the *arche*. A similar belief was attributed by some ancient sources to Diogenes Apolloniates (late 5th century BCE), who also linked air with intelligence and soul (*psyche*), but other sources claim that his *arche* was a substance between air and fire. Aristophanes parodied such teachings in his play *The Clouds* by putting a prayer to air in the mouth of Socrates.
Air was one of many *archai* proposed by the Pre-socratics, most of whom tried to reduce all things to a single substance. However, Empedocles of Acragas (c. 495-c. 435 BCE) selected four *archai* for his four roots: air, fire, water, and earth. Ancient and modern opinions differ as to whether he identified air by the divine name Hera, Aidoneus or even Zeus. Empedocles' roots became the four classical elements of Greek philosophy. Plato (427--347 BCE) took over the four elements of Empedocles. In the *Timaeus*, his major cosmological dialogue, the Platonic solid associated with air is the octahedron which is formed from eight equilateral triangles. This places air between fire and water which Plato regarded as appropriate because it is intermediate in its mobility, sharpness, and ability to penetrate. He also said of air that its minuscule components are so smooth that one can barely feel them.
Plato\'s student Aristotle (384--322 BCE) developed a different explanation for the elements based on pairs of qualities. The four elements were arranged concentrically around the center of the universe to form the sublunary sphere. According to Aristotle, air is both hot and wet and occupies a place between fire and water among the elemental spheres. Aristotle definitively separated air from aether. For him, aether was an unchanging, almost divine substance that was found only in the heavens, where it formed celestial spheres.
### Humorism and temperaments {#humorism_and_temperaments}
------------- ------------ ----------- ------------- ------------- ---------------- -----------------
**Humour** **Season** **Ages** **Element** **Organ** **Qualities** **Temperament**
Blood spring infancy air liver moist and warm sanguine
Yellow bile summer youth fire gallbladder warm and dry choleric
Black bile autumn adulthood earth spleen dry and cold melancholic
Phlegm winter old age water brain/lungs cold and moist phlegmatic
------------- ------------ ----------- ------------- ------------- ---------------- -----------------
In ancient Greek medicine, each of the four humours became associated with an element. Blood was the humor identified with air, since both were hot and wet. Other things associated with air and blood in ancient and medieval medicine included the season of spring, since it increased the qualities of heat and moisture; the sanguine temperament (of a person dominated by the blood humour); hermaphrodite (combining the masculine quality of heat with the feminine quality of moisture); and the northern point of the compass.
### Alchemy
thumb\|upright=0.4\|Alchemical symbol for air The alchemical symbol for air is an upward-pointing triangle, bisected by a horizontal line.
## Modern reception {#modern_reception}
The Hermetic Order of the Golden Dawn, founded in 1888, incorporates air and the other Greek classical elements into its teachings. The elemental weapon of air is the dagger which must be painted yellow with magical names and sigils written upon it in violet. Each of the elements has several associated spiritual beings. The archangel of air is Raphael, the angel is Chassan, the ruler is Ariel, the king is Paralda, and the air elementals (following Paracelsus) are called sylphs. Air is considerable and it is referred to the upper left point of the pentagram in the Supreme Invoking Ritual of the Pentagram. Many of these associations have since spread throughout the occult community.
In the Golden Dawn and many other magical systems, each element is associated with one of the cardinal points and is placed under the care of guardian Watchtowers. The Watchtowers derive from the Enochian system of magic founded by Dee. In the Golden Dawn, they are represented by the Enochian elemental tablets. Air is associated with the east, which is guarded by the First Watchtower.
Air is one of the five elements that appear in most Wiccan and Pagan traditions. Wicca in particular was influenced by the Golden Dawn system of magic and Aleister Crowley\'s mysticism.
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# Air (classical element)
## Parallels in non-Western traditions {#parallels_in_non_western_traditions}
Air is not one of the traditional five Chinese classical elements. Nevertheless, the ancient Chinese concept of *Qi* or *chi* is believed to be close to that of air. *Qi* is believed to be part of every living thing that exists, as a kind of \"life force\" or \"spiritual energy\". It is frequently translated as \"energy flow\", or literally as \"air\" or \"breath\". (For example, *tiānqì*, literally \"sky breath\", is the Chinese word for \"weather\"). The concept of qi is often reified, however no scientific evidence supports its existence.
The element air also appears as a concept in the Buddhist philosophy which has an ancient history in China.
Some Western modern occultists equate the Chinese classical element of metal with *air*, others with wood due to the elemental association of wind and wood in the bagua.
Enlil was the god of air in ancient Sumer. Shu was the ancient Egyptian deity of air and the husband of Tefnut, goddess of moisture. He became an emblem of strength by virtue of his role in separating Nut from Geb. Shu played a primary role in the Coffin Texts, which were spells intended to help the deceased reach the realm of the afterlife safely. On the way to the sky, the spirit had to travel through the air as one spell indicates: \"I have gone up in Shu, I have climbed on the sunbeams.\"
According to Jain beliefs, the element air is inhabited by one-sensed beings or spirits called vāyukāya ekendriya, sometimes said to inhabit various kinds of winds such as whirlwinds, cyclones, monsoons, west winds and trade winds. Prior to reincarnating into another lifeform, spirits can remain as vāyukāya ekendriya from anywhere between one instant to up to three-thousand years, depending on the karma of the spirits
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# Earth (classical element)
**Earth** is one of the classical elements, in some systems being one of the four along with air, fire, and water.
## European tradition {#european_tradition}
thumb\|left\|upright=0.7\|*Earth* (1681) by Benoît Massou, a statue of the *Grande Commande*, with allegorical attributes inspired by Cesare Ripa's *Iconologia*. Earth is one of the four classical elements in ancient Greek philosophy and science. It was commonly associated with qualities of heaviness, matter and the terrestrial world. Due to the hero cults, and chthonic underworld deities, the element of *earth* is also associated with the sensual aspects of both life and death in later occultism.
Empedocles of Acragas `{{nowrap|(c. 495 –}}`{=mediawiki} `{{nowrap|c. 435 BCE)}}`{=mediawiki} proposed four *archai* by which to understand the cosmos: *fire*,*air*, *water*, and *earth*. Plato (427--347 BCE) believed the elements were geometric forms (the platonic solids) and he assigned the cube to the element of *earth* in his dialogue *Timaeus*. Aristotle (384--322 BCE) believed *earth* was the heaviest element, and his theory of *natural place* suggested that any *earth--laden* substances, would fall quickly, straight down, towards the center of the *cosmos*.
In Classical Greek and Roman myth, various goddesses represented the Earth, seasons, crops and fertility, including Demeter and Persephone; Ceres; the Horae (goddesses of the seasons), and Proserpina; and Hades (Pluto) who ruled the souls of dead in the Underworld.
In ancient Greek medicine, each of the four humours became associated with an element. Black bile was the humor identified with earth, since both were cold and dry. Other things associated with earth and black bile in ancient and medieval medicine included the season of fall, since it increased the qualities of cold and aridity; the melancholic temperament (of a person dominated by the black bile humour); the feminine; and the southern point of the compass.
thumb\|left\|upright=0.4\|Alchemical symbol for earth
In alchemy, earth was believed to be primarily dry, and secondarily cold, (as per Aristotle). Beyond those classical attributes, the chemical substance salt, was associated with earth and its alchemical symbol was a downward-pointing triangle, bisected by a horizontal line.
## Indian tradition {#indian_tradition}
**Prithvi** (Sanskrit: *`{{IAST|pṛthvī}}`{=mediawiki}*, also *`{{IAST|pṛthivī}}`{=mediawiki}*) is the Hindu *earth* and mother goddess. According to one such tradition, she is the personification of the Earth itself; according to another, its actual mother, being *Prithvi Tattwa*, the essence of the element earth.
As *Prithvi Mata*, or \"Mother Earth\", she contrasts with *Dyaus Pita*, \"father sky\". In the Rigveda, *earth* and sky are frequently addressed as a duality, often indicated by the idea of two complementary \"half-shells.\" In addition, the element Earth is associated with Budha or Mercury who represents communication, business, mathematics and other practical matters.
Jainism mentions one-sensed beings or spirits believed to inhabit the element earth sometimes classified as pṛthvīkāya ekendriya.
## Ceremonial magic {#ceremonial_magic}
Earth and the other Greek classical elements were incorporated into the Golden Dawn system. Zelator is the elemental grade attributed to earth; this grade is also attributed to the Sephirot of Malkuth. The elemental weapon of earth is the Pentacle. Each of the elements has several associated spiritual beings. The archangel of earth is Uriel, the angel is Phorlakh, the ruler is Kerub, the king is Ghob, and the earth elementals (following Paracelsus) are called gnomes. Earth is considered to be passive; it is represented by the symbol for Taurus, and it is referred to the lower left point of the pentagram in the Supreme Invoking Ritual of the Pentagram. Many of these associations have since spread throughout the occult community.
It is sometimes represented by its Tattva or by a downward pointing triangle with a horizontal line through it.
## Modern witchcraft {#modern_witchcraft}
Earth is one of the five elements that appear in most Wiccan and Pagan traditions. Wicca in particular was influenced by the Golden Dawn system of magic, and Aleister Crowley\'s mysticism which was in turn inspired by the Golden Dawn.
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# Earth (classical element)
## Other traditions {#other_traditions}
*Earth* is represented in the Aztec religion by a house; to the Hindus, a lotus; to the Scythians, a plough; to the Greeks, a wheel; and in Christian iconography; bulls and birds
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# Blue Jam
***Blue Jam*** was an ambient, surreal dark comedy and horror radio programme created and directed by Chris Morris. It was broadcast on BBC Radio 1 in the early hours of the morning, for three series from 1997 to 1999.
The programme gained cult status due to its unique mix of surreal monologue, ambient soundtrack, synthesised voices, heavily edited broadcasts and recurring sketches. It featured vocal performances of Kevin Eldon, Julia Davis, Mark Heap, David Cann and Amelia Bullmore, with Morris himself delivering disturbing monologues, one of which was revamped and made into the BAFTA-winning short film *My Wrongs #8245--8249 & 117*. Writers who contributed to the programme included Graham Linehan, Arthur Mathews, Peter Baynham, David Quantick, Jane Bussmann, Robert Katz and the cast.
The programme was adapted into the TV series *Jam*, which aired in 2000.
## Production
On his inspiration for making the show, Morris commented: \"It was so singular, and it came from a mood, quite a desolate mood. I had this misty, autumnal, boggy mood anyway, so I just went with that. But no doubt getting to the end of something like *Brass Eye*, where you\'ve been forced to be a sort of surrogate lawyer, well, that\'s the most creatively stifling thing you could possibly do.\" Morris also described the show as being \"like the nightmares you have when you fall asleep listening to the BBC World Service\" (a reference to the World Service also appears in one of the monologues read by Morris).
Morris originally requested that the show be broadcast at 3 a.m. on Radio 1 \"because at that hour, on insomniac radio, the amplitude of terrible things is enormously overblown\". As a compromise, the show was broadcast at midnight without much promotion. Morris reportedly included sketches too graphic or transgressive for radio that he knew would be cut so as to make his other material seem less transgressive in comparison. During the airing of episode 6 of series one, a re-editing of the Archbishop of Canterbury\'s speech at Princess Diana\'s funeral was deemed too offensive for broadcast, and was switched with a different episode as it aired.
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# Blue Jam
## Format and style {#format_and_style}
Each episode opened (and closed) with a short spoken monologue (delivered by Morris) describing, in surreal, broken language, various bizarre feelings and situations (for example: \"when you sick so sad you cry, and in crying cry a whole leopard from your eye\"), set to ambient music interspersed with short clips of other songs and sounds. The introduction would always end with \"welcome in Blue Jam\", inviting the listener, who is presumably experiencing such feelings, to get lost in the program. (This format was replicated in the television adaptation *Jam*, often reusing opening monologues from series 3 of the radio series.) The sketches within dealt with heavy and taboo topics, such as murder, suicide, missing or dead children, and rape.
### Common recurring sketches {#common_recurring_sketches}
- **Doctor** (played by David Cann): \"The Doctor\" is a seemingly \"normal\" physician working in a standard British medical practice. However, he has a habit of treating his patients in bizarre and often disturbing ways, such as prescribing heroin for a sore jaw, kissing patients on various body parts to make swellings go away, making a man with a headache jump up and down to make his penis swing (while mirroring the patient\'s bewildered jumping himself) and making a patient leave and go into the next room so he can examine him over the telephone. His name is revealed to be **Michael Perlin** in several sketches.
- **The Monologue Man** (played by Chris Morris): Short stories, often up to 10 minutes in length, written from the perspective of a lonely and socially inept man. Each story usually involves the protagonist\'s acquaintance Suzy in some capacity.
- **Michael Alexander St. John**: A parody of hyperbolic and pun-laden radio presenting, St. John presents items such as the top 10 singles charts and the weekend\'s gigs in an incongruous upper class English accent
- **Bad Sex**: Short clips of two lovers (played by Julia Davis and Kevin Eldon) making increasingly bizarre erotic requests of one another, such as to \"shit your leg off\" and \"make your spunk come out green\".
- **The Interviewer** (played by Chris Morris): conducting real interviews with celebrities such as Andrew Morton and Jerry Springer, Morris confuses and mocks his subjects with ambiguous and odd questions.
- **Mr. Ventham** (played by Mark Heap): An extremely awkward man who requires one-to-one consultations with **Mr. Reilly** (played by David Cann), who seems to be his psychologist, for the most banal of matters.
- **Unflustered Parents** (David Cann and Julia Davis): A middle class couple that seem quite ambivalent to the fact their young son has been abducted from school or that their pet lions are eating their neighbours
The sketches not listed are often in the style of a documentary; characters speak as if being interviewed about a recent event. In one sketch, a character voiced by Morris describes a man attempting to commit suicide by jumping off a second-story balcony repeatedly; in another, an angry man (Eldon) shouts about how his car, after being picked up from the garage, is only four feet long.
### Radio stings {#radio_stings}
Morris included a series of \'radio stings\', bizarre sequences of sounds and prose as a parody of modern DJs\' own soundbites and self-advertising pieces. Each one revolves around a contemporary DJ, such as Chris Moyles, Jo Whiley and Mark Goodier, typically involving each DJ dying in a graphic way or going mad in some form -- for example, Chris Moyles covering himself in jam and hanging himself from the top of a building.
## Episodes
Three series were produced, with a total of eighteen episodes. All episodes were originally broadcast weekly on BBC Radio 1. Series 1 was broadcast from 14 November to 19 December 1997; series 2 was broadcast from 27 March to 1 May 1998; and series 3 broadcast from 21 January to 25 February 1999.
- Series 1 -- (Fridays) 14 November 1997 to 19 December 1997, from 00:00 to 01:00.
- Series 2 -- (Fridays) 27 March 1998 to 1 May 1998, from 01:00 to 02:00.
- Series 3 -- (Thursdays) 21 January 1999 to 25 February 1999, from 00:00 to 01:00.
The first five episodes of series 1 of *Blue Jam* were repeated by BBC Radio 4 Extra in February and March 2014, and series 2 was rebroadcast in December.
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# Blue Jam
## Music
*Blue Jam* features songs, generally of a downtempo nature, interspersed between (and sometimes during) sketches. Artists featured includes Massive Attack, Air, Morcheeba, The Chemical Brothers, Björk, Aphex Twin, Everything But the Girl and Dimitri from Paris, as well as various non-electronic artists including Sly and the Family Stone, Serge Gainsbourg, The Cardigans and Eels.
## Reception
*Blue Jam* was favourably reviewed on several occasions by *The Guardian* and also received a positive review by *The Independent*.
Digital Spy wrote in 2014: \"It\'s a heady cocktail that provokes an odd, unsettling reaction in the listener, yet *Blue Jam* is still thumpingly and frequently laugh-out-loud hilarious.\" *Hot Press* called it \"as odd as comedy gets\".
## CD release {#cd_release}
A CD of a number of *Blue Jam* sketches was released on 23 October 2000 by record label Warp. Although the CD claims to have 22 tracks, the last one, \"www.bishopslips.com\", is not a track, but rather a reference to the \"Bishopslips\" sketch, which was cut in the middle of a broadcast. Most of the sketches on the CD were remade for *Jam*.
Track listing
1. \"Blue Jam Intro\"
2. \"Doc Phone\"
3. \"Lamacq sting\"
4. \"4 ft Car\"
5. \"Suicide Journalist\"
6. \"Acupuncture\"
7. \"Bad Sex\"
8. \"Mayo Sting\"
9. \"Unflustered Parents\"
10. \"Moyles Sting\"
11. \"TV Lizards\"
12. \"Doc Cock\"
13. \"Hobbs Sting\"
14. \"Morton Interview\"
15. \"Fix It Girl\"
16. \"Porn\"
17. \"Kids Party\"
18. \"Club News\"
19. \"Whiley Sting\"
20. \"Little Girl Balls\"
21. \"Blue Jam Outro\"
22. \"www.bishopslips.com\" (not a real track)
## Related shows {#related_shows}
*Blue Jam* was later made for television and broadcast on Channel 4 as *Jam*. It used unusual editing techniques to achieve an unnerving ambience in keeping with the radio show. Many of the sketches were lifted from the radio version, even to the extent of simply setting images to the radio soundtrack. A subsequent \"re-mixed\" airing, called *Jaaaaam* was even more extreme in its use of post-production gadgetry, often heavily distorting the footage
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# Clement Martyn Doke
**Clement Martyn Doke** (16 May 1893 in Bristol, United Kingdom -- 24 February 1980 in East London, South Africa) was a South African linguist working mainly on African languages. Realizing that the grammatical structures of Bantu languages are quite different from those of European languages, he was one of the first African linguists of his time to abandon the Euro-centric approach to language description for a more locally grounded one. A most prolific writer, he published a string of grammars, several dictionaries, comparative work, and a history of Bantu linguistics.
## Early life and career {#early_life_and_career}
The Doke family had been engaged in missionary activity for the Baptist Church for some generations. His father, Reverend Joseph J. Doke, left England and travelled to South Africa in 1882, where he met and married Agnes Biggs. They returned to England, where Clement was born as the third of four children. The family moved to New Zealand and eventually returned to South Africa in 1903, where it later settled in Johannesburg.
At the age of 18, Clement received a bachelor\'s degree from Transvaal University College in Pretoria (now the University of Pretoria). He decided to devote his life to missionary activity. In 1913, he accompanied his father on a tour of north-western Rhodesia, to an area called Lambaland, now known as Ilamba. It is at the watershed of the Congo and Zambesi rivers. Part of the district lay in Northern Rhodesia and part of the Belgian Congo. The Cape-Cairo Railway threaded through its eastern portion; otherwise, most travel had to be on foot.
The Reverend William Arthur Phillips of the Nyasa Industrial Mission in Blantyre had established a Baptist mission there in 1905; it served an area of 25000 sqmi and 50,000 souls. The Dokes were supposed to investigate whether the mission in Lambaland could be taken over by the Baptist Union of South Africa. It was on that trip that Doke\'s father contracted enteric fever and died soon afterwards. Mahatma Gandhi attended the memorial service and addressed the congregation. Clement assumed his father\'s role.
The South African Baptists decided to take over Kafulafuta Mission, and its founder, Reverend Phillips, remained as superintendent. Clement Doke returned to Kafulafuta as missionary in 1914, followed by his sister Olive two years later.
## Study of Lamba {#study_of_lamba}
At first, Clement Doke was frustrated by his inability to communicate with the Lamba. The only written material available at the time was a translation of Jonah and a collection of 47 hymns. Soon, however, he mastered the language and published his first book, *Ifintu Fyakwe Lesa* (\"The Things of God, a Primer of Scripture Knowledge\") in 1917. He enrolled in Johannesburg as the extension of Transvaal University College for an MA degree. His thesis was published as *The Grammar of the Lamba language*. The book is couched in traditional grammatical terms, as Doke had not yet established his innovative method to analyse and describe the Bantu languages. His later *Textbook of Lamba Grammar* is far superior in that respect.
Doke was also interested in ethnology. In 1931 he compiled *The Lambas of Northern Rhodesia*, which remains one of the outstanding ethnographic descriptions of the peoples of Central Africa. For Doke, literacy was part of evangelisation since it was required so that people to appreciate the Bible\'s message, but it was only after his retirement that he completed the translation of the Bible into Lamba. It was published under the title of *Amasiwi AwaLesa* (\"The Words of God\") in 1959.
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# Clement Martyn Doke
## University of the Witwatersrand {#university_of_the_witwatersrand}
In 1919, Doke married Hilda Lehmann, who accompanied him back to Lambaland. Both contracted malaria during their work, and she was forbidden to return to Lambaland. Clement Doke also realised that his field work could not continue much longer, and he left in 1921. He was recruited by the newly founded University of the Witwatersrand. So that he could secure a qualification as a lecturer, the family moved to England, where he registered at the School of Oriental and African Studies. His major languages were Lamba and Luba, but as no suitable examiner was available, he eventually had to change his language to Zulu.
Doke took up his appointment in the new Department of Bantu Studies at the University of Witwatersrand in 1923. In 1925 he received his D.Litt. for his doctoral thesis *The Phonetics of the Zulu Language* and was promoted to Senior Lecturer. In 1931 he was appointed to the Chair of Bantu Studies and thus headed the Department of Bantu Studies. The department acted as a catalyst for the admission of Africans to the university. As early as 1925 a limited number were admitted to the vacation course in African Studies. Doke supported the appointment of Benedict Wallet Vilakazi as member of the staff, as he believed a native speaker was essential for acquiring a language. That provoked a storm of criticism and controversy from the public. Both of them collaborated on the *Zulu-English Dictionary*. First published in 1948, it is still one of the best examples of lexicography for any Bantu language.
At the request of the government of Southern Rhodesia, Doke investigated the range of dialect diversity among the languages of the country and made recommendations for *Unified Shona*, which formed the basis for Standard Shona. He devised a unified orthography based on the Zezuru, Karanga and Manyika dialects. However, Doke\'s orthography was never fully accepted, and the South African government introduced an alternative, which left Shona with two competing orthographies between 1935 and 1955.
During his tenure, Doke developed and promoted a method of linguistic analysis and description of the Bantu languages that was based upon the structure of these languages. The \"Dokean model\" continues to be one of the dominant models of linguistic description in Southern and Central Africa. His classification of the Bantu languages was for many years the dominant view of the interrelations among the African languages. He was also an early describer of Khoisan and Bantu click consonants, devising phonetic symbols for a number of them.
Doke served the University of the Witwatersrand until his retirement in 1953. He was awarded the honorary degree of Doctor of Letters by Rhodes University and the honorary degree of Doctor of Laws by the University of the Witwatersrand in 1972.
The former missionary always remained devoted to the Baptist Church. He was elected President of the South African Baptist Union in 1949 and spent a year visiting churches and mission stations. He used his presidential address in condemning the recently established apartheid policy: *I solemnly warn the Government that the spirit behind their apartheid legislation, and the way in which they are introducing discriminatory measures of all types today, will bring disaster upon this fair land of ours.*
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# Clement Martyn Doke
## Selected publications {#selected_publications}
- *Ifintu Fyakwe Lesa* (The Things of God, a Primer of Scripture Knowledge in Lamba), 1917.
- An outline of the phonetics of the language of the ʗhũ̬꞉ Bushman of the North-West Kalahari. *Bantu Studies*. 2: 129--166, 1925. `{{doi|10.1080/02561751.1923.9676181}}`{=mediawiki} [1](https://archive.org/details/african-studies_1923-1926_2/page/129/mode/1up)
- *The phonetics of the Zulu language*. University of the Witwatersrand Press, 1969 \[1926\]. [2](https://storage.lib.uchicago.edu/pres/2009/pres2009-0344.pdf)
- *The Lambas of Northern Rhodesia: A Study of their Customs and Beliefs*. London: George G. Harrap, 1931.
- *Report on the Unification of the Shona Dialects*. Government of Southern Rhodesia: Government Blue Book, 1931.
- *Bantu linguistic terminology*. London; New York Longmans, Green, 1935.
- *Textbook of Lamba Grammar*. Johannesburg: Witwatersrand University Press, 1938.
- *Outline grammar of Bantu*. Johannesburg: University of the Witwatersrand, 1943.
- *Zulu--English Dictionary*. Johannesburg: Witwatersrand University Press, 1948. (with Benedict Wallet Vilakazi)
- *The Southern Bantu languages*. London; New York: Oxford University Press, 1954.
- *Amasiwi AwaLesa* (The Words of God in Lamba), 1959.
- *Contributions to the history of Bantu linguistics*. Johannesburg: Witwatersrand University Press, 1961 (with D. T. Cole).
- *Trekking in South Central Africa 1913--1919*. Johannesburg: Witwatersrand University Press, 1993
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# Carl Meinhof
**Carl Friedrich Michael Meinhof** (23 July 1857 -- 11 February 1944) was a German linguist and one of the first linguists to study African languages.
## Early years and career {#early_years_and_career}
Meinhof was born in Barzwitz near Rügenwalde in the Province of Pomerania, Kingdom of Prussia. He studied at the University of Tübingen and at the University of Greifswald. In 1905 he became professor at the School of Oriental Studies in Berlin. On 5 May 1933 he became a member of the Nazi Party.
## Works
His most notable work was developing comparative grammar studies of the Bantu languages, building on the pioneering work of Wilhelm Bleek. In his work, Meinhof looked at the common Bantu languages such as Swahili and Zulu to determine similarities and differences.
In his work, Meinhof looked at noun classes with all Bantu languages having at least 10 classes and with 22 classes of nouns existing throughout the Bantu languages, though his definition of noun class differs slightly from the accepted one, considering the plural form of a word as belonging to a different class from the singular form (thus leading, for example, to consider a language like French as having four classes instead of two). While no language has all 22 (later: 23) classes active, Venda has 20, Lozi has 18, and Ganda has 16 or 17 (depending on whether the locative class 23 *e-* is included). All Bantu languages have a noun class specifically for humans (sometimes including other animate beings).
Meinhof also examined other African languages, including groups classified at the time as Kordofanian, Bushman, Khoikhoi, and Hamitic.
Meinhof developed a comprehensive classification scheme for African languages. His classification was the standard one for many years (Greenberg 1955:3). It was replaced by those of Joseph Greenberg in 1955 and in 1963. His ideas influenced the notation of African-language phonetics as advanced in the mid-nineteenth century by the Egyptologist Karl Richard Lepsius and gave rise to what some called the \"Meinhof-Lepsius system\" of diacritical markers.
In 1902, Meinhof made recordings of East African music. These are among the first recordings made of traditional African music.
## Controversial views {#controversial_views}
In 1912, Carl Meinhof published *Die Sprachen der Hamiten* (The Languages of the Hamites). He used the term Hamitic. Meinhof\'s system of classification of the Hamitic languages was based on a belief that \"speakers of Hamitic became largely coterminous with cattle herding peoples with essentially Caucasian origins, intrinsically different from and superior to the \'Negroes of Africa\'.\" However, in the case of the so-called Nilo-Hamitic languages (a concept he introduced), it was based on the typological feature of gender and a \"fallacious theory of language mixture.\" Meinhof did this in spite of earlier work by scholars such as Lepsius and Johnston demonstrating that the languages which he would later dub \"Nilo-Hamitic\" were in fact Nilotic languages with numerous similarities in vocabulary with other Nilotic languages.
## Family
Carl Meinhof was the great-uncle (the brother of the grandfather) of Ulrike Meinhof, a well known German journalist, who later became a founding member of the Red Army Faction (RAF), a left-wing militant group operating chiefly in West Germany in the 1970s and 1980s
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# Chorded keyboard
A **keyset** or **chorded keyboard** (also called a chorded keyset, *chord keyboard* or *chording keyboard*) is a computer input device that allows the user to enter characters or commands formed by pressing several keys together, like playing a \"chord\" on a piano. The large number of combinations available from a small number of keys allows text or commands to be entered with one hand, leaving the other hand free. A secondary advantage is that it can be built into a device (such as a pocket-sized computer or a bicycle handlebar) that is too small to contain a normal-sized keyboard.
A chorded keyboard minus the board, typically designed to be used while held in the hand, is called a keyer. Douglas Engelbart introduced the chorded keyset as a computer interface in 1968 at what is often called \"The Mother of All Demos\".
## Principles of operation {#principles_of_operation}
Each key is mapped to a number and then can be mapped to a corresponding letter or command. By pressing two or more keys together the user can generate many combinations. In Engelbart\'s original mapping, he used five keys: 1, 2, 4, 8, 16. The keys were mapped as follows: a = 1, b = 2, c = 3, d = 4, and so on. If the user pressed keys 1 and 2 simultaneously, and then released the keys, 1 and 2 would be added to 3, and since C is the 3rd letter of the alphabet, and the letter \"c\" appeared. Unlike pressing a chord on a piano, the chord is recognized only after all the keys or mouse buttons are released. Since Engelbart introduced the keyset, several different designs have been developed based on similar concepts.
As a crude example, each finger might control one key which corresponds to one bit in a byte, so that using seven keys and seven fingers, one could enter any character in the ASCII set---if the user could remember the binary codes. Due to the small number of keys required, chording is easily adapted from a desktop to mobile environment.
Practical devices generally use simpler chords for common characters (*e.g.,* Baudot), or may have ways to make it easier to remember the chords (*e.g.,* Microwriter), but the same principles apply. These portable devices first became popular with the wearable computer movement in the 1980s.
Thad Starner from Georgia Institute of Technology and others published numerous studies showing that two-handed chorded text entry was faster and yielded fewer errors than on a QWERTY keyboard. Currently stenotype machines hold the record for fastest word entry. Many stenotype users can reach 300 words per minute. However, stenographers typically train for three years before reaching professional levels of speed and accuracy.
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# Chorded keyboard
## History
The earliest known chord keyboard was part of the \"five-needle\" telegraph operator station, designed by Wheatstone and Cooke in 1836, in which any two of the five needles could point left or right to indicate letters on a grid. It was designed to be used by untrained operators (who would determine which keys to press by looking at the grid), and was not used where trained telegraph operators were available. The first widespread use of a chord keyboard was in the stenotype machine used by court reporters, which was invented in 1868 and is still in use. The output of the stenotype was originally a phonetic code that had to be transcribed later (usually by the same operator who produced the original output), rather than arbitrary text---automatic conversion software is now commonplace.
In 1874, the five-bit Baudot telegraph code and a matching 5-key chord keyboard was designed to be used with the operator forming the codes manually. The code is optimized for speed and low wear: chords were chosen so that the most common characters used the simplest chords. But telegraph operators were already using typewriters with QWERTY keyboards to \"copy\" received messages, and at the time it made more sense to build a typewriter that could generate the codes automatically, rather than making them learn to use a new input device.
Some early keypunch machines used a keyboard with 12 labeled keys to punch the correct holes in paper cards. The numbers 0 through 9 were represented by one punch; 26 letters were represented by combinations of two punches, and symbols were represented by combinations of two or three punches. Braille (a writing system for the blind) uses either 6 or 8 tactile \'points\' from which all letters and numbers are formed. When Louis Braille invented it, it was produced with a needle holing successively all needed points in a cardboard sheet. In 1892, Frank Haven Hall, superintendent of the Illinois Institute for the Education of the Blind, created the Hall Braille Writer, which was like a typewriter with 6 keys, one for each dot in a braille cell. The Perkins Brailler, first manufactured in 1951, uses a 6-key chord keyboard (plus a spacebar) to produce braille output, and has been very successful as a mass market affordable product. Braille, like Baudot, uses a number symbol and a shift symbol, which may be repeated for shift lock, to fit numbers and upper case into the 63 codes that 6 bits offer.
After World War II, with the arrival of electronics for reading chords and looking in tables of \"codes\", the postal sorting offices started to research chordic solutions to be able to employ people other than trained and expensive typists. In 1954, an important concept was discovered: chordic production is easier to master when the production is done at the release of the keys instead of when they are pressed.
Researchers at IBM investigated chord keyboards for both typewriters and computer data entry as early as 1959, with the idea that it might be faster than touch-typing if some chords were used to enter whole words or parts of words. A 1975 design by IBM Fellow Nat Rochester had 14 keys that were dimpled on the edges as well as the top, so one finger could press two adjacent keys for additional combinations. Their results were inconclusive, but research continued until at least 1978.
Doug Engelbart began experimenting with keysets to use with the mouse in the mid-1960s. In a famous 1968 demonstration, Engelbart introduced a computer human interface that included the QWERTY keyboard, a three button mouse, and a five key keyset. Engelbart used the keyset with his left hand and the mouse with his right to type text and enter commands. The mouse buttons marked selections and confirmed or aborted commands.
Users in Engelbart\'s Augmentation Research Center at SRI became proficient with the mouse and keyset. In the 1970s the funding Engelbart\'s group received from the Advanced Research Projects Agency (ARPA) was cut and many key members of Engelbart\'s team went to work for Xerox PARC where they continued to experiment with the mouse and keyset. Keychord sets were used at Xerox PARC in the early 1980s, along with mice, GUIs, on the Xerox Star and Alto workstations. A one-button version of the mouse was incorporated into the Apple Macintosh but Steve Jobs decided against incorporating the chorded keyset.
In the early 1980s, Philips Research labs at Redhill, Surrey did a brief study into small, cheap keyboards for entering text on a telephone. One solution used a grid of hexagonal keys with symbols inscribed into dimples in the keys that were either in the center of a key, across the boundary of two keys, or at the joining of three keys. Pressing down on one of the dimples would cause either one, two or three of the hexagonal buttons to be depressed at the same time, forming a chord that would be unique to that symbol. With this arrangement, a nine button keyboard with three rows of three hexagonal buttons could be fitted onto a telephone and could produce up to 33 different symbols. By choosing widely separated keys, one could employ one dimple as a \'shift\' key to allow both letters and numbers to be produced. With eleven keys in a 3/4/4 arrangement, 43 symbols could be arranged allowing for lowercase text, numbers and a modest number of punctuation symbols to be represented along with a \'shift\' function for accessing uppercase letters. While this had the advantage of being usable by untrained users via \'hunt and peck\' typing and requiring one less key switch than a conventional 12 button keypad, it had the disadvantage that some symbols required three times as much force to depress them as others which made it hard to achieve any speed with the device. That solution is still alive and proposed by Fastap and Unitap among others, and a commercial phone has been produced and promoted in Canada during 2006.
## Standards
Historically, the baudot and braille keyboards were standardized to some extent, but they are unable to replicate the full character set of a modern keyboard. Braille comes closest, as it has been extended to eight bits.
The only proposed modern standard, GKOS (or Global Keyboard Open Standard) can support most characters and functions found on a computer keyboard but has had little commercial development. There is, however, a GKOS keyboard application available for iPhone since May 8, 2010, for Android since October 3, 2010 and for MeeGo Harmattan since October 27, 2011.
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# Chorded keyboard
## Stenography
Stenotype machines, sometimes used by court reporters, use a chording keyboard to represent sounds: on the standard keyboard, the U represents the sound and word, \'you\', and the three-key trigraph KAT represents the sound and word \'cat\'. The stenotype keyboard is explicitly ordered: in KAT, K, on the left, is the starting sound. P, S, and T, which are common starting sounds and also common ending sounds, are available on both sides of the keyboard: POP is a 3-key chord, using both P keys.
## Open-source designs {#open_source_designs}
Multiple open-source keyer/keyset designs are available, such as the pickey, a PS/2 device based on the PIC microcontroller; the spiffchorder, a USB device based on the Atmel AVR family of microcontrollers; the FeatherChorder, a BLE chorder based on the Adafruit Feather, an all-in-one board incorporating an Arduino-compatible microcontroller; and the GKOS keypad driver for Linux as well as the Gkos library for the Atmel/Arduino open-source board.
Plover is a free, open-source, cross-platform program intended to bring real-time stenographic technology not just to stenographers, but also to hobbyists using anything from professional Stenotype machines to low-cost NKRO gaming keyboards. It is available for Linux, Windows, and macOS.
Joy2chord is a chorded keyboard driver for Linux. With a configuration file, any joystick or gamepad can be turned into a chorded keyboard. This design philosophy was decided on to lower the cost of building devices, and in turn lower the entry barrier to becoming familiar with chorded keyboards. Macro keys, and multiple modes are also easily implemented with a user space driver.
## Commercial devices {#commercial_devices}
One minimal chordic keyboard example is Edgar Matias\' Half-Qwerty keyboard described in patent `{{patent|US|5288158}}`{=mediawiki} circa 1992 that produces the letters of the missing half when the user simultaneously presses the space bar along with the mirror key. INTERCHI \'93 published a study by Matias, MacKenzie and Buxton showing that people who have already learned to touch-type can quickly recover 50 to 70% of their two-handed typing speed. The loss contributes to the speed discussion above. It is implemented on two popular mobile phones, each provided with software disambiguation, which allows users to avoid using the space-bar.
\"Multiambic\" keyers for use with wearable computers were invented in Canada in the 1970s. Multiambic keyers are similar to chording keyboards but without the board, in that the keys are grouped in a cluster for being handheld, rather than for sitting on a flat surface.
Chording keyboards are also used as portable but two handed input devices for the visually impaired (either combined with a refreshable braille display or vocal synthesis). Such keyboards use a minimum of seven keys, where each key corresponds to an individual braille point, except one key which is used as a spacebar. In some applications, the spacebar is used to produce additional chords which enable the user to issue editing commands, such as moving the cursor, or deleting words. Note that the number of points used in braille computing is not 6, but 8, as this allows the user, among other things, to distinguish between small and capital letters, as well as identify the position of the cursor. As a result, most newer chorded keyboards for braille input include at least nine keys.
Touch screen chordic keyboards are available to smartphone users as an optional way of entering text. As the number of keys is low, the button areas can be made bigger and easier to hit on the small screen. The most common letters do not necessarily require chording as is the case with the GKOS keyboard optimised layouts (Android app) where the twelve most frequent characters only require single keys.
The DecaTxt one-handed Bluetooth Chord keyboard, by IN10DID, Inc. has ten keys, two at each finger and is able to replace all standard keystrokes with chords of four keys or less. It is small at 3.25\"x 2.25\" and weighs about 2 ounces, making it quite wearable strapped to either hand for use while walking. DecaTxt is generally considered as assistive technology since it works with a variety of issues such as limited vision, limb loss, shaking and poor motor skills.
The company CharaChorder commercially sells chorded entry devices. Their first commercially available device is the CharaChorder One, which features a split design with each having access to 9 switches that can be moved in five directions (up, down, left, right, and pressed) in contrast to typical keyboards. This device allows for both chorded entry as well as traditional character entry. The set of words that can be chorded can be dynamically changed by the user in real time, but by default includes the 300 most common words in the English language. This chorded entry feature allows for potentially extremely fast typing speeds, so much so the founder of the company has been banned from online typing competitions. Additionally, they create the Charachorder Lite with a more traditional keyboard design. The manufacturer claimed that users of the Charachorder One can reach speeds of 300 words per minute, while users of the Charachorder Lite can reach 250 words per minute.
### Historical
The WriteHander, a 12-key chord keyboard from NewO Company, appeared in 1978 issues of ROM Magazine, an early microcomputer applications magazine.
Another early commercial model was the six-button Microwriter, designed by Cy Endfield and Chris Rainey, and first sold in 1980. Microwriting is the system of chord keying and is based on a set of mnemonics. It was designed only for right-handed use.
In 1982 the Octima 8 keys cord keyboard was presented by Ergoplic Kebords Ltd an Israeli Startup that was founded by Israeli researcher with intensive experience in Man Machine Interface design. The keyboard had 8 keys one for each finger and additional 3 keys that enabled the production of numbers, punctuations and control functions. The keyboard was fully compatible with the IBM PC and AT keyboards and had an Apple IIe version as well. Its key combinations were based on a mnemonic system that enabled fast and easy touch type learning. Within a few hours the user could achieve a typing speed similar to hand writing speed. The unique design also gave a relief from hand stress (Carpal Tunnel Syndrome) and allowed longer typing sessions than traditional keyboards. It was multi-lingual supporting English, German, French and Hebrew.
The BAT is a 7-key hand-sized device from Infogrip, and has been sold since 1985. It provides one key for each finger and three for the thumb. It is proposed for the hand which does not hold the mouse, in an exact continuation of Engelbart\'s vision
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# Centaurus
**Centaurus** (`{{IPAc-en|s|ɛ|n|ˈ|t|ɔːr|ə|s|,_|-|ˈ|t|ɑːr|-}}`{=mediawiki}) is a bright constellation in the southern sky. One of the largest constellations, Centaurus was included among the 48 constellations listed by the 2nd-century astronomer Ptolemy, and it remains one of the 88 modern constellations. In Greek mythology, Centaurus represents a centaur; a creature that is half human, half horse (another constellation named after a centaur is one from the zodiac: Sagittarius). Notable stars include Alpha Centauri, the nearest star system to the Solar System, its neighbour in the sky Beta Centauri, and HR 5171, one of the largest stars yet discovered. The constellation also contains Omega Centauri, the brightest globular cluster as visible from Earth and the largest identified in the Milky Way, possibly a remnant of a dwarf galaxy.
## Notable features {#notable_features}
thumb\|left\|upright=1.1\|Centaurus in the southwestern sky, shortly after sunset. right\|thumb\|upright=1.2\|The two bright stars are (left) Alpha Centauri and (right) Beta Centauri. The faint red star in the center of the red circle is Proxima Centauri. thumb\|right\|upright=1.4\|Centaurus in the *Firmamentum Sobiescianum* of Johannes Hevelius. `{{Abbreviation|N.B.|Nota bene (note well)}}`{=mediawiki} This image is reversed from what one sees looking at the sky --- it is as though one is looking at the \"celestial sphere\" from the outside.
### Stars
Centaurus contains several very bright stars. Its alpha and beta stars are used as \"pointer stars\" to help observers find the constellation Crux. Centaurus has 281 stars above magnitude 6.5, meaning that they are visible to the unaided eye, the most of any constellation. Alpha Centauri, the closest star system to the Sun, has a high proper motion; it will be a mere half-degree from Beta Centauri in approximately 4000 years.
Alpha Centauri is a triple star system composed of a binary system orbited by Proxima Centauri, currently the nearest star to the Sun. Traditionally called Rigil Kentaurus (from Arabic رجل قنطورس, meaning \"foot of the centaur\") or Toliman (from Arabic الظليمين meaning \"two male ostriches\"), the system has an overall magnitude of −0.28 and is 4.4 light-years from Earth. The primary and secondary are both yellow-hued stars; the first is of magnitude −0.01 and the second: 1.35. Proxima, the tertiary star, is a red dwarf of magnitude 11.0; it appears almost 2 degrees away from the close pairing of Alpha and has a period of approximately one million years. Also a flare star, Proxima has minutes-long outbursts where it brightens by over a magnitude. The Alpha couple revolve in 80-year periodicity and will next appear closest as seen from Earth\'s telescopes in 2037 and 2038, together as they appear to the naked eye they present the third-brightest \"star\" in the night sky.
One other first magnitude star Beta Centauri is in the constellation in a position beyond Proxima and toward the narrow axis of Crux, thus with Alpha forming a far-south limb of the constellation. Also called Hadar and Agena, it is a double star; the primary is a blue-hued giant star of magnitude 0.6, 525 light-years from Earth. The secondary is of magnitude 4.0 and has a modest separation, appearing only under intense magnification due to its distance.
The northerly star Theta Centauri, officially named Menkent, is an orange giant star of magnitude 2.06. It is the only bright star of Centaurus that is easily visible from mid-northern latitudes.
The next bright object is Gamma Centauri, a binary star which appears to the naked eye at magnitude 2.2. The primary and secondary are both blue-white hued stars of magnitude 2.9; their period is 84 years.
Centaurus also has many dimmer double stars and binary stars. 3 Centauri is a double star with a blue-white hued primary of magnitude 4.5 and a secondary of magnitude 6.0. The primary is 344 light-years away.
Centaurus is home to many variable stars. R Centauri is a Mira variable star with a minimum magnitude of 11.8 and a maximum magnitude of 5.3; it is about 1,250 light-years from Earth and has a period of 18 months. V810 Centauri is a semiregular variable.
BPM 37093 is a white dwarf star whose carbon atoms are thought to have formed a crystalline structure. Since diamond also consists of carbon arranged in a crystalline lattice (though of a different configuration), scientists have nicknamed this star \"Lucy\" after the Beatles song \"*Lucy in the Sky with Diamonds*.\"
PDS 70, (V1032 Centauri) a low mass T Tauri star is found in the constellation Centaurus. In July 2018 astronomers captured the first conclusive image of a protoplanetary disk containing a nascent exoplanet, named PDS 70b.
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# Centaurus
## Notable features {#notable_features}
### Deep-sky objects {#deep_sky_objects}
ω Centauri (NGC 5139), despite being listed as the constellation\'s \"omega\" star, is in fact a naked-eye globular cluster, 17,000 light-years away with a diameter of 150 light-years. It is the largest and brightest globular cluster in the Milky Way; at ten times the size of the next-largest cluster, it has a magnitude of 3.7. It is also the most luminous globular cluster in the Milky Way, at over one million solar luminosities. Omega Centauri is classified as a Shapley class VIII cluster, which means that its center is loosely concentrated. It is also one of only two globular clusters to be given a stellar designation; in its case a Bayer letter. The other is 47 Tucanae (Xi Tucanae), which has a Flamsteed number. Omega Centauri contains several million stars, most of which are yellow dwarf stars, but also possesses red giants and blue-white stars; the stars have an average age of 12 billion years. This has prompted suspicion that Omega Centauri was the core of a dwarf galaxy that had been absorbed by the Milky Way. Omega Centauri was determined to be nonstellar in 1677 by the English astronomer Edmond Halley, though it was visible as a star to the ancients. Its status as a globular cluster was determined by James Dunlop in 1827. To the unaided eye, Omega Centauri appears fuzzy and is obviously non-circular; it is approximately half a degree in diameter, the same size as the full Moon.
Centaurus is also home to open clusters. NGC 3766 is an open cluster 6,300 light-years from Earth that is visible to the unaided eye. It contains approximately 100 stars, the brightest of which are 7th magnitude. NGC 5460 is another naked-eye open cluster, 2,300 light-years from Earth, that has an overall magnitude of 6 and contains approximately 40 stars.
There is one bright planetary nebula in Centaurus, NGC 3918, also known as the Blue Planetary. It has an overall magnitude of 8.0 and a central star of magnitude 11.0; it is 2600 light-years from Earth. The Blue Planetary was discovered by John Herschel and named for its color\'s similarity to Uranus, though the nebula is apparently three times larger than the planet.
Centaurus is rich in galaxies as well. NGC 4622 is a face-on spiral galaxy located 200 million light-years from Earth (redshift 0.0146). Its spiral arms wind in both directions, which makes it nearly impossible for astronomers to determine the rotation of the galaxy. Astronomers theorize that a collision with a smaller companion galaxy near the core of the main galaxy could have led to the unusual spiral structure. NGC 5253, a peculiar irregular galaxy, is located near the border with Hydra and M83, with which it likely had a close gravitational interaction 1--2 billion years ago. This may have sparked the galaxy\'s high rate of star formation, which continues today and contributes to its high surface brightness. NGC 5253 includes a large nebula and at least 12 large star clusters. In the eyepiece, it is a small galaxy of magnitude 10 with dimensions of 5 arcminutes by 2 arcminutes and a bright nucleus. NGC 4945 is a spiral galaxy seen edge-on from Earth, 13 million light-years away. It is visible with any amateur telescope, as well as binoculars under good conditions; it has been described as \"shaped like a candle flame\", being long and thin (16\' by 3\'). In the eyepiece of a large telescope, its southeastern dust lane becomes visible. Another galaxy is NGC 5102, found by star-hopping from Iota Centauri. In the eyepiece, it appears as an elliptical object 9 arcminutes by 2.5 arcminutes tilted on a southwest--northeast axis.
One of the closest active galaxies to Earth is the Centaurus A galaxy, NGC 5128, at 11 million light-years away (redshift 0.00183). It has a supermassive black hole at its core, which expels massive jets of matter that emit radio waves due to synchrotron radiation. Astronomers posit that its dust lanes, not common in elliptical galaxies, are due to a previous merger with another galaxy, probably a spiral galaxy. NGC 5128 appears in the optical spectrum as a fairly large elliptical galaxy with a prominent dust lane. Its overall magnitude is 7.0 and it has been seen under perfect conditions with the naked eye, making it one of the most distant objects visible to the unaided observer. In equatorial and southern latitudes, it is easily found by star hopping from Omega Centauri. In small telescopes, the dust lane is not visible; it begins to appear with about 4 inches of aperture under good conditions. In large amateur instruments, above about 12 inches in aperture, the dust lane\'s west-northwest to east-southeast direction is easily discerned. Another dim dust lane on the east side of the 12-arcminute-by-15-arcminute galaxy is also visible. ESO 270-17, also called the Fourcade-Figueroa Object, is a low-surface brightness object believed to be the remnants of a galaxy; it does not have a core and is very difficult to observe with an amateur telescope. It measures 7 arcminutes by 1 arcminute. It likely originated as a spiral galaxy and underwent a catastrophic gravitational interaction with Centaurus A around 500 million years ago, stopping its rotation and destroying its structure.
NGC 4650A is a polar-ring galaxy 136 million light-years from Earth (redshift 0.01). It has a central core made of older stars that resembles an elliptical galaxy, and an outer ring of young stars that orbits around the core. The plane of the outer ring is distorted, which suggests that NGC 4650A is the result of a galaxy collision about a billion years ago. This galaxy has also been cited in studies of dark matter, because the stars in the outer ring orbit too quickly for their collective mass. This suggests that the galaxy is surrounded by a dark matter halo, which provides the necessary mass.
One of the closest galaxy clusters to Earth is the Centaurus Cluster at c. 160 million light-years away, having redshift 0.0114. It has a cooler, denser central region of gas and a hotter, more diffuse outer region. The intracluster medium in the Centaurus Cluster has a high concentration of metals (elements heavier than helium) due to a large number of supernovae. This cluster also possesses a plume of gas whose origin is unknown.
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# Centaurus
## History
While Centaurus now has a high southern latitude, at the dawn of civilization it was an equatorial constellation. Precession has been slowly shifting it southward for millennia, and it is now close to its maximal southern declination. In a little over 7000 years it will be at maximum visibility for those in the northern hemisphere, visible at times in the year up to quite a high northern latitude. The figure of Centaurus can be traced back to a Babylonian constellation known as the Bison-man (MUL.GUD.ALIM). This being was depicted in two major forms: firstly, as a 4-legged bison with a human head, and secondly, as a being with a man\'s head and torso attached to the rear legs and tail of a bull or bison. It has been closely associated with the Sun god Utu-Shamash from very early times.
The Greeks depicted the constellation as a centaur and gave it its current name. It was mentioned by Eudoxus in the 4th century BC and Aratus in the 3rd century BC. In the 2nd century AD, Claudius Ptolemy catalogued 37 stars in Centaurus, including Alpha Centauri. Large as it is now, in earlier times it was even larger, as the constellation Lupus was treated as an asterism within Centaurus, portrayed in illustrations as an unspecified animal either in the centaur\'s grasp or impaled on its spear. The Southern Cross, which is now regarded as a separate constellation, was treated by the ancients as a mere asterism formed of the stars composing the centaur\'s legs. Additionally, what is now the minor constellation Circinus was treated as undefined stars under the centaur\'s front hooves.
According to the Roman poet Ovid (*Fasti* v.379), the constellation honors the centaur Chiron, who was tutor to many of the earlier Greek heroes including Heracles (Hercules), Theseus, and Jason, the leader of the Argonauts. It is not to be confused with the more warlike centaur represented by the zodiacal constellation Sagittarius. The legend associated with Chiron says that he was accidentally poisoned with an arrow shot by Hercules, and was subsequently placed in the heavens.
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# Centaurus
## Equivalents
In Chinese astronomy, the stars of Centaurus are found in three areas: the Azure Dragon of the East (東方青龍, *Dōng Fāng Qīng Lóng*), the Vermillion Bird of the South (南方朱雀, *Nán Fāng Zhū Què*), and the Southern Asterisms (近南極星區, *Jìnnánjíxīngōu*). Not all of the stars of Centaurus can be seen from China, and the unseen stars were classified among the Southern Asterisms by Xu Guangqi, based on his study of western star charts. However, most of the brightest stars of Centaurus, including α Centauri, θ Centauri (or Menkent), ε Centauri and η Centauri, can be seen in the Chinese sky.
Some Polynesian peoples considered the stars of Centaurus to be a constellation as well. On Pukapuka, Centaurus had two names: *Na Mata-o-te-tokolua* and *Na Lua-mata-o-Wua-ma-Velo*. In Tonga, the constellation was called by four names: *O-nga-tangata*, *Tautanga-ufi*, *Mamangi-Halahu*, and *Mau-kuo-mau*. Alpha and Beta Centauri were not named specifically by the people of Pukapuka or Tonga, but they were named by the people of Hawaii and the Tuamotus. In Hawaii, the name for Alpha Centauri was either *Melemele* or *Ka Maile-hope* and the name for Beta Centauri was either *Polapola* or *Ka Maile-mua*. In the Tuamotu islands, Alpha was called *Na Kuhi* and Beta was called *Tere*.
The Pointer (α Centauri and β Centauri) is one of the asterisms used by Bugis sailors for navigation, called *bintoéng balué*, meaning \"the widowed-before-marriage\". It is also called *bintoéng sallatang* meaning \"southern star\".
## Namesakes
Two United States Navy ships, `{{USS|Centaurus|AKA-17}}`{=mediawiki} and `{{USS|Centaurus|AK-264}}`{=mediawiki}, were named after Centaurus, the constellation
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# Impact crater
An **impact crater** is a depression in the surface of a solid astronomical body formed by the hypervelocity impact of a smaller object. In contrast to volcanic craters, which result from explosion or internal collapse, impact craters typically have raised rims and floors that are lower in elevation than the surrounding terrain. Impact craters are typically circular, though they can be elliptical in shape or even irregular due to events such as landslides. Impact craters range in size from microscopic craters seen on lunar rocks returned by the Apollo Program to simple bowl-shaped depressions and vast, complex, multi-ringed impact basins. Meteor Crater is a well-known example of a small impact crater on Earth.
Impact craters are the dominant geographic features on many solid Solar System objects including the Moon, Mercury, Callisto, Ganymede, and most small moons and asteroids. On other planets and moons that experience more active surface geological processes, such as Earth, Venus, Europa, Io, Titan, and Triton, visible impact craters are less common because they become eroded, buried, or transformed by tectonic and volcanic processes over time. Where such processes have destroyed most of the original crater topography, the terms impact structure or astrobleme are more commonly used. In early literature, before the significance of impact cratering was widely recognised, the terms cryptoexplosion or cryptovolcanic structure were often used to describe what are now recognised as impact-related features on Earth.
The cratering records of very old surfaces, such as Mercury, the Moon, and the southern highlands of Mars, record a period of intense early bombardment in the inner Solar System around 3.9 billion years ago. The rate of crater production on Earth has since been considerably lower, but it is appreciable nonetheless. Earth experiences, on average, from one to three impacts large enough to produce a 20 km crater every million years. This indicates that there should be far more relatively young craters on the planet than have been discovered so far. The cratering rate in the inner solar system fluctuates as a consequence of collisions in the asteroid belt that create a family of fragments that are often sent cascading into the inner solar system. Formed in a collision 80 million years ago, the Baptistina family of asteroids is thought to have caused a large spike in the impact rate. The rate of impact cratering in the outer Solar System could be different from the inner Solar System.
Although Earth\'s active surface processes quickly destroy the impact record, about 190 terrestrial impact craters have been identified. These range in diameter from a few tens of meters up to about 300 km, and they range in age from recent times (e.g. the Sikhote-Alin craters in Russia whose creation was witnessed in 1947) to more than two billion years, though most are less than 500 million years old because geological processes tend to obliterate older craters. They are also selectively found in the stable interior regions of continents. Few undersea craters have been discovered because of the difficulty of surveying the sea floor, the rapid rate of change of the ocean bottom, and the subduction of the ocean floor into Earth\'s interior by processes of plate tectonics.
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# Impact crater
## History
Daniel M. Barringer, a mining engineer, was convinced already in 1903 that the crater he owned, Meteor Crater, was of cosmic origin. Most geologists at the time assumed it formed as the result of a volcanic steam eruption.
In the 1920s, the American geologist Walter H. Bucher studied a number of sites now recognized as impact craters in the United States. He concluded they had been created by some great explosive event, but believed that this force was probably volcanic in origin. However, in 1936, the geologists John D. Boon and Claude C. Albritton Jr. revisited Bucher\'s studies and concluded that the craters that he studied were probably formed by impacts.
Grove Karl Gilbert suggested in 1893 that the Moon\'s craters were formed by large asteroid impacts. Ralph Baldwin in 1949 wrote that the Moon\'s craters were mostly of impact origin. Around 1960, Gene Shoemaker revived the idea. According to David H. Levy, Shoemaker \"saw the craters on the Moon as logical impact sites that were formed not gradually, in eons, but explosively, in seconds.\" For his PhD degree at Princeton University (1960), under the guidance of Harry Hammond Hess, Shoemaker studied the impact dynamics of Meteor Crater. Shoemaker noted that Meteor Crater had the same form and structure as two explosion craters created from atomic bomb tests at the Nevada Test Site, notably Jangle U in 1951 and Teapot Ess in 1955. In 1960, Edward C. T. Chao and Shoemaker identified coesite (a form of silicon dioxide) at Meteor Crater, proving the crater was formed from an impact generating extremely high temperatures and pressures. They followed this discovery with the identification of coesite within suevite at Nördlinger Ries, proving its impact origin.
Armed with the knowledge of shock-metamorphic features, Carlyle S. Beals and colleagues at the Dominion Astrophysical Observatory in Victoria, British Columbia, Canada and Wolf von Engelhardt of the University of Tübingen in Germany began a methodical search for impact craters. By 1970, they had tentatively identified more than 50. Although their work was controversial, the American Apollo Moon landings, which were in progress at the time, provided supportive evidence by recognizing the rate of impact cratering on the Moon. Because the processes of erosion on the Moon are minimal, craters persist. Since the Earth could be expected to have roughly the same cratering rate as the Moon, it became clear that the Earth had suffered far more impacts than could be seen by counting evident craters.
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# Impact crater
## Crater formation {#crater_formation}
Impact cratering involves high velocity collisions between solid objects, typically much greater than the speed of sound in those objects. Such hyper-velocity impacts produce physical effects such as melting and vaporization that do not occur in familiar sub-sonic collisions. On Earth, ignoring the slowing effects of travel through the atmosphere, the lowest impact velocity with an object from space is equal to the gravitational escape velocity of about 11 km/s. The fastest impacts occur at about 72 km/s in the \"worst case\" scenario in which an object in a retrograde near-parabolic orbit hits Earth. The median impact velocity on Earth is about 20 km/s.
However, the slowing effects of travel through the atmosphere rapidly decelerate any potential impactor, especially in the lowest 12 kilometres where 90% of the Earth\'s atmospheric mass lies. Meteors of up to 7,000 kg lose all their cosmic velocity due to atmospheric drag at a certain altitude (retardation point), and start to accelerate again due to Earth\'s gravity until the body reaches its terminal velocity of 0.09 to 0.16 km/s. The larger the meteoroid (i.e. asteroids and comets) the more of its initial cosmic velocity it preserves. While an object of 9,000 kg maintains about 6% of its original velocity, one of 900,000 kg already preserves about 70%. Extremely large bodies (about 100,000 tonnes) are not slowed by the atmosphere at all, and impact with their initial cosmic velocity if no prior disintegration occurs.
Impacts at these high speeds produce shock waves in solid materials, and both impactor and the material impacted are rapidly compressed to high density. Following initial compression, the high-density, over-compressed region rapidly depressurizes, exploding violently, to set in train the sequence of events that produces the impact crater. Impact-crater formation is therefore more closely analogous to cratering by high explosives than by mechanical displacement. Indeed, the energy density of some material involved in the formation of impact craters is many times higher than that generated by high explosives. Since craters are caused by explosions, they are nearly always circular -- only very low-angle impacts cause significantly elliptical craters.
This describes impacts on solid surfaces. Impacts on porous surfaces, such as that of Hyperion, may produce internal compression without ejecta, punching a hole in the surface without filling in nearby craters. This may explain the \'sponge-like\' appearance of that moon.
It is convenient to divide the impact process conceptually into three distinct stages: (1) initial contact and compression, (2) excavation, (3) modification and collapse. In practice, there is overlap between the three processes with, for example, the excavation of the crater continuing in some regions while modification and collapse is already underway in others.
### Contact and compression {#contact_and_compression}
In the absence of atmosphere, the impact process begins when the impactor first touches the target surface. This contact accelerates the target and decelerates the impactor. Because the impactor is moving so rapidly, the rear of the object moves a significant distance during the short-but-finite time taken for the deceleration to propagate across the impactor. As a result, the impactor is compressed, its density rises, and the pressure within it increases dramatically. Peak pressures in large impacts exceed 1 T Pa to reach values more usually found deep in the interiors of planets, or generated artificially in nuclear explosions.
In physical terms, a shock wave originates from the point of contact. As this shock wave expands, it decelerates and compresses the impactor, and it accelerates and compresses the target. Stress levels within the shock wave far exceed the strength of solid materials; consequently, both the impactor and the target close to the impact site are irreversibly damaged. Many crystalline minerals can be transformed into higher-density phases by shock waves; for example, the common mineral quartz can be transformed into the higher-pressure forms coesite and stishovite. Many other shock-related changes take place within both impactor and target as the shock wave passes through, and some of these changes can be used as diagnostic tools to determine whether particular geological features were produced by impact cratering.
As the shock wave decays, the shocked region decompresses towards more usual pressures and densities. The damage produced by the shock wave raises the temperature of the material. In all but the smallest impacts this increase in temperature is sufficient to melt the impactor, and in larger impacts to vaporize most of it and to melt large volumes of the target. As well as being heated, the target near the impact is accelerated by the shock wave, and it continues moving away from the impact behind the decaying shock wave.
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# Impact crater
## Crater formation {#crater_formation}
### Excavation
Contact, compression, decompression, and the passage of the shock wave all occur within a few tenths of a second for a large impact. The subsequent excavation of the crater occurs more slowly, and during this stage the flow of material is largely subsonic. During excavation, the crater grows as the accelerated target material moves away from the point of impact. The target\'s motion is initially downwards and outwards, but it becomes outwards and upwards. The flow initially produces an approximately hemispherical cavity that continues to grow, eventually producing a paraboloid (bowl-shaped) crater in which the centre has been pushed down, a significant volume of material has been ejected, and a topographically elevated crater rim has been pushed up. When this cavity has reached its maximum size, it is called the transient cavity.
The depth of the transient cavity is typically a quarter to a third of its diameter. Ejecta thrown out of the crater do not include material excavated from the full depth of the transient cavity; typically the depth of maximum excavation is only about a third of the total depth. As a result, about one third of the volume of the transient crater is formed by the ejection of material, and the remaining two thirds is formed by the displacement of material downwards, outwards and upwards, to form the elevated rim. For impacts into highly porous materials, a significant crater volume may also be formed by the permanent compaction of the pore space. Such compaction craters may be important on many asteroids, comets and small moons.
In large impacts, as well as material displaced and ejected to form the crater, significant volumes of target material may be melted and vaporized together with the original impactor. Some of this impact melt rock may be ejected, but most of it remains within the transient crater, initially forming a layer of impact melt coating the interior of the transient cavity. In contrast, the hot dense vaporized material expands rapidly out of the growing cavity, carrying some solid and molten material within it as it does so. As this hot vapor cloud expands, it rises and cools much like the archetypal mushroom cloud generated by large nuclear explosions. In large impacts, the expanding vapor cloud may rise to many times the scale height of the atmosphere, effectively expanding into free space.
Most material ejected from the crater is deposited within a few crater radii, but a small fraction may travel large distances at high velocity, and in large impacts it may exceed escape velocity and leave the impacted planet or moon entirely. The majority of the fastest material is ejected from close to the center of impact, and the slowest material is ejected close to the rim at low velocities to form an overturned coherent flap of ejecta immediately outside the rim. As ejecta escapes from the growing crater, it forms an expanding curtain in the shape of an inverted cone. The trajectory of individual particles within the curtain is thought to be largely ballistic.
Small volumes of un-melted and relatively un-shocked material may be spalled at very high relative velocities from the surface of the target and from the rear of the impactor. Spalling provides a potential mechanism whereby material may be ejected into inter-planetary space largely undamaged, and whereby small volumes of the impactor may be preserved undamaged even in large impacts. Small volumes of high-speed material may also be generated early in the impact by jetting. This occurs when two surfaces converge rapidly and obliquely at a small angle, and high-temperature highly shocked material is expelled from the convergence zone with velocities that may be several times larger than the impact velocity.
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# Impact crater
## Crater formation {#crater_formation}
### Modification and collapse {#modification_and_collapse}
In most circumstances, the transient cavity is not stable and collapses under gravity. In small craters, less than about 4 km diameter on Earth, there is some limited collapse of the crater rim coupled with debris sliding down the crater walls and drainage of impact melts into the deeper cavity. The resultant structure is called a simple crater, and it remains bowl-shaped and superficially similar to the transient crater. In simple craters, the original excavation cavity is overlain by a lens of collapse breccia, ejecta and melt rock, and a portion of the central crater floor may sometimes be flat.
Above a certain threshold size, which varies with planetary gravity, the collapse and modification of the transient cavity is much more extensive, and the resulting structure is called a complex crater. The collapse of the transient cavity is driven by gravity, and involves both the uplift of the central region and the inward collapse of the rim. The central uplift is not the result of elastic rebound, which is a process in which a material with elastic strength attempts to return to its original geometry; rather the collapse is a process in which a material with little or no strength attempts to return to a state of gravitational equilibrium.
Complex craters have uplifted centers, and they have typically broad flat shallow crater floors, and terraced walls. At the largest sizes, one or more exterior or interior rings may appear, and the structure may be labeled an impact basin rather than an impact crater. Complex-crater morphology on rocky planets appears to follow a regular sequence with increasing size: small complex craters with a central topographic peak are called central peak craters, for example Tycho; intermediate-sized craters, in which the central peak is replaced by a ring of peaks, are called peak-ring craters, for example Schrödinger; and the largest craters contain multiple concentric topographic rings, and are called multi-ringed basins, for example Orientale. On icy (as opposed to rocky) bodies, other morphological forms appear that may have central pits rather than central peaks, and at the largest sizes may contain many concentric rings. Valhalla on Callisto is an example of this type.
### Subsequent modification {#subsequent_modification}
Long after an impact event, a crater may be further modified by erosion, mass wasting processes, viscous relaxation, or erased entirely. These effects are most prominent on geologically and meteorologically active bodies such as Earth, Titan, Triton, and Io. However, heavily modified craters may be found on more primordial bodies such as Callisto, where many ancient craters flatten into bright ghost craters, or palimpsests.
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# Impact crater
## Identifying impact craters {#identifying_impact_craters}
Non-explosive volcanic craters can usually be distinguished from impact craters by their irregular shape and the association of volcanic flows and other volcanic materials. Impact craters produce melted rocks as well, but usually in smaller volumes with different characteristics.
The distinctive mark of an impact crater is the presence of rock that has undergone shock-metamorphic effects, such as shatter cones, melted rocks, and crystal deformations. The problem is that these materials tend to be deeply buried, at least for simple craters. They tend to be revealed in the uplifted center of a complex crater, however.
Impacts produce distinctive shock-metamorphic effects that allow impact sites to be distinctively identified. Such shock-metamorphic effects can include:
- A layer of shattered or \"brecciated\" rock under the floor of the crater. This layer is called a \"breccia lens\".
- Shatter cones, which are chevron-shaped impressions in rocks. Such cones are formed most easily in fine-grained rocks.
- High-temperature rock types, including laminated and welded blocks of sand, spherulites and tektites, or glassy spatters of molten rock. The impact origin of tektites has been questioned by some researchers; they have observed some volcanic features in tektites not found in impactites. Tektites are also drier (contain less water) than typical impactites. While rocks melted by the impact resemble volcanic rocks, they incorporate unmelted fragments of bedrock, form unusually large and unbroken fields, and have a much more mixed chemical composition than volcanic materials spewed up from within the Earth. They also may have relatively large amounts of trace elements that are associated with meteorites, such as nickel, platinum, iridium, and cobalt. Note: scientific literature has reported that some \"shock\" features, such as small shatter cones, which are often associated only with impact events, have been found also in terrestrial volcanic ejecta.
- Microscopic pressure deformations of minerals. These include fracture patterns in crystals of quartz and feldspar, and formation of high-pressure materials such as diamond, derived from graphite and other carbon compounds, or stishovite and coesite, varieties of shocked quartz.
- Buried craters, such as the Decorah crater, can be identified through drill coring, aerial electromagnetic resistivity imaging, and airborne gravity gradiometry.
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# Impact crater
## Economic importance {#economic_importance}
On Earth, impact craters have resulted in useful minerals. Some of the ores produced from impact related effects on Earth include ores of iron, uranium, gold, copper, and nickel. It is estimated that the value of materials mined from impact structures is five billion dollars/year just for North America. The eventual usefulness of impact craters depends on several factors, especially the nature of the materials that were impacted and when the materials were affected. In some cases, the deposits were already in place and the impact brought them to the surface. These are called \"progenetic economic deposits.\" Others were created during the actual impact. The great energy involved caused melting. Useful minerals formed as a result of this energy are classified as \"syngenetic deposits.\" The third type, called \"epigenetic deposits,\" is caused by the creation of a basin from the impact. Many of the minerals that our modern lives depend on are associated with impacts in the past. The Vredeford Dome in the center of the Witwatersrand Basin is the largest goldfield in the world, which has supplied about 40% of all the gold ever mined in an impact structure (though the gold did not come from the bolide). The asteroid that struck the region was 6 mi wide. The Sudbury Basin was caused by an impacting body over 6 mi in diameter. This basin is famous for its deposits of nickel, copper, and platinum group elements. An impact was involved in making the Carswell structure in Saskatchewan, Canada; it contains uranium deposits. Hydrocarbons are common around impact structures. Fifty percent of impact structures in North America in hydrocarbon-bearing sedimentary basins contain oil/gas fields.
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# Impact crater
## Lists of craters {#lists_of_craters}
### Impact craters on Earth {#impact_craters_on_earth}
On Earth, the recognition of impact craters is a branch of geology, and is related to planetary geology in the study of other worlds. Out of many proposed craters, relatively few are confirmed. The following twenty are a sample of articles of confirmed and well-documented impact sites.
See the Earth Impact Database, a website concerned with 190 (`{{as of|2019|07|lc=y}}`{=mediawiki}) scientifically confirmed impact craters on Earth.
### Some extraterrestrial craters {#some_extraterrestrial_craters}
- Caloris Basin (Mercury)
- Hellas Basin (Mars)
- Herschel crater (Mimas)
- Mare Orientale (Moon)
- Petrarch crater (Mercury)
- South Pole -- Aitken basin (Moon)
### Largest named craters in the Solar System {#largest_named_craters_in_the_solar_system}
1. North Polar Basin/Borealis Basin (disputed) -- Mars -- Diameter: 10,600 km
2. South Pole-Aitken basin -- Moon -- Diameter: 2,500 km
3. Hellas Basin -- Mars -- Diameter: 2,100 km
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1. Caloris Basin -- Mercury -- Diameter: 1,550 km
2. Sputnik Planitia -- Pluto -- Diameter: 1,300 km
3. Imbrium Basin -- Moon -- Diameter: 1,100 km
4. Isidis Planitia -- Mars -- Diameter: 1,100 km
5. Mare Tranquilitatis -- Moon -- Diameter: 870 km
6. Argyre Planitia -- Mars -- Diameter: 800 km
7. Rembrandt -- Mercury -- Diameter: 715 km
8. Serenitatis Basin -- Moon -- Diameter: 700 km
9. Mare Nubium -- Moon -- Diameter: 700 km
10. Beethoven -- Mercury -- Diameter: 625 km
11. Valhalla -- Callisto -- Diameter: 600 km, with rings to 4,000 km diameter
12. Hertzsprung -- Moon -- Diameter: 590 km
13. Turgis -- Iapetus -- Diameter: 580 km
14. Apollo -- Moon -- Diameter: 540 km
15. Engelier -- Iapetus -- Diameter: 504 km
16. Mamaldi -- Rhea -- Diameter: 480 km
17. Huygens -- Mars -- Diameter: 470 km
18. Schiaparelli -- Mars -- Diameter: 470 km
19. Rheasilvia -- 4 Vesta -- Diameter: 460 km
20. Gerin -- Iapetus -- Diameter: 445 km
21. Odysseus -- Tethys -- Diameter: 445 km
22. Korolev -- Moon -- Diameter: 430 km
23. Falsaron -- Iapetus -- Diameter: 424 km
24. Dostoevskij -- Mercury -- Diameter: 400 km
25. Menrva -- Titan -- Diameter: 392 km
26. Tolstoj -- Mercury -- Diameter: 390 km
27. Goethe -- Mercury -- Diameter: 380 km
28. Malprimis -- Iapetus -- Diameter: 377 km
29. Tirawa -- Rhea -- Diameter: 360 km
30. Orientale Basin -- Moon -- Diameter: 350 km, with rings to 930 km diameter
31. Evander -- Dione -- Diameter: 350 km
32. Epigeus -- Ganymede -- Diameter: 343 km
33. Gertrude -- Titania -- Diameter: 326 km
34. Telemus -- Tethys -- Diameter: 320 km
35. Asgard -- Callisto -- Diameter: 300 km, with rings to 1,400 km diameter
36. Vredefort impact structure -- Earth -- Diameter: 300 km
37. Burney -- Pluto -- Diameter: 296 km
There are approximately twelve more impact craters/basins larger than 300 km on the Moon, five on Mercury, and four on Mars. Large basins, some unnamed but mostly smaller than 300 km, can also be found on Saturn\'s moons Dione, Rhea and Iapetus
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# Charles Farrar Browne
**Charles Farrar Browne** (April 26, 1834 -- March 6, 1867) was an American humor writer, better known under his *nom de plume*, **Artemus Ward**, which as a character, an illiterate rube with \"Yankee common sense\", Browne also played in public performances. He is considered to be America\'s first stand-up comedian. His birth name was Brown but he added the \"e\" after he became famous.
## Biography
Browne was born on 26 April 1834,
in Waterford, Maine to Caroline (née Farrar)
\"a descendant of the first Puritans\" and Levi Brown,
who \"operated a store in Waterford, engaged in farming and did some surveying\",
and was a justice of the peace.
He began his career at the age of fourteen, \"learned the printer\'s trade\"`{{cite news |title=ARTEMUS WARD.; An Old Friend's Reminiscences of the Genial American Humorist. |url=https://www.nytimes.com/1905/01/04/archives/artemus-ward-an-old-friends-reminiscences-of-the-genial-american.html |access-date=14 June 2025 |work=[[nytimes.com]] |date=Jan 4, 1905}}`{=mediawiki}
at *The Advertiser* in Norway, Maine, and later apprenticed in the printing office of *The Skowhegan Clarion*,
Skowhegan, Maine, then, as a compositor and occasional contributor to the daily and weekly journals. In 1858, in *The Plain Dealer* newspaper (Cleveland, Ohio), he published the first of the \"Artemus Ward\" series (\"a barely literate circus sideshow manager who toured the country and wrote about the people and events he saw.\"`{{cite news |title=Coastal History: Maine's Charles F. Browne, a.k.a. Artemus Ward, and the birth of stand-up |url=https://www.pressherald.com/2019/04/24/coastal-history-maines-charles-f-browne-a-k-a-artemus-ward-and-the-birth-of-stand-up/ |access-date=14 June 2025 |work=[[Portland Press Herald]] |date=24 April 2019}}`{=mediawiki}
\"loosely based on P.T. Barnum\"
), which, in collected form, achieved great popularity in both America and England.
Browne\'s companion at the *Plain Dealer*, George Hoyt, wrote:
> \"his desk was a rickety table which had been whittled and gashed until it looked as if it had been the victim of lightning. His chair was a fit companion thereto, a wabbling, unsteady affair, sometimes with four and sometimes with three legs. But Browne saw neither the table, nor the chair, nor any person who might be near, nothing, in fact, but the funny pictures which were tumbling out of his brain. When writing, his gaunt form looked ridiculous enough. One leg hung over the arm of his chair like a great hook, while he would write away, sometimes laughing to himself, and then slapping the table in the excess of his mirth.\"
In 1860, he became editor of the first *Vanity Fair*, a humorous New York weekly that failed in 1863. At about the same time, he began to appear as a lecturer who, by his droll and eccentric humor, attracted large audiences. Browne was also known as a member of the New York bohemian set which included leader Henry Clapp Jr., Walt Whitman, Fitz Hugh Ludlow, and actress Adah Isaacs Menken.
> Though his books were popular, it was his lecturing, delivered with deadpan expression, that brought him fame.
In 1863, Browne came to San Francisco to perform as Artemus Ward. An early expert at show business publicity, Browne sent his manager ahead by several weeks to buy advertising in the local papers and promote the show among prominent citizens for endorsements. On November 13, 1863, Browne stood before a packed crowd at Platt\'s Music Hall, in Waterford, Maine.
## Legacy
In Cleveland, where Browne started his comedy career, an elementary school is named after him, known as **Artemus Ward Elementary** on W. 140th Street. In the American Garden of the Cleveland Cultural Gardens in Rockefeller Park, a monument of him was erected, next to Mark Twain.
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# Charles Farrar Browne
## Works
### Short stories {#short_stories}
- A Visit to Brigham Young
- Women\'s Rights
- One of Mr Ward\'s Business Letters
- On \"Forts\"
- Fourth of July Oration
- High-Handed Outrage at Utica
- Artemus Ward and the Prince of Wales
- Interview with Lincoln
- Letters to his Wife
### *Artemus Ward* books {#artemus_ward_books}
- [Artemus Ward His Book](https://babel.hathitrust.org/cgi/pt?id=miun.ack0410.0001.001) (1862) (full text online)
- [Artemus Ward His Travels](https://babel.hathitrust.org/cgi/pt?id=miun.abs0367.0001.001) (1865) (full text online)
- [Artemus Ward Among the Mormons](https://babel.hathitrust.org/cgi/pt?id=mdp.39015006946506) (1865) (full text online)
- [Artemus Ward in London](https://babel.hathitrust.org/cgi/pt?id=uc2.ark:/13960/t84j0k81v) (1867) (full text online)
- [Artemus Ward\'s Panorama](https://archive.org/details/artemuswardspan00hinggoog/page/n9) (1869) (full text online)
- [Artemus Ward\'s Lecture](https://babel.hathitrust.org/cgi/pt?id=dul1
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# Chojnów
Piast Castle \| image3 = Chojnów, wieża kościoła Świętych Apostołów Piotra i Pawła (3).jpg{{!}}Gothic Saints Peter and Paul Church \| image4 = Baszta-tkaczy-chojnow.jpg{{!}}Medieval Weavers Tower \| caption1 = Market Square \| caption2 = Piast Castle \| caption3 = Saints Peter and Paul Church \| caption4 = Weavers Tower }} \| image_flag = POL Chojnów flag.svg \| image_shield = POL Chojnów COA.svg \| pushpin_map = Poland \| pushpin_label_position = right \| subdivision_type = Country \| subdivision_name = `{{POL}}`{=mediawiki} \| subdivision_type1 = Voivodeship \| subdivision_name1 = `{{flag|Lower Silesian Voivodeship|name=Lower Silesian}}`{=mediawiki} \| subdivision_type2 = County \| subdivision_name2 = Legnica \| subdivision_type3 = Gmina \| subdivision_name3 = Chojnów (urban gmina) \| leader_title = Mayor \| leader_name = Jan Serkies \| established_title = Established \| established_date = 14th century \| established_title3 = Town rights \| established_date3 = 1333 \| area_total_km2 = 5.32 \| population_as_of = 31 December 2021 \| population_total = 13002 \| population_density_km2 = auto \| timezone = CET \| utc_offset = +1 \| timezone_DST = CEST \| utc_offset_DST = +2 \| coordinates = 51 16 N 15 56 E region:PL display=title,inline \| elevation_m = 170 \| postal_code_type = Postal code \| postal_code = 59-224, 59-225 \| area_code = +48 76 \| blank_name = Car plates \| blank_info = DLE \| website = `{{URL|chojnow.eu}}`{=mediawiki} }} **Chojnów** `{{IPAc-pl|AUD|Pl-Chojnów.ogg|'|h|o|j|n|u|f}}`{=mediawiki} (*Haynau*) is a small town in Legnica County, Lower Silesian Voivodeship, in south-western Poland.`{{TERYT}}`{=mediawiki} It is located on the Skora river, a tributary of the Kaczawa at an average altitude of 170 m above sea level. Chojnów is the administrative seat of the rural gmina called Gmina Chojnów, although the town is not part of its territory and forms a separate urban gmina. As of December 2021, the town has 13,002 inhabitants.
Chojnów is located 18 km west of Legnica, 26 km east from Bolesławiec and 18 km north of Złotoryja, 5 km from the A4 motorway. It has railroad connections to Bolesławiec and Legnica.
## Heraldry
The Chojnów coat of arms is a blue escutcheon featuring a white castle with three towers. To the right side of the central tower is a silver crescent moon and to its left side a golden sun. In the gate of the castle is a Silesian Eagle on a yellow background. Chojnów\'s motto is \"Friendly Town\".
## Geography
Chojnów is located in the Central-Western part of the Lower Silesia region. The Skora (Leather) River flows through the town in a westerly direction. The city of Chojnów is 5.32 km2 in area, including 41% agricultural land.
Chojnów has a connection with the major cities of the country (road and rail) and located 5 km south of Chojnów has the A4 Autostrada. To the South of the town is the surrounding Chojnowska Plain.
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# Chojnów
## History
The town is first mentioned in a Latin mediaeval document issued in Wrocław on February 26, 1253, stating, the Silesian Duke Henry III when the town is mentioned under the name Honowo. Possible the name of nearby Hainau Island. The name is of Polish origin, and in more modern records from the 19th century, the Polish name appears as *Hajnów*, while *Haynau* is the Germanized version of the original Polish name.
The settlement of *Haynow* was mentioned in a 1272 deed. It was already called a *civitas* in a 1288 document issued by the Piast duke Henry V of Legnica, and officially received town privileges in 1333 from Duke Bolesław III the Generous. It was part of the duchies of Wrocław, Głogów and Legnica of fragmented Poland and remained under the rule of the Piast dynasty until 1675. Its population was predominantly Polish. In 1292 the first castellan of Chojnów, Bronisław Budziwojowic, was mentioned. In the 14th and early 15th centuries Chojnów was granted various privileges, including staple right and gold mining right, thanks to which it flourished.
The town survived the Hussites, who burned almost the entire town center and castle, but it quickly helped recover its former glory. The largest boom Chojnów experienced was in the 16th century, however by the end of that century began to decline due to fires and epidemic, which claimed many victims in 1613. During the Thirty Years\' War (1618--1648), there was another outbreak in the city, it was occupied by the Austrians and Swedes and in 1642 it was also plundered by the Swedes. It remained part of the Piast-ruled Duchy of Legnica until its dissolution in 1675, when it was incorporated to Habsburg-ruled Bohemia. In the 18th century, cloth production developed and a clothmaking school was established in the town. One of two main routes connecting Warsaw and Dresden ran through the town in the 18th century and Kings Augustus II the Strong and Augustus III of Poland traveled that route numerous times. In 1740 the town was captured by Prussia and subsequently annexed in 1742. In 1804 it suffered a flood. During the Napoleonic wars there were more epidemics. In 1813 in Chojnów, Napoleon Bonaparte issued instructions regarding the reorganization of the 8th Polish Corps of Prince Józef Poniatowski. The event is commemorated by a plaque in the facade of the Piast Castle. A railway line was opened in the 19th century. Sewer, Gas lighting a Newspaper and a hospital soon followed as the towns economy improved.
The city was not spared in World War II, with 30% of the town being destroyed on February 10, 1945, when Soviet Red Army troops took the abandoned town. After World War II and the implementation of the Oder-Neisse line in 1945, the town passed to the Republic of Poland. It was repopulated by Poles, expelled from former eastern Poland annexed by the Soviet Union. In 1946 it was renamed *Chojnów*, a more modern version of the old Polish *Hajnów*. Also Greeks, refugees of the Greek Civil War, settled in Chojnów.
## Population
## Economy
Chojnów is an industrial and agricultural town. Among local products are: paper, agricultural machinery, chains, metal furniture for hospitals, equipment for the meat industry, beer, wine, leather clothing, and clothing for infants, children and adults.
## Sights and nature {#sights_and_nature}
Among the interesting monuments of Chojnów are the 13th-century castle of the Dukes of Legnica (currently used as a museum), two old churches, the *Baszta Tkaczy* (*Weavers\' Tower*) and preserved fragments of city walls.
The biggest green area in Chojnów is small forest *Park Piastowski* (*Piast\'s Park*), named after Piast dynasty. Wild animals that can be found in the Chojnów area are roe deer, foxes, rabbits and wild domestic animals, especially cats.
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# Chojnów
## Culture and sport {#culture_and_sport}
Every year in the first days of June, the *Days of Chojnów* (*Dni Chojnowa*) are celebrated. The Whole-Poland bike race *Masters* has been organized yearly in Chojnów for the past few years.
Chojnów has a Municipal sports and recreation center formed in 2008 holding various events, festivals, reviews, exhibitions, and competitions. The regional Museum is housed in the old Piast era castle. The collections include tiles, relics, and the castle garden. Next to the Museum there is a municipal library. In śródmiejskim Park, near the Town Hall is the amphitheatre.
The local government-run weekly newspaper is Gazeta Chojnowska, which has been published since 1992. It is published biweekly. Editions have a run of 900 copies and it is one of the oldest newspapers in Poland issued without interruption. The *Chojnów* is the official newspaper of Chojnów with copy run of 750 copies.
## Education
In Chojnów, there are two kindergartens, two elementary schools and two middle schools.
- Mary Konopnickiej is the smallest elementary school in Chojnów, and is located in the northern part of the city, close to the train station and founded in 1962.
- Janusz Korczak is the largest primary school in Chojnów in the southern part of the town.
- Middle School No. (Pope John Paul II), it is situated in the north-western part of the city next to the \"Small Church\".
- Gimnazjum nr 2 im. Nicolaus Copernicus is the largest high school in Chojnów.
- Liceum Ogólnokształcące im. Nicolaus Copernicus
## Religion
Chojnów is in the Catholic deanery of Chojnów and has two parishes, Immaculate Conception of the Blessed Virgin Mary and also the Holy Apostles Peter and Paul. Both parishes have active congregations. There are also two Congregations of Jehovah\'s witnesses.
## Notable people {#notable_people}
- Johann Wilhelm Ritter (1776--1810), chemist and physicist
- Georg Michaelis (1857--1936), politician, Chancellor of Germany (1917).
- Edith Jacobson (1897--1978), German psychoanalyst
- Oswald Lange (1912--2000), German--American aerospace engineer
- Horst Mahler (born 1936), German lawyer, former Red Army Faction militant, now Neo-Nazi activist
## Twin towns -- sister cities {#twin_towns_sister_cities}
Chojnów is twinned with:
- Commentry, France
- Egelsbach, Germany
- Mnichovo Hradiště, Czech Republic
## Gallery
Chojnow(js).jpg\|Entrance to the Piast Castle Chojnów, Ab-047.JPG\|Flower beds in Chojnów Chojnów, Wzgórze Chmielowe.jpg\|Park Piastowski Chojnów, Ratusz (2).jpg\|Town hall SM Chojnów kościół Niepokalanego Poczęcia NMP (5) ID 593383.jpg\|Immaculate Conception Church SM Chojnów Konarskiego4 (0).jpg\|Nicolaus Copernicus Gymnasium No. 2 Chojnów, Ab-057.JPG\|Monument to Polish soldiers killed in World War II and murdered in labour camps and exiled to Siberia Łabędzi Staw.jpg\|Swan\'s Pond (*Łabędzi Staw*) in winter Chojnów, Dworzec kolejowy (2).jpg\|Chojnów Railway Station Chojnow 055 most kolejowy
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# Clock
A **clock** or **chronometer** is a device that measures and displays time. The clock is one of the oldest human inventions, meeting the need to measure intervals of time shorter than the natural units such as the day, the lunar month, and the year. Devices operating on several physical processes have been used over the millennia.
Some predecessors to the modern clock may be considered \"clocks\" that are based on movement in nature: A sundial shows the time by displaying the position of a shadow on a flat surface. There is a range of duration timers, a well-known example being the hourglass. Water clocks, along with sundials, are possibly the oldest time-measuring instruments. A major advance occurred with the invention of the verge escapement, which made possible the first mechanical clocks around 1300 in Europe, which kept time with oscillating timekeepers like balance wheels.
Traditionally, in horology (the study of timekeeping), the term *clock* was used for a striking clock, while a clock that did not strike the hours audibly was called a **timepiece**. This distinction is not generally made any longer. Watches and other timepieces that can be carried on one\'s person are usually not referred to as clocks. Spring-driven clocks appeared during the 15th century. During the 15th and 16th centuries, clockmaking flourished. The next development in accuracy occurred after 1656 with the invention of the pendulum clock by Christiaan Huygens. A major stimulus to improving the accuracy and reliability of clocks was the importance of precise time-keeping for navigation. The mechanism of a timepiece with a series of gears driven by a spring or weights is referred to as clockwork; the term is used by extension for a similar mechanism not used in a timepiece. The electric clock was patented in 1840, and electronic clocks were introduced in the 20th century, becoming widespread with the development of small battery-powered semiconductor devices.
The timekeeping element in every modern clock is a harmonic oscillator, a physical object (resonator) that vibrates or oscillates at a particular frequency. This object can be a pendulum, a balance wheel, a tuning fork, a quartz crystal, or the vibration of electrons in atoms as they emit microwaves, the last of which is so precise that it serves as the formal definition of the second.
Clocks have different ways of displaying the time. Analog clocks indicate time with a traditional clock face and moving hands. Digital clocks display a numeric representation of time. Two numbering systems are in use: 12-hour time notation and 24-hour notation. Most digital clocks use electronic mechanisms and LCD, LED, or VFD displays. For the blind and for use over telephones, speaking clocks state the time audibly in words. There are also clocks for the blind that have displays that can be read by touch.
## Etymology
The word *clock* derives from the medieval Latin word for \'bell\'---*clocca*---and has cognates in many European languages. Clocks spread to England from the Low Countries, so the English word came from the Middle Low German and Middle Dutch *Klocke*. The word is also derived from the Middle English *clokke*, Old North French *cloque*, or Middle Dutch *clocke*, all of which mean \'bell\'.
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# Clock
## History of time-measuring devices {#history_of_time_measuring_devices}
### Sundials
The apparent position of the Sun in the sky changes over the course of each day, reflecting the rotation of the Earth. Shadows cast by stationary objects move correspondingly, so their positions can be used to indicate the time of day. A sundial shows the time by displaying the position of a shadow on a (usually) flat surface that has markings that correspond to the hours. Sundials can be horizontal, vertical, or in other orientations. Sundials were widely used in ancient times. With knowledge of latitude, a well-constructed sundial can measure local solar time with reasonable accuracy, within a minute or two. Sundials continued to be used to monitor the performance of clocks until the 1830s, when the use of the telegraph and trains standardized time and time zones between cities.
### Devices that measure duration, elapsed time and intervals {#devices_that_measure_duration_elapsed_time_and_intervals}
Many devices can be used to mark the passage of time without respect to reference time (time of day, hours, minutes, etc.) and can be useful for measuring duration or intervals. Examples of such duration timers are candle clocks, incense clocks, and the hourglass. Both the candle clock and the incense clock work on the same principle, wherein the consumption of resources is more or less constant, allowing reasonably precise and repeatable estimates of time passages. In the hourglass, fine sand pouring through a tiny hole at a constant rate indicates an arbitrary, predetermined passage of time. The resource is not consumed, but re-used.
### Water clocks {#water_clocks}
Water clocks, along with sundials, are possibly the oldest time-measuring instruments, with the only exception being the day-counting tally stick. Given their great antiquity, where and when they first existed is not known and is perhaps unknowable. The bowl-shaped outflow is the simplest form of a water clock and is known to have existed in Babylon and Egypt around the 16th century BC. Other regions of the world, including India and China, also have early evidence of water clocks, but the earliest dates are less certain. Some authors, however, write about water clocks appearing as early as 4000 BC in these regions of the world.
The Macedonian astronomer Andronicus of Cyrrhus supervised the construction of the Tower of the Winds in Athens in the 1st century BC, which housed a large clepsydra inside as well as multiple prominent sundials outside, allowing it to function as a kind of early clocktower. The Greek and Roman civilizations advanced water clock design with improved accuracy. These advances were passed on through Byzantine and Islamic times, eventually making their way back to Europe. Independently, the Chinese developed their own advanced water clocks (*水鐘*) by 725 AD, passing their ideas on to Korea and Japan.
Some water clock designs were developed independently, and some knowledge was transferred through the spread of trade. Pre-modern societies do not have the same precise timekeeping requirements that exist in modern industrial societies, where every hour of work or rest is monitored and work may start or finish at any time regardless of external conditions. Instead, water clocks in ancient societies were used mainly for astrological reasons. These early water clocks were calibrated with a sundial. While never reaching the level of accuracy of a modern timepiece, the water clock was the most accurate and commonly used timekeeping device for millennia until it was replaced by the more accurate pendulum clock in 17th-century Europe.
Islamic civilization is credited with further advancing the accuracy of clocks through elaborate engineering. In 797 (or possibly 801), the Abbasid caliph of Baghdad, Harun al-Rashid, presented Charlemagne with an Asian elephant named Abul-Abbas together with a \"particularly elaborate example\" of a water clock. Pope Sylvester II introduced clocks to northern and western Europe around 1000 AD.
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# Clock
## History of time-measuring devices {#history_of_time_measuring_devices}
### Mechanical water clocks {#mechanical_water_clocks}
The first known geared clock was invented by the great mathematician, physicist, and engineer Archimedes during the 3rd century BC. Archimedes created his astronomical clock,`{{fact|reason=I can't find any record of this book, ISBN doesn't even seem to exist|date=April 2024}}`{=mediawiki} which was also a cuckoo clock with birds singing and moving every hour. It is the first carillon clock as it plays music simultaneously with a person blinking his eyes, surprised by the singing birds. The Archimedes clock works with a system of four weights, counterweights, and strings regulated by a system of floats in a water container with siphons that regulate the automatic continuation of the clock. The principles of this type of clock are described by the mathematician and physicist Hero, who says that some of them work with a chain that turns a gear in the mechanism. Another Greek clock probably constructed at the time of Alexander was in Gaza, as described by Procopius. The Gaza clock was probably a Meteoroskopeion, i.e., a building showing celestial phenomena and the time. It had a pointer for the time and some automations similar to the Archimedes clock. There were 12 doors opening one every hour, with Hercules performing his labors, the Lion at one o\'clock, etc., and at night a lamp becomes visible every hour, with 12 windows opening to show the time.
The Tang dynasty Buddhist monk Yi Xing along with government official Liang Lingzan made the escapement in 723 (or 725) to the workings of a water-powered armillary sphere and clock drive, which was the world\'s first clockwork escapement. The Song dynasty polymath and genius Su Song (1020--1101) incorporated it into his monumental innovation of the astronomical clock tower of Kaifeng in 1088.`{{Page needed|date=July 2011}}`{=mediawiki} His astronomical clock and rotating armillary sphere still relied on the use of either flowing water during the spring, summer, and autumn seasons or liquid mercury during the freezing temperatures of winter (i.e., hydraulics). In Su Song\'s waterwheel linkwork device, the action of the escapement\'s arrest and release was achieved by gravity exerted periodically as the continuous flow of liquid-filled containers of a limited size. In a single line of evolution, Su Song\'s clock therefore united the concepts of the clepsydra and the mechanical clock into one device run by mechanics and hydraulics. In his memorial, Su Song wrote about this concept:
> According to your servant\'s opinion there have been many systems and designs for astronomical instruments during past dynasties all differing from one another in minor respects. But the principle of the use of water-power for the driving mechanism has always been the same. The heavens move without ceasing but so also does water flow (and fall). Thus if the water is made to pour with perfect evenness, then the comparison of the rotary movements (of the heavens and the machine) will show no discrepancy or contradiction; for the unresting follows the unceasing.
Song was also strongly influenced by the earlier armillary sphere created by Zhang Sixun (976 AD), who also employed the escapement mechanism and used liquid mercury instead of water in the waterwheel of his astronomical clock tower. The mechanical clockworks for Su Song\'s astronomical tower featured a great driving-wheel that was 11 feet in diameter, carrying 36 scoops, into each of which water was poured at a uniform rate from the \"constant-level tank\". The main driving shaft of iron, with its cylindrical necks supported on iron crescent-shaped bearings, ended in a pinion, which engaged a gear wheel at the lower end of the main vertical transmission shaft. This great astronomical hydromechanical clock tower was about ten metres high (about 30 feet), featured a clock escapement, and was indirectly powered by a rotating wheel either with falling water or liquid mercury. A full-sized working replica of Su Song\'s clock exists in the Republic of China (Taiwan)\'s National Museum of Natural Science, Taichung city. This full-scale, fully functional replica, approximately 12 meters (39 feet) in height, was constructed from Su Song\'s original descriptions and mechanical drawings. The Chinese escapement spread west and was the source for Western escapement technology. In the 12th century, Al-Jazari, an engineer from Mesopotamia (lived 1136--1206) who worked for the Artuqid king of Diyar-Bakr, Nasir al-Din, made numerous clocks of all shapes and sizes. The most reputed clocks included the elephant, scribe, and castle clocks, some of which have been successfully reconstructed. As well as telling the time, these grand clocks were symbols of the status, grandeur, and wealth of the Urtuq State. Knowledge of these mercury escapements may have spread through Europe with translations of Arabic and Spanish texts.
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# Clock
## History of time-measuring devices {#history_of_time_measuring_devices}
### Fully mechanical {#fully_mechanical}
The word *horologia* (from the Greek *ὥρα*---\'hour\', and *λέγειν*---\'to tell\') was used to describe early mechanical clocks, but the use of this word (still used in several Romance languages) for all timekeepers conceals the true nature of the mechanisms. For example, there is a record that in 1176, Sens Cathedral in France installed an \'horologe\', but the mechanism used is unknown. According to Jocelyn de Brakelond, in 1198, during a fire at the abbey of St Edmundsbury (now Bury St Edmunds), the monks \"ran to the clock\" to fetch water, indicating that their water clock had a reservoir large enough to help extinguish the occasional fire. The word *clock* (via Medieval Latin *clocca* from Old Irish *clocc*, both meaning \'bell\'), which gradually supersedes \"horologe\", suggests that it was the sound of bells that also characterized the prototype mechanical clocks that appeared during the 13th century in Europe.
thumb\|upright=1.2\|A 17th-century weight-driven clock in Läckö Castle, Sweden
In Europe, between 1280 and 1320, there was an increase in the number of references to clocks and horologes in church records, and this probably indicates that a new type of clock mechanism had been devised. Existing clock mechanisms that used water power were being adapted to take their driving power from falling weights. This power was controlled by some form of oscillating mechanism, probably derived from existing bell-ringing or alarm devices. This controlled release of power -- the escapement -- marks the beginning of the true mechanical clock, which differed from the previously mentioned cogwheel clocks. The verge escapement mechanism appeared during the surge of true mechanical clock development, which did not need any kind of fluid power, like water or mercury, to work.
These mechanical clocks were intended for two main purposes: for signalling and notification (e.g., the timing of services and public events) and for modeling the Solar System. The former purpose is administrative; the latter arises naturally given the scholarly interests in astronomy, science, and astrology and how these subjects integrated with the religious philosophy of the time. The astrolabe was used both by astronomers and astrologers, and it was natural to apply a clockwork drive to the rotating plate to produce a working model of the solar system.
Simple clocks intended mainly for notification were installed in towers and did not always require faces or hands. They would have announced the canonical hours or intervals between set times of prayer. Canonical hours varied in length as the times of sunrise and sunset shifted. The more sophisticated astronomical clocks would have had moving dials or hands and would have shown the time in various time systems, including Italian hours, canonical hours, and time as measured by astronomers at the time. Both styles of clocks started acquiring extravagant features, such as automata.
In 1283, a large clock was installed at Dunstable Priory in Bedfordshire in southern England; its location above the rood screen suggests that it was not a water clock. In 1292, Canterbury Cathedral installed a \'great horloge\'. Over the next 30 years, there were mentions of clocks at a number of ecclesiastical institutions in England, Italy, and France. In 1322, a new clock was installed in Norwich, an expensive replacement for an earlier clock installed in 1273. This had a large (2 metre) astronomical dial with automata and bells. The costs of the installation included the full-time employment of two clockkeepers for two years.
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# Clock
## History of time-measuring devices {#history_of_time_measuring_devices}
### Astronomical
An elaborate water clock, the \'Cosmic Engine\', was invented by Su Song, a Chinese polymath, designed and constructed in China in 1092. This great astronomical hydromechanical clock tower was about ten metres high (about 30 feet) and was indirectly powered by a rotating wheel with falling water and liquid mercury, which turned an armillary sphere capable of calculating complex astronomical problems.
In Europe, there were the clocks constructed by Richard of Wallingford in Albans by 1336, and by Giovanni de Dondi in Padua from 1348 to 1364. They no longer exist, but detailed descriptions of their design and construction survive, and modern reproductions have been made. They illustrate how quickly the theory of the mechanical clock had been translated into practical constructions, and also that one of the many impulses to their development had been the desire of astronomers to investigate celestial phenomena.
The Astrarium of Giovanni Dondi dell\'Orologio was a complex astronomical clock built between 1348 and 1364 in Padua, Italy, by the doctor and clock-maker Giovanni Dondi dell\'Orologio. The Astrarium had seven faces and 107 moving gears; it showed the positions of the Sun, the Moon and the five planets then known, as well as religious feast days. The astrarium stood about 1 metre high, and consisted of a seven-sided brass or iron framework resting on 7 decorative paw-shaped feet. The lower section provided a 24-hour dial and a large calendar drum, showing the fixed feasts of the church, the movable feasts, and the position in the zodiac of the Moon\'s ascending node. The upper section contained 7 dials, each about 30 cm in diameter, showing the positional data for the Primum Mobile, Venus, Mercury, the Moon, Saturn, Jupiter, and Mars. Directly above the 24-hour dial is the dial of the Primum Mobile, so called because it reproduces the diurnal motion of the stars and the annual motion of the Sun against the background of stars. Each of the \'planetary\' dials used complex clockwork to produce reasonably accurate models of the planets\' motion. These agreed reasonably well both with Ptolemaic theory and with observations.
Wallingford\'s clock had a large astrolabe-type dial, showing the Sun, the Moon\'s age, phase, and node, a star map, and possibly the planets. In addition, it had a wheel of fortune and an indicator of the state of the tide at London Bridge. Bells rang every hour, the number of strokes indicating the time. Dondi\'s clock was a seven-sided construction, 1 metre high, with dials showing the time of day, including minutes, the motions of all the known planets, an automatic calendar of fixed and movable feasts, and an eclipse prediction hand rotating once every 18 years. It is not known how accurate or reliable these clocks would have been. They were probably adjusted manually every day to compensate for errors caused by wear and imprecise manufacture. Water clocks are sometimes still used, and can be examined in places such as ancient castles and museums. The Salisbury Cathedral clock, built in 1386, is considered to be the world\'s oldest surviving mechanical clock that strikes the hours.
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# Clock
## History of time-measuring devices {#history_of_time_measuring_devices}
### Spring-driven {#spring_driven}
Matthew Norman carriage clock with winding key.jpg\|Matthew Norman carriage clock with winding key 1908 Gilbert mantel clock decorated with Memento Mori decoupage.JPG\|Decorated William Gilbert mantel clock
Clockmakers developed their art in various ways. Building smaller clocks was a technical challenge, as was improving accuracy and reliability. Clocks could be impressive showpieces to demonstrate skilled craftsmanship, or less expensive, mass-produced items for domestic use. The escapement in particular was an important factor affecting the clock\'s accuracy, so many different mechanisms were tried.
Spring-driven clocks appeared during the 15th century, although they are often erroneously credited to Nuremberg watchmaker Peter Henlein (or Henle, or Hele) around 1511. The earliest existing spring driven clock is the chamber clock given to Phillip the Good, Duke of Burgundy, around 1430, now in the Germanisches Nationalmuseum. Spring power presented clockmakers with a new problem: how to keep the clock movement running at a constant rate as the spring ran down. This resulted in the invention of the *stackfreed* and the fusee in the 15th century, and many other innovations, down to the invention of the modern *going barrel* in 1760.
Early clock dials did not indicate minutes and seconds. A clock with a dial indicating minutes was illustrated in a 1475 manuscript by Paulus Almanus, and some 15th-century clocks in Germany indicated minutes and seconds. An early record of a seconds hand on a clock dates back to about 1560 on a clock now in the Fremersdorf collection.
During the 15th and 16th centuries, clockmaking flourished, particularly in the metalworking towns of Nuremberg and Augsburg, and in Blois, France. Some of the more basic table clocks have only one time-keeping hand, with the dial between the hour markers being divided into four equal parts making the clocks readable to the nearest 15 minutes. Other clocks were exhibitions of craftsmanship and skill, incorporating astronomical indicators and musical movements. The cross-beat escapement was invented in 1584 by Jost Bürgi, who also developed the remontoire. Bürgi\'s clocks were a great improvement in accuracy as they were correct to within a minute a day. These clocks helped the 16th-century astronomer Tycho Brahe to observe astronomical events with much greater precision than before.`{{how|date=November 2014}}`{=mediawiki}
### Pendulum
The next development in accuracy occurred after 1656 with the invention of the pendulum clock. Galileo had the idea to use a swinging bob to regulate the motion of a time-telling device earlier in the 17th century. Christiaan Huygens, however, is usually credited as the inventor. He determined the mathematical formula that related pendulum length to time (about 99.4 cm or 39.1 inches for the one second movement) and had the first pendulum-driven clock made. The first model clock was built in 1657 in the Hague, but it was in England that the idea was taken up. The longcase clock (also known as the *grandfather clock*) was created to house the pendulum and works by the English clockmaker William Clement in 1670 or 1671. It was also at this time that clock cases began to be made of wood and clock faces to use enamel as well as hand-painted ceramics.
In 1670, William Clement created the anchor escapement, an improvement over Huygens\' crown escapement. Clement also introduced the pendulum suspension spring in 1671. The concentric minute hand was added to the clock by Daniel Quare, a London clockmaker and others, and the second hand was first introduced.
### Hairspring
In 1675, Huygens and Robert Hooke invented the spiral balance spring, or the hairspring, designed to control the oscillating speed of the balance wheel. This crucial advance finally made accurate pocket watches possible. The great English clockmaker Thomas Tompion, was one of the first to use this mechanism successfully in his pocket watches, and he adopted the minute hand which, after a variety of designs were trialled, eventually stabilised into the modern-day configuration. The rack and snail striking mechanism for striking clocks, was introduced during the 17th century and had distinct advantages over the \'countwheel\' (or \'locking plate\') mechanism. During the 20th century there was a common misconception that Edward Barlow invented *rack and snail* striking. In fact, his invention was connected with a repeating mechanism employing the rack and snail. The repeating clock, that chimes the number of hours (or even minutes) on demand was invented by either Quare or Barlow in 1676. George Graham invented the deadbeat escapement for clocks in 1720.
### Marine chronometer {#marine_chronometer}
A major stimulus to improving the accuracy and reliability of clocks was the importance of precise time-keeping for navigation. The position of a ship at sea could be determined with reasonable accuracy if a navigator could refer to a clock that lost or gained less than about 10 seconds per day. This clock could not contain a pendulum, which would be virtually useless on a rocking ship. In 1714, the British government offered large financial rewards to the value of 20,000 pounds for anyone who could determine longitude accurately. John Harrison, who dedicated his life to improving the accuracy of his clocks, later received considerable sums under the Longitude Act.
In 1735, Harrison built his first chronometer, which he steadily improved on over the next thirty years before submitting it for examination. The clock had many innovations, including the use of bearings to reduce friction, weighted balances to compensate for the ship\'s pitch and roll in the sea and the use of two different metals to reduce the problem of expansion from heat. The chronometer was tested in 1761 by Harrison\'s son and by the end of 10 weeks the clock was in error by less than 5 seconds.
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# Clock
## History of time-measuring devices {#history_of_time_measuring_devices}
### Mass production {#mass_production}
The British had dominated watch manufacture for much of the 17th and 18th centuries, but maintained a system of production that was geared towards high quality products for the elite. Although there was an attempt to modernise clock manufacture with mass-production techniques and the application of duplicating tools and machinery by the British Watch Company in 1843, it was in the United States that this system took off. In 1816, Eli Terry and some other Connecticut clockmakers developed a way of mass-producing clocks by using interchangeable parts. Aaron Lufkin Dennison started a factory in 1851 in Massachusetts that also used interchangeable parts, and by 1861 was running a successful enterprise incorporated as the Waltham Watch Company.
### Early electric {#early_electric}
In 1815, the English scientist Francis Ronalds published the first electric clock powered by dry pile batteries. Alexander Bain, a Scottish clockmaker, patented the electric clock in 1840. The electric clock\'s mainspring is wound either with an electric motor or with an electromagnet and armature. In 1841, he first patented the electromagnetic pendulum. By the end of the nineteenth century, the advent of the dry cell battery made it feasible to use electric power in clocks. Spring or weight-driven clocks that use electricity, either alternating current (AC) or direct current (DC), to rewind the spring or raise the weight of a mechanical clock would be classified as an electromechanical clock. This classification would also apply to clocks that employ an electrical impulse to propel the pendulum. In electromechanical clocks, electricity serves no time-keeping function. These types of clocks were made as individual timepieces but are more commonly used in synchronized time installations in schools, businesses, factories, railroads and government facilities as a master clock and slave clocks.
Where an AC electrical supply of stable frequency is available, timekeeping can be maintained very reliably by using a synchronous motor, essentially counting the cycles. The supply current alternates with an accurate frequency of 50 hertz in many countries, and 60 hertz in others. While the frequency may vary slightly during the day as the load changes, generators are designed to maintain an accurate number of cycles over a day, so the clock may be a fraction of a second slow or fast at any time, but will be perfectly accurate over a long time. The rotor of the motor rotates at a speed that is related to the alternation frequency. Appropriate gearing converts this rotation speed to the correct ones for the hands of the analog clock. Time in these cases is measured in several ways, such as by counting the cycles of the AC supply, vibration of a tuning fork, the behaviour of quartz crystals, or the quantum vibrations of atoms. Electronic circuits divide these high-frequency oscillations into slower ones that drive the time display.
### Quartz
The piezoelectric properties of crystalline quartz were discovered by Jacques and Pierre Curie in 1880. The first crystal oscillator was invented in 1917 by Alexander M. Nicholson, after which the first quartz crystal oscillator was built by Walter G. Cady in 1921. In 1927 the first quartz clock was built by Warren Marrison and J.W. Horton at Bell Telephone Laboratories in Canada. The following decades saw the development of quartz clocks as precision time measurement devices in laboratory settings---the bulky and delicate counting electronics, built with vacuum tubes at the time, limited their practical use elsewhere. The National Bureau of Standards (now NIST) based the time standard of the United States on quartz clocks from late 1929 until the 1960s, when it changed to atomic clocks. In 1969, Seiko produced the world\'s first quartz wristwatch, the Astron. Their inherent accuracy and low cost of production resulted in the subsequent proliferation of quartz clocks and watches.
### Atomic
Currently, atomic clocks are the most accurate clocks in existence. They are considerably more accurate than quartz clocks as they can be accurate to within a few seconds over trillions of years. Atomic clocks were first theorized by Lord Kelvin in 1879. In the 1930s the development of magnetic resonance created practical method for doing this. A prototype ammonia maser device was built in 1949 at the U.S. National Bureau of Standards (NBS, now NIST). Although it was less accurate than existing quartz clocks, it served to demonstrate the concept. The first accurate atomic clock, a caesium standard based on a certain transition of the caesium-133 atom, was built by Louis Essen in 1955 at the National Physical Laboratory in the UK. Calibration of the caesium standard atomic clock was carried out by the use of the astronomical time scale *ephemeris time* (ET). As of 2013, the most stable atomic clocks are ytterbium clocks, which are stable to within less than two parts in 1 quintillion (`{{val|2|e=-18}}`{=mediawiki}).
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# Clock
## Operation
The invention of the mechanical clock in the 13th century initiated a change in timekeeping methods from continuous processes, such as the motion of the gnomon\'s shadow on a sundial or the flow of liquid in a water clock, to periodic oscillatory processes, such as the swing of a pendulum or the vibration of a quartz crystal, which had the potential for more accuracy. All modern clocks use oscillation.
Although the mechanisms they use vary, all oscillating clocks, mechanical, electric, and atomic, work similarly and can be divided into analogous parts. They consist of an object that repeats the same motion over and over again, an *oscillator*, with a precisely constant time interval between each repetition, or \'beat\'. Attached to the oscillator is a *controller* device, which sustains the oscillator\'s motion by replacing the energy it loses to friction, and converts its oscillations into a series of pulses. The pulses are then counted by some type of *counter*, and the number of counts is converted into convenient units, usually seconds, minutes, hours, etc. Finally some kind of *indicator* displays the result in human readable form.
### Power source {#power_source}
### Oscillator
The timekeeping element in every modern clock is a harmonic oscillator, a physical object (resonator) that vibrates or oscillates repetitively at a precisely constant frequency.
- In mechanical clocks, this is either a pendulum or a balance wheel.
- In some early electronic clocks and watches such as the Accutron, they use a tuning fork.
- In quartz clocks and watches, it is a quartz crystal.
- In atomic clocks, it is the vibration of electrons in atoms as they emit microwaves.
- In early mechanical clocks before 1657, it was a crude balance wheel or foliot which was not a harmonic oscillator because it lacked a balance spring. As a result, they were very inaccurate, with errors of perhaps an hour a day.
The advantage of a harmonic oscillator over other forms of oscillator is that it employs resonance to vibrate at a precise natural resonant frequency or \"beat\" dependent only on its physical characteristics, and resists vibrating at other rates. The possible precision achievable by a harmonic oscillator is measured by a parameter called its Q, or quality factor, which increases (other things being equal) with its resonant frequency. This is why there has been a long-term trend toward higher frequency oscillators in clocks. Balance wheels and pendulums always include a means of adjusting the rate of the timepiece. Quartz timepieces sometimes include a rate screw that adjusts a capacitor for that purpose. Atomic clocks are primary standards, and their rate cannot be adjusted.
#### Synchronized or slave clocks {#synchronized_or_slave_clocks}
Some clocks rely for their accuracy on an external oscillator; that is, they are automatically synchronized to a more accurate clock:
- Slave clocks, used in large institutions and schools from the 1860s to the 1970s, kept time with a pendulum, but were wired to a master clock in the building, and periodically received a signal to synchronize them with the master, often on the hour. Later versions without pendulums were triggered by a pulse from the master clock and certain sequences used to force rapid synchronization following a power failure.
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- Synchronous electric clocks do not have an internal oscillator, but count cycles of the 50 or 60 Hz oscillation of the AC power line, which is synchronized by the utility to a precision oscillator. The counting may be done electronically, usually in clocks with digital displays, or, in analog clocks, the AC may drive a synchronous motor which rotates an exact fraction of a revolution for every cycle of the line voltage, and drives the gear train. Although changes in the grid line frequency due to load variations may cause the clock to temporarily gain or lose several seconds during the course of a day, the total number of cycles per 24 hours is maintained extremely accurately by the utility company, so that the clock keeps time accurately over long periods.
- Computer real-time clocks keep time with a quartz crystal, but can be periodically (usually weekly) synchronized over the Internet to atomic clocks (UTC), using the Network Time Protocol (NTP).
- Radio clocks keep time with a quartz crystal, but are periodically synchronized to time signals transmitted from dedicated standard time radio stations or satellite navigation signals, which are set by atomic clocks.
### Controller
This has the dual function of keeping the oscillator running by giving it \'pushes\' to replace the energy lost to friction, and converting its vibrations into a series of pulses that serve to measure the time.
- In mechanical clocks, this is the escapement, which gives precise pushes to the swinging pendulum or balance wheel, and releases one gear tooth of the *escape wheel* at each swing, allowing all the clock\'s wheels to move forward a fixed amount with each swing.
- In electronic clocks this is an electronic oscillator circuit that gives the vibrating quartz crystal or tuning fork tiny \'pushes\', and generates a series of electrical pulses, one for each vibration of the crystal, which is called the clock signal.
- In atomic clocks the controller is an evacuated microwave cavity attached to a microwave oscillator controlled by a microprocessor. A thin gas of caesium atoms is released into the cavity where they are exposed to microwaves. A laser measures how many atoms have absorbed the microwaves, and an electronic feedback control system called a phase-locked loop tunes the microwave oscillator until it is at the frequency that causes the atoms to vibrate and absorb the microwaves. Then the microwave signal is divided by digital counters to become the clock signal.
In mechanical clocks, the low Q of the balance wheel or pendulum oscillator made them very sensitive to the disturbing effect of the impulses of the escapement, so the escapement had a great effect on the accuracy of the clock, and many escapement designs were tried. The higher Q of resonators in electronic clocks makes them relatively insensitive to the disturbing effects of the drive power, so the driving oscillator circuit is a much less critical component.
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# Clock
## Operation
### Counter chain {#counter_chain}
This counts the pulses and adds them up to get traditional time units of seconds, minutes, hours, etc. It usually has a provision for *setting* the clock by manually entering the correct time into the counter.
- In mechanical clocks this is done mechanically by a gear train, known as the wheel train. The gear train scales the rotation speed to give a shaft rotating once per hour to which the minute hand of the clock is attached, a shaft rotating once per 12 hours to which the hour hand of the clock is attached, and in some clocks a shaft rotating once per minute, to which the second hand is attached. The gear train also has a second function; to transmit mechanical power from the power source to run the oscillator. There is a friction coupling called the \'cannon pinion\' between the gears driving the hands and the rest of the clock, allowing the hands to be turned to set the time.
- In digital clocks a series of integrated circuit counters or dividers add the pulses up digitally, using binary logic. Often pushbuttons on the case allow the hour and minute counters to be incremented and decremented to set the time.
### Indicator
This displays the count of seconds, minutes, hours, etc. in a human readable form.
- The earliest mechanical clocks in the 13th century did not have a visual indicator and signalled the time audibly by striking bells. Many clocks to this day are striking clocks which strike the hour.
- Analog clocks display time with an analog clock face, which consists of a dial with the numbers 1 through 12 or 24, the hours in the day, around the outside. The hours are indicated with an hour hand, which makes one or two revolutions in a day, while the minutes are indicated by a minute hand, which makes one revolution per hour. In mechanical clocks a gear train drives the hands; in electronic clocks the circuit produces pulses every second which drive a stepper motor and gear train, which move the hands.
- Digital clocks display the time in periodically changing digits on a digital display. A common misconception is that a digital clock is more accurate than an analog wall clock, but the indicator type is separate and apart from the accuracy of the timing source.
- Talking clocks and the speaking clock services provided by telephone companies speak the time audibly, using either recorded or digitally synthesized voices.
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# Clock
## Types
Clocks can be classified by the type of time display, as well as by the method of timekeeping.
### Time display methods {#time_display_methods}
#### Analog
Analog clocks usually use a clock face which indicates time using rotating pointers called \"hands\" on a fixed numbered dial or dials. The standard clock face, known universally throughout the world, has a short \"hour hand\" which indicates the hour on a circular dial of 12 hours, making two revolutions per day, and a longer \"minute hand\" which indicates the minutes in the current hour on the same dial, which is also divided into 60 minutes. It may also have a \"second hand\" which indicates the seconds in the current minute. The only other widely used clock face today is the 24 hour analog dial, because of the use of 24 hour time in military organizations and timetables. Before the modern clock face was standardized during the Industrial Revolution, many other face designs were used throughout the years, including dials divided into 6, 8, 10, and 24 hours. During the French Revolution the French government tried to introduce a 10-hour clock, as part of their decimal-based metric system of measurement, but it did not achieve widespread use. An Italian 6 hour clock was developed in the 18th century, presumably to save power (a clock or watch striking 24 times uses more power).
Another type of analog clock is the sundial, which tracks the sun continuously, registering the time by the shadow position of its gnomon. Because the sun does not adjust to daylight saving time, users must add an hour during that time. Corrections must also be made for the equation of time, and for the difference between the longitudes of the sundial and of the central meridian of the time zone that is being used (i.e. 15 degrees east of the prime meridian for each hour that the time zone is ahead of GMT). Sundials use some or part of the 24 hour analog dial. There also exist clocks which use a digital display despite having an analog mechanism---these are commonly referred to as flip clocks. Alternative systems have been proposed. For example, the \"Twelv\" clock indicates the current hour using one of twelve colors, and indicates the minute by showing a proportion of a circular disk, similar to a moon phase.
#### Digital
Kanazawa Station Water Clock.jpg\|Digital clock displaying time by controlling valves on the fountain Digital-clock-radio-basic hf.jpg\|Simplistic digital clock radio Analog clock with digital display.png\|Diagram of a mechanical digital display of a flip clock Cifra 5 digital flip clock designed by Gino Valle (1957).jpg\|Cifra 5 digital flip clock (1957) SAMSUNG Galaxy S22 Ultra BLACK.jpg\|A digital clock on a Samsung Galaxy smartphone
Digital clocks display a numeric representation of time. Two numeric display formats are commonly used on digital clocks:
- the 24-hour notation with hours ranging 00--23;
- the 12-hour notation with AM/PM indicator, with hours indicated as 12AM, followed by 1AM--11AM, followed by 12PM, followed by 1PM--11PM (a notation mostly used in domestic environments).
Most digital clocks use electronic mechanisms and LCD, LED, or VFD displays; many other display technologies are used as well (cathode-ray tubes, nixie tubes, etc.). After a reset, battery change or power failure, these clocks without a backup battery or capacitor either start counting from 12:00, or stay at 12:00, often with blinking digits indicating that the time needs to be set. Some newer clocks will reset themselves based on radio or Internet time servers that are tuned to national atomic clocks. Since the introduction of digital clocks in the 1960s, there has been a notable decline in the use of analog clocks.
Some clocks, called \'flip clocks\', have digital displays that work mechanically. The digits are painted on sheets of material which are mounted like the pages of a book. Once a minute, a page is turned over to reveal the next digit. These displays are usually easier to read in brightly lit conditions than LCDs or LEDs. Also, they do not go back to 12:00 after a power interruption. Flip clocks generally do not have electronic mechanisms. Usually, they are driven by AC-synchronous motors.
#### Hybrid (analog-digital) {#hybrid_analog_digital}
Clocks with analog quadrants, with a digital component, usually minutes and hours displayed analogously and seconds displayed in digital mode.
#### Auditory
For convenience, distance, telephony or blindness, auditory clocks present the time as sounds. The sound is either spoken natural language, (e.g. \"The time is twelve thirty-five\"), or as auditory codes (e.g. number of sequential bell rings on the hour represents the number of the hour like the bell, Big Ben). Most telecommunication companies also provide a speaking clock service as well.
#### Word
Word clocks are clocks that display the time visually using sentences. E.g.: \"It\'s about three o\'clock.\" These clocks can be implemented in hardware or software.
#### Projection
Some clocks, usually digital ones, include an optical projector that shines a magnified image of the time display onto a screen or onto a surface such as an indoor ceiling or wall. The digits are large enough to be easily read, without using glasses, by persons with moderately imperfect vision, so the clocks are convenient for use in their bedrooms. Usually, the timekeeping circuitry has a battery as a backup source for an uninterrupted power supply to keep the clock on time, while the projection light only works when the unit is connected to an A.C. supply. Completely battery-powered portable versions resembling flashlights are also available.
#### Tactile
Auditory and projection clocks can be used by people who are blind or have limited vision. There are also clocks for the blind that have displays that can be read by using the sense of touch. Some of these are similar to normal analog displays, but are constructed so the hands can be felt without damaging them. Another type is essentially digital, and uses devices that use a code such as Braille to show the digits so that they can be felt with the fingertips.
#### Multi-display {#multi_display}
Some clocks have several displays driven by a single mechanism, and some others have several completely separate mechanisms in a single case. Clocks in public places often have several faces visible from different directions, so that the clock can be read from anywhere in the vicinity; all the faces show the same time. Other clocks show the current time in several time-zones. Watches that are intended to be carried by travellers often have two displays, one for the local time and the other for the time at home, which is useful for making pre-arranged phone calls. Some equation clocks have two displays, one showing mean time and the other solar time, as would be shown by a sundial. Some clocks have both analog and digital displays. Clocks with Braille displays usually also have conventional digits so they can be read by sighted people.
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# Clock
## Purposes
Clocks are in homes, offices and many other places; smaller ones (watches) are carried on the wrist or in a pocket; larger ones are in public places, e.g. a railway station or church. A small clock is often shown in a corner of computer displays, mobile phones and many MP3 players.
The primary purpose of a clock is to *display* the time. Clocks may also have the facility to make a loud alert signal at a specified time, typically to waken a sleeper at a preset time; they are referred to as *alarm clocks*. The alarm may start at a low volume and become louder, or have the facility to be switched off for a few minutes then resume. Alarm clocks with visible indicators are sometimes used to indicate to children too young to read the time that the time for sleep has finished; they are sometimes called *training clocks*.
A clock mechanism may be used to *control* a device according to time, e.g. a central heating system, a VCR, or a time bomb (see: digital counter). Such mechanisms are usually called timers. Clock mechanisms are also used to drive devices such as solar trackers and astronomical telescopes, which have to turn at accurately controlled speeds to counteract the rotation of the Earth.
Most digital computers depend on an internal signal at constant frequency to synchronize processing; this is referred to as a clock signal. (A few research projects are developing CPUs based on asynchronous circuits.) Some equipment, including computers, also maintains time and date for use as required; this is referred to as time-of-day clock, and is distinct from the system clock signal, although possibly based on counting its cycles.
### Time standards {#time_standards}
For some scientific work timing of the utmost accuracy is essential. It is also necessary to have a standard of the maximum accuracy against which working clocks can be calibrated. An ideal clock would give the time to unlimited accuracy, but this is not realisable. Many physical processes, in particular including some transitions between atomic energy levels, occur at exceedingly stable frequency; counting cycles of such a process can give a very accurate and consistent time---clocks which work this way are usually called atomic clocks. Such clocks are typically large, very expensive, require a controlled environment, and are far more accurate than required for most purposes; they are typically used in a standards laboratory.
### Navigation
Until advances in the late twentieth century, navigation depended on the ability to measure latitude and longitude. Latitude can be determined through celestial navigation; the measurement of longitude requires accurate knowledge of time. This need was a major motivation for the development of accurate mechanical clocks. John Harrison created the first highly accurate marine chronometer in the mid-18th century. The Noon gun in Cape Town still fires an accurate signal to allow ships to check their chronometers. Many buildings near major ports used to have (some still do) a large ball mounted on a tower or mast arranged to drop at a pre-determined time, for the same purpose. While satellite navigation systems such as GPS require unprecedentedly accurate knowledge of time, this is supplied by equipment on the satellites; vehicles no longer need timekeeping equipment.
### Sports and games {#sports_and_games}
Clocks can be used to measure varying periods of time in games and sports. Stopwatches can be used to time the performance of track athletes. Chess clocks are used to limit the board game players\' time to make a move. In various sports, *`{{Vanchor|Game clock|text=game clocks}}`{=mediawiki}* measure the duration the game or subdivisions of the game, while other clocks may be used for tracking different durations; these include play clocks, shot clocks, and pitch clocks.
## Culture
### Folklore and superstition {#folklore_and_superstition}
In the United Kingdom, clocks are associated with various beliefs, many involving death or bad luck. In legends, clocks have reportedly stopped of their own accord upon a nearby person\'s death, especially those of monarchs. The clock in the House of Lords supposedly stopped at \"nearly\" the hour of George III\'s death in 1820, the one at Balmoral Castle stopped during the hour of Queen Victoria\'s death, and similar legends are related about clocks associated with William IV and Elizabeth I. Many superstitions exist about clocks. One stopping before a person has died may foretell coming death. Similarly, if a clock strikes during a church hymn or a marriage ceremony, death or calamity is prefigured for the parishioners or a spouse, respectively. Death or ill events are foreshadowed if a clock strikes the wrong time. It may also be unlucky to have a clock face a fire or to speak while a clock is striking.
In Chinese culture, giving a clock (`{{zh|t=送鐘|s=送钟|first=t|p=sòng zhōng}}`{=mediawiki}) is often taboo, especially to the elderly, as it is a homophone of the act of attending another\'s funeral (`{{zh|t={{linktext|送終}}|s={{linktext|送终}}|first=t|p=sòngzhōng}}`{=mediawiki})
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# Charles Edward Jones
**Charles Edward** \"**Chuck**\" **Jones** (November 8, 1952 -- September 11, 2001) was a United States Air Force officer, an aeronautical engineer, computer programmer, and an astronaut in the USAF Manned Spaceflight Engineer Program. He was killed during the September 11 attacks, aboard American Airlines Flight 11.
## Life
Charles Edward Jones was born November 8, 1952, in Clinton, Indiana. He graduated from Wichita East High School in 1970, earned a Bachelor of Science degree in Astronautical Engineering from the United States Air Force Academy in 1974, and received a Master of Science degree in astronautics from Massachusetts Institute of Technology in 1980. He entered the USAF Manned Spaceflight Engineer Program in 1982, and was scheduled to fly on mission STS-71-B in December 1986, but the mission was canceled after the *Challenger* disaster in January 1986. He left the Manned Spaceflight Engineer program in 1987.
He later worked for Defense Intelligence Agency, Bolling Air Force Base in Washington, D.C., and was Systems Program Director for Intelligence and Information Systems, Hanscom Air Force Base, Massachusetts. Jones later was the manager of space programs for BAE Systems.
Jones was killed at the age of 48 in the attacks of September 11, 2001, aboard American Airlines Flight 11. Jones was flying that day on a routine business trip for BAE Systems, and had been living as a retired U.S. Air Force colonel in Bedford, Massachusetts, at the time of his death. He was survived by his wife Jeanette.
At the National 9/11 Memorial, Jones is memorialized at the North Pool, on Panel N-74.
thumb\|upright=1.0\|left\|Jones\' name is located on Panel N-74 of the National September 11 Memorial\'s North Pool, along with those of other passengers of Flight 11
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# Ceramic
A **ceramic** is any of the various hard, brittle, heat-resistant, and corrosion-resistant materials made by shaping and then firing an inorganic, nonmetallic material, such as clay, at a high temperature. Common examples are earthenware, porcelain, and brick.
The earliest ceramics made by humans were fired clay bricks used for building house walls and other structures. Other pottery objects such as pots, vessels, vases and figurines were made from clay, either by itself or mixed with other materials like silica, hardened by sintering in fire. Later, ceramics were glazed and fired to create smooth, colored surfaces, decreasing porosity through the use of glassy, amorphous ceramic coatings on top of the crystalline ceramic substrates. Ceramics now include domestic, industrial, and building products, as well as a wide range of materials developed for use in advanced ceramic engineering, such as semiconductors.
The word *ceramic* comes from the Ancient Greek word `{{wikt-lang|grc|κεραμικός}}`{=mediawiki} (`{{grc-transl|κεραμικός}}`{=mediawiki}), meaning \"of or for pottery\" (`{{etymology||''{{wikt-lang|grc|κέραμος}}'' ({{grc-transl|κέραμος}})|potter's clay, tile, pottery}}`{=mediawiki}). The earliest known mention of the root *ceram-* is the Mycenaean Greek `{{nowrap|{{Transliteration|gmy|ke-ra-me-we}}}}`{=mediawiki}, workers of ceramic, written in Linear B syllabic script. The word *ceramic* can be used as an adjective to describe a material, product, or process, or it may be used as a noun, either singular or, more commonly, as the plural noun *ceramics*.
## Materials
thumb\|upright=1.35\|right\|Silicon nitride rocket thruster. Left: Mounted in test stand. Right: Being tested with H~2~/O~2~ propellants. Ceramic material is an inorganic, metallic oxide, nitride, or carbide material. Some elements, such as carbon or silicon, may be considered ceramics. Ceramic materials are brittle, hard, strong in compression, and weak in shearing and tension. They withstand the chemical erosion that occurs in other materials subjected to acidic or caustic environments. Ceramics generally can withstand very high temperatures, ranging from 1,000 °C to 1,600 °C (1,800 °F to 3,000 °F).
thumb\|upright=1.2\|A low magnification SEM micrograph of an advanced ceramic material. The properties of ceramics make fracturing an important inspection method. The crystallinity of ceramic materials varies widely. Most often, fired ceramics are either vitrified or semi-vitrified, as is the case with earthenware, stoneware, and porcelain. Varying crystallinity and electron composition in the ionic and covalent bonds cause most ceramic materials to be good thermal and electrical insulators (researched in ceramic engineering). With such a large range of possible options for the composition/structure of a ceramic (nearly all of the elements, nearly all types of bonding, and all levels of crystallinity), the breadth of the subject is vast, and identifiable attributes (hardness, toughness, electrical conductivity) are difficult to specify for the group as a whole. General properties such as high melting temperature, high hardness, poor conductivity, high moduli of elasticity, chemical resistance, and low ductility are the norm, with known exceptions to each of these rules (piezoelectric ceramics, low glass transition temperature ceramics, superconductive ceramics).
Composites such as fiberglass and carbon fiber, while containing ceramic materials, are not considered to be part of the ceramic family.
Highly oriented crystalline ceramic materials are not amenable to a great range of processing. Methods for dealing with them tend to fall into one of two categories: either making the ceramic in the desired shape by reaction *in situ* or \"forming\" powders into the desired shape and then sintering to form a solid body. Ceramic forming techniques include shaping by hand (sometimes including a rotation process called \"throwing\"), slip casting, tape casting (used for making very thin ceramic capacitors), injection molding, dry pressing, and other variations.
Many ceramics experts do not consider materials with an amorphous (noncrystalline) character (i.e., glass) to be ceramics, even though glassmaking involves several steps of the ceramic process and its mechanical properties are similar to those of ceramic materials. However, heat treatments can convert glass into a semi-crystalline material known as glass-ceramic.
Traditional ceramic raw materials include clay minerals such as kaolinite, whereas more recent materials include aluminium oxide, more commonly known as alumina. Modern ceramic materials, which are classified as advanced ceramics, include silicon carbide and tungsten carbide. Both are valued for their abrasion resistance and are therefore used in applications such as the wear plates of crushing equipment in mining operations. Advanced ceramics are also used in the medical, electrical, electronics, and armor industries.
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# Ceramic
## History
thumb\|upright=0.55 \|Earliest known ceramics are the Gravettian figurines that date to 29,000--25,000 BC. Human beings appear to have been making their own ceramics for at least 26,000 years, subjecting clay and silica to intense heat to fuse and form ceramic materials. The earliest found so far were in southern central Europe and were sculpted figures, not dishes. The earliest known pottery was made by mixing animal products with clay and firing it at up to 800 °C. While pottery fragments have been found up to 19,000 years old, it was not until about 10,000 years later that regular pottery became common. An early people that spread across much of Europe is named after its use of pottery: the Corded Ware culture. These early Indo-European peoples decorated their pottery by wrapping it with rope while it was still wet. When the ceramics were fired, the rope burned off but left a decorative pattern of complex grooves on the surface.
The invention of the wheel eventually led to the production of smoother, more even pottery using the wheel-forming (throwing) technique, like the pottery wheel. Early ceramics were porous, absorbing water easily. It became useful for more items with the discovery of glazing techniques, which involved coating pottery with silicon, bone ash, or other materials that could melt and reform into a glassy surface, making a vessel less pervious to water.
### Archaeology
Ceramic artifacts have an important role in archaeology for understanding the culture, technology, and behavior of peoples of the past. They are among the most common artifacts to be found at an archaeological site, generally in the form of small fragments of broken pottery called sherds. The processing of collected sherds can be consistent with two main types of analysis: technical and traditional.
The traditional analysis involves sorting ceramic artifacts, sherds, and larger fragments into specific types based on style, composition, manufacturing, and morphology. By creating these typologies, it is possible to distinguish between different cultural styles, the purpose of the ceramic, and the technological state of the people, among other conclusions. Besides, by looking at stylistic changes in ceramics over time, it is possible to separate (seriate) the ceramics into distinct diagnostic groups (assemblages). A comparison of ceramic artifacts with known dated assemblages allows for a chronological assignment of these pieces.
The technical approach to ceramic analysis involves a finer examination of the composition of ceramic artifacts and sherds to determine the source of the material and, through this, the possible manufacturing site. Key criteria are the composition of the clay and the temper used in the manufacture of the article under study: the temper is a material added to the clay during the initial production stage and is used to aid the subsequent drying process. Types of temper include shell pieces, granite fragments, and ground sherd pieces called \'grog\'. Temper is usually identified by microscopic examination of the tempered material. Clay identification is determined by a process of refiring the ceramic and assigning a color to it using Munsell Soil Color notation. By estimating both the clay and temper compositions and locating a region where both are known to occur, an assignment of the material source can be made. Based on the source assignment of the artifact, further investigations can be made into the site of manufacture.
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# Ceramic
## Properties
The physical properties of any ceramic substance are a direct result of its crystalline structure and chemical composition. Solid-state chemistry reveals the fundamental connection between microstructure and properties, such as localized density variations, grain size distribution, type of porosity, and second-phase content, which can all be correlated with ceramic properties such as mechanical strength σ by the Hall-Petch equation, hardness, toughness, dielectric constant, and the optical properties exhibited by transparent materials.
Ceramography is the art and science of preparation, examination, and evaluation of ceramic microstructures. Evaluation and characterization of ceramic microstructures are often implemented on similar spatial scales to that used commonly in the emerging field of nanotechnology: from nanometers to tens of micrometers (µm). This is typically somewhere between the minimum wavelength of visible light and the resolution limit of the naked eye.
The microstructure includes most grains, secondary phases, grain boundaries, pores, micro-cracks, structural defects, and hardness micro indentions. Most bulk mechanical, optical, thermal, electrical, and magnetic properties are significantly affected by the observed microstructure. The fabrication method and process conditions are generally indicated by the microstructure. The root cause of many ceramic failures is evident in the cleaved and polished microstructure. Physical properties which constitute the field of materials science and engineering include the following:
### Mechanical properties {#mechanical_properties}
Mechanical properties are important in structural and building materials as well as textile fabrics. In modern materials science, fracture mechanics is an important tool in improving the mechanical performance of materials and components. It applies the physics of stress and strain, in particular the theories of elasticity and plasticity, to the microscopic crystallographic defects found in real materials in order to predict the macroscopic mechanical failure of bodies. Fractography is widely used with fracture mechanics to understand the causes of failures and also verify the theoretical failure predictions with real-life failures.
Ceramic materials are usually ionic or covalent bonded materials. A material held together by either type of bond will tend to fracture before any plastic deformation takes place, which results in poor toughness and brittle behavior in these materials. Additionally, because these materials tend to be porous, pores and other microscopic imperfections act as stress concentrators, decreasing the toughness further, and reducing the tensile strength. These combine to give catastrophic failures, as opposed to the more ductile failure modes of metals.
These materials do show plastic deformation. However, because of the rigid structure of crystalline material, there are very few available slip systems for dislocations to move, and so they deform very slowly.
To overcome the brittle behavior, ceramic material development has introduced the class of ceramic matrix composite materials, in which ceramic fibers are embedded and with specific coatings are forming fiber bridges across any crack. This mechanism substantially increases the fracture toughness of such ceramics. Ceramic disc brakes are an example of using a ceramic matrix composite material manufactured with a specific process.
Scientists are working on developing ceramic materials that can withstand significant deformation without breaking. A first such material that can deform in room temperature was found in 2024.
#### Toughening Mechanisms {#toughening_mechanisms}
Many strategies are employed to improve the toughness of ceramics to prevent fracture. This includes crack deflection, microcrack toughening, crack bridging, incorporation of ductile particles, and transformation toughening.
Crack deflection is a toughening mechanism that involves deflecting cracks away from more rapid crack propagation paths, preventing catastrophic sudden failure. Cracks may be deflected using microstructures such as whiskers, as in the use of silicon carbide whiskers to reinforce molybdenum disilicide ceramic material in a 1987 paper. Crack deflecting second phases may also take the form of platelets, particles, or fibers.
Microcrack toughening involves nucleation (creation) of microcracks near a macroscopic crack tip where the crack propagates, which lowers the stress experienced by the tip and therefore the urgency of crack propagation. To improve toughness, second phase particles can be incorporated into ceramic such that they are subject to microcracking, which relieves stress to prevent fracture.
Crack bridging occurs when a strong discontinuous reinforcing phase applies a force behind the propagating tip of the crack that discourages further cracking. These second phase bridges essentially pin the crack to discourage its extension. Crack bridging can be used to improve toughness via the incorporation of second phase whiskers in the ceramic, as well as other shapes, to bridge cracks.
Ductile particle ceramic matrix composites are composed of ductile particles such as metals distributed in a ceramic matrix. These particles boost toughness by deforming plastically to absorb energy, and by bridging advancing cracks. To be most effective, the particles should be isolated from each other. The most studied iterations of these composites consist of an alumina matrix, and nickel, iron, molybdenum, copper, or silver metal particles.
Transformation toughening occurs when a material undergoes stress-induced phase transformation. Some ceramics are capable of undergoing stress-induced martensitic transformation, which involves an energy barrier that must be overcome by absorbing energy. Martensitic transformations are diffusionless shear transformations involving the transition between an \"austenite\" or \"parent\" phase that is stable at higher temperatures and a \"martensitic\" phase that is stable at lower temperatures. Because the transformation absorbs energy, stress-induced martensitic transformations can hinder crack progression and increases toughness. A key example of this phenomenon is zirconia, whose martensitic transformation involves a crystal structure transformation from a tetragonal crystal structure (the austenite phase) to a monoclinic structure. The volume increase associated with transformation from tetragonal to monoclinic also relieves tensile stress at the crack, tip, further discouraging cracking and increasing toughness. When zirconia particles in a ceramic matrix undergo transformation during fabrication due to cooling , the stress fields around the particles lead to nucleation and extension of microcracks, which can also improve toughness of the material. These stress fields, as well as the particles themselves, can also contribute to crack deflection.
#### Ice-templating for enhanced mechanical properties {#ice_templating_for_enhanced_mechanical_properties}
If a ceramic is subjected to substantial mechanical loading, it can undergo a process called ice-templating, which allows some control of the microstructure of the ceramic product and therefore some control of the mechanical properties. Ceramic engineers use this technique to tune the mechanical properties to their desired application. Specifically, the strength is increased when this technique is employed. Ice templating allows the creation of macroscopic pores in a unidirectional arrangement. The applications of this oxide strengthening technique are important for solid oxide fuel cells and water filtration devices.
To process a sample through ice templating, an aqueous colloidal suspension is prepared to contain the dissolved ceramic powder evenly dispersed throughout the colloid,`{{clarify|reason=is the powder in suspension or actually dissolved?|date=December 2019}}`{=mediawiki} for example yttria-stabilized zirconia (YSZ). The solution is then cooled from the bottom to the top on a platform that allows for unidirectional cooling. This forces ice crystals to grow in compliance with the unidirectional cooling, and these ice crystals force the dissolved YSZ particles to the solidification front`{{clarify|date=June 2023}}`{=mediawiki} of the solid-liquid interphase boundary, resulting in pure ice crystals lined up unidirectionally alongside concentrated pockets of colloidal particles. The sample is then heated and at the same the pressure is reduced enough to force the ice crystals to sublime and the YSZ pockets begin to anneal together to form macroscopically aligned ceramic microstructures. The sample is then further sintered to complete the evaporation of the residual water and the final consolidation of the ceramic microstructure.
During ice-templating, a few variables can be controlled to influence the pore size and morphology of the microstructure. These important variables are the initial solids loading of the colloid, the cooling rate, the sintering temperature and duration, and the use of certain additives which can influence the microstructural morphology during the process. A good understanding of these parameters is essential to understanding the relationships between processing, microstructure, and mechanical properties of anisotropically porous materials.
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# Ceramic
## Properties
### Electrical properties {#electrical_properties}
#### Semiconductors
Some ceramics are semiconductors. Most of these are transition metal oxides that are II-VI semiconductors, such as zinc oxide. While there are prospects of mass-producing blue light-emitting diodes (LED) from zinc oxide, ceramicists are most interested in the electrical properties that show grain boundary effects. One of the most widely used of these is the varistor. These are devices that exhibit the property that resistance drops sharply at a certain threshold voltage. Once the voltage across the device reaches the threshold, there is a breakdown of the electrical structure`{{clarification needed|date=November 2021}}`{=mediawiki} in the vicinity of the grain boundaries, which results in its electrical resistance dropping from several megohms down to a few hundred ohms. The major advantage of these is that they can dissipate a lot of energy, and they self-reset; after the voltage across the device drops below the threshold, its resistance returns to being high. This makes them ideal for surge-protection applications; as there is control over the threshold voltage and energy tolerance, they find use in all sorts of applications. The best demonstration of their ability can be found in electrical substations, where they are employed to protect the infrastructure from lightning strikes. They have rapid response, are low maintenance, and do not appreciably degrade from use, making them virtually ideal devices for this application. Semiconducting ceramics are also employed as gas sensors. When various gases are passed over a polycrystalline ceramic, its electrical resistance changes. With tuning to the possible gas mixtures, very inexpensive devices can be produced.
#### Superconductivity
Under some conditions, such as extremely low temperatures, some ceramics exhibit high-temperature superconductivity (in superconductivity, \"high temperature\" means above 30 K). The reason for this is not understood, but there are two major families of superconducting ceramics.
#### Ferroelectricity and supersets {#ferroelectricity_and_supersets}
Piezoelectricity, a link between electrical and mechanical response, is exhibited by a large number of ceramic materials, including the quartz used to measure time in watches and other electronics. Such devices use both properties of piezoelectrics, using electricity to produce a mechanical motion (powering the device) and then using this mechanical motion to produce electricity (generating a signal). The unit of time measured is the natural interval required for electricity to be converted into mechanical energy and back again.
The piezoelectric effect is generally stronger in materials that also exhibit pyroelectricity, and all pyroelectric materials are also piezoelectric. These materials can be used to inter-convert between thermal, mechanical, or electrical energy; for instance, after synthesis in a furnace, a pyroelectric crystal allowed to cool under no applied stress generally builds up a static charge of thousands of volts. Such materials are used in motion sensors, where the tiny rise in temperature from a warm body entering the room is enough to produce a measurable voltage in the crystal.
In turn, pyroelectricity is seen most strongly in materials that also display the ferroelectric effect, in which a stable electric dipole can be oriented or reversed by applying an electrostatic field. Pyroelectricity is also a necessary consequence of ferroelectricity. This can be used to store information in ferroelectric capacitors, elements of ferroelectric RAM.
The most common such materials are lead zirconate titanate and barium titanate. Aside from the uses mentioned above, their strong piezoelectric response is exploited in the design of high-frequency loudspeakers, transducers for sonar, and actuators for atomic force and scanning tunneling microscopes.
#### Positive thermal coefficient {#positive_thermal_coefficient}
Temperature increases can cause grain boundaries to suddenly become insulating in some semiconducting ceramic materials, mostly mixtures of heavy metal titanates. The critical transition temperature can be adjusted over a wide range by variations in chemistry. In such materials, current will pass through the material until joule heating brings it to the transition temperature, at which point the circuit will be broken and current flow will cease. Such ceramics are used as self-controlled heating elements in, for example, the rear-window defrost circuits of automobiles.
At the transition temperature, the material\'s dielectric response becomes theoretically infinite. While a lack of temperature control would rule out any practical use of the material near its critical temperature, the dielectric effect remains exceptionally strong even at much higher temperatures. Titanates with critical temperatures far below room temperature have become synonymous with \"ceramic\" in the context of ceramic capacitors for just this reason.
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# Ceramic
## Properties
### Optical properties {#optical_properties}
Optically transparent materials focus on the response of a material to incoming light waves of a range of wavelengths. Frequency selective optical filters can be utilized to alter or enhance the brightness and contrast of a digital image. Guided lightwave transmission via frequency selective waveguides involves the emerging field of fiber optics and the ability of certain glassy compositions as a transmission medium for a range of frequencies simultaneously (multi-mode optical fiber) with little or no interference between competing wavelengths or frequencies. This resonant mode of energy and data transmission via electromagnetic (light) wave propagation, though low powered, is virtually lossless. Optical waveguides are used as components in Integrated optical circuits (e.g. light-emitting diodes, LEDs) or as the transmission medium in local and long haul optical communication systems. Also of value to the emerging materials scientist is the sensitivity of materials to radiation in the thermal infrared (IR) portion of the electromagnetic spectrum. This heat-seeking ability is responsible for such diverse optical phenomena as night-vision and IR luminescence.
Thus, there is an increasing need in the military sector for high-strength, robust materials which have the capability to transmit light (electromagnetic waves) in the visible (0.4 -- 0.7 micrometers) and mid-infrared (1 -- 5 micrometers) regions of the spectrum. These materials are needed for applications requiring transparent armor, including next-generation high-speed missiles and pods, as well as protection against improvised explosive devices (IED).
In the 1960s, scientists at General Electric (GE) discovered that under the right manufacturing conditions, some ceramics, especially aluminium oxide (alumina), could be made translucent. These translucent materials were transparent enough to be used for containing the electrical plasma generated in high-pressure sodium street lamps. During the past two decades, additional types of transparent ceramics have been developed for applications such as nose cones for heat-seeking missiles, windows for fighter aircraft, and scintillation counters for computed tomography scanners. Other ceramic materials, generally requiring greater purity in their make-up than those above, include forms of several chemical compounds, including:
1. Barium titanate**:** (often mixed with strontium titanate) displays ferroelectricity, meaning that its mechanical, electrical, and thermal responses are coupled to one another and also history-dependent. It is widely used in electromechanical transducers, ceramic capacitors, and data storage elements. Grain boundary conditions can create PTC effects in heating elements.
2. Sialon (silicon aluminium oxynitride) has high strength; resistance to thermal shock, chemical and wear resistance, and low density. These ceramics are used in non-ferrous molten metal handling, weld pins, and the chemical industry.
3. Silicon carbide (SiC) is used as a susceptor in microwave furnaces, a commonly used abrasive, and as a refractory material.
4. Silicon nitride (Si~3~N~4~) is used as an abrasive powder.
5. Steatite (magnesium silicates) is used as an electrical insulator.
6. Titanium carbide Used in space shuttle re-entry shields and scratchproof watches.
7. Uranium oxide (UO~2~), used as fuel in nuclear reactors.
8. Yttrium barium copper oxide (YBa~2~Cu~3~O~7−x~), a high-temperature superconductor.
9. Zinc oxide (ZnO), which is a semiconductor, and used in the construction of varistors.
10. Zirconium dioxide (zirconia), which in pure form undergoes many phase changes between room temperature and practical sintering temperatures, can be chemically \"stabilized\" in several different forms. Its high oxygen ion conductivity recommends it for use in fuel cells and automotive oxygen sensors. In another variant, metastable structures can impart transformation toughening for mechanical applications; most ceramic knife blades are made of this material. Partially stabilised zirconia (PSZ) is much less brittle than other ceramics and is used for metal forming tools, valves and liners, abrasive slurries, kitchen knives and bearings subject to severe abrasion.
## Products
### By usage {#by_usage}
For convenience, ceramic products are usually divided into four main types; these are shown below with some examples:
1. Structural, including bricks, pipes, floor and roof tiles, vitrified tile
2. Refractories, such as kiln linings, gas fire radiants, steel and glass making crucibles
3. Whitewares, including tableware, cookware, wall tiles, pottery products and sanitary ware
4. Technical, also known as engineering, advanced, special, and fine ceramics. Such items include:
- gas burner nozzles
- ballistic protection, vehicle armor
- nuclear fuel uranium oxide pellets
- biomedical implants
- coatings of jet engine turbine blades
- ceramic matrix composite gas turbine parts
- reinforced carbon--carbon ceramic disc brakes
- missile nose cones
- bearings
- thermal insulation tiles used on the Space Shuttle orbiter
### Ceramics made with clay {#ceramics_made_with_clay}
Frequently, the raw materials of modern ceramics do not include clays. Those that do have been classified as:
1. Earthenware, fired at lower temperatures than other types
2. Stoneware, vitreous or semi-vitreous
3. Porcelain, which contains a high content of kaolin
4. Bone china
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# Ceramic
## Properties
### Classification
Ceramics can also be classified into three distinct material categories:
1. Oxides**:** alumina, beryllia, ceria, zirconia
2. Non-oxides**:** carbide, boride, nitride, silicide
3. Composite materials**:** particulate reinforced, fiber reinforced, combinations of oxides and non-oxides.
Each one of these classes can be developed into unique material properties.
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# Ceramic
## Applications
1. Knife blades**:** the blade of a ceramic knife will stay sharp for much longer than that of a steel knife, although it is more brittle and susceptible to breakage.
2. Carbon-ceramic brake disks for vehicles: highly resistant to brake fade at high temperatures.
3. Advanced composite ceramic and metal matrices have been designed for most modern armoured fighting vehicles because they offer superior penetrating resistance against shaped charge (HEAT rounds) and kinetic energy penetrators.
4. Ceramics such as alumina and boron carbide have been used as plates in ballistic armored vests to repel high-velocity rifle fire. Such plates are known commonly as small arms protective inserts, or SAPIs. Similar low-weight material is used to protect the cockpits of some military aircraft.
5. Ceramic ball bearings can be used in place of steel. Their greater hardness results in lower susceptibility to wear. Ceramic bearings typically last triple the lifetime of steel bearings. They deform less than steel under load, resulting in less contact with the bearing retainer walls and lower friction. In very high-speed applications, heat from friction causes more problems for metal bearings than ceramic bearings. Ceramics are chemically resistant to corrosion and are preferred for environments where steel bearings would rust. In some applications their electricity-insulating properties are advantageous. Drawbacks to ceramic bearings include significantly higher cost, susceptibility to damage under shock loads, and the potential to wear steel parts due to ceramics\' greater hardness.
6. In the early 1980s Toyota researched production of an adiabatic engine using ceramic components in the hot gas area. The use of ceramics would have allowed temperatures exceeding 1650 °C. Advantages would include lighter materials and a smaller cooling system (or no cooling system at all), leading to major weight reduction. The expected increase of fuel efficiency (due to higher operating temperatures, demonstrated in Carnot\'s theorem) could not be verified experimentally. It was found that heat transfer on the hot ceramic cylinder wall was greater than the heat transfer to a cooler metal wall. This is because the cooler gas film on a metal surface acts as a thermal insulator. Thus, despite the desirable properties of ceramics, prohibitive production costs and limited advantages have prevented widespread ceramic engine component adoption. In addition, small imperfections in ceramic material along with low fracture toughness can lead to cracking and potentially dangerous equipment failure. Such engines are possible experimentally, but mass production is not feasible with current technology.
7. Experiments with ceramic parts for gas turbine engines are being conducted. Currently, even blades made of advanced metal alloys used in the engines\' hot section require cooling and careful monitoring of operating temperatures. Turbine engines made with ceramics could operate more efficiently, providing for greater range and payload.
8. Recent advances have been made in ceramics which include bioceramics such as dental implants and synthetic bones. Hydroxyapatite, the major mineral component of bone, has been made synthetically from several biological and chemical components and can be formed into ceramic materials. Orthopedic implants coated with these materials bond readily to bone and other tissues in the body without rejection or inflammatory reaction. They are of great interest for gene delivery and tissue engineering scaffolding. Most hydroxyapatite ceramics are quite porous and lack mechanical strength and are therefore used solely to coat metal orthopedic devices to aid in forming a bond to bone or as bone fillers. They are also used as fillers for orthopedic plastic screws to aid in reducing inflammation and increase the absorption of these plastic materials. Work is being done to make strong, fully dense nanocrystalline hydroxyapatite ceramic materials for orthopedic weight bearing devices, replacing foreign metal and plastic orthopedic materials with a synthetic but naturally occurring bone mineral. Ultimately, these ceramic materials may be used as bone replacement, or with the incorporation of protein collagens, the manufacture of synthetic bones.
9. Applications for actinide-containing ceramic materials include nuclear fuels for burning excess plutonium (Pu), or a chemically inert source of alpha radiation in power supplies for uncrewed space vehicles or microelectronic devices. Use and disposal of radioactive actinides require immobilization in a durable host material. Long half-life radionuclides such as actinide are immobilized using chemically durable crystalline materials based on polycrystalline ceramics and large single crystals.
10. High-tech ceramics are used for producing watch cases. The material is valued by watchmakers for its light weight, scratch resistance, durability, and smooth touch. IWC is one of the brands that pioneered the use of ceramic in watchmaking.
11. Ceramics are used in the design of mobile phone bodies due to their high hardness, resistance to scratches, and ability to dissipate heat. Ceramic\'s thermal management properties help in maintaining optimal device temperatures during heavy use enhancing performance. Additionally, ceramic materials can support wireless charging and offer better signal transmission compared to metals, which can interfere with antennas. Companies like Apple and Samsung have incorporated ceramic in their devices.
12. Ceramics made of silicon carbide are used in pump and valve components because of their corrosion resistance characteristics. It is also used in nuclear reactors as fuel cladding materials due to their ability to withstand radiation and thermal stress. Other uses of Silicon carbide ceramics include paper manufacturing, ballistics, chemical production, and as pipe system components
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# Canon law
**Canon law** (from *κανών*, *kanon*, a \'straight measuring rod, ruler\') is a set of ordinances and regulations made by ecclesiastical authority (church leadership) for the government of a Christian organization or church and its members.
Canon law includes the internal ecclesiastical law, or operational policy, governing the Catholic Church (both the Latin Church and the Eastern Catholic Churches), the Eastern Orthodox and Oriental Orthodox churches, and the individual national churches within the Anglican Communion. The way that such church law is legislated, interpreted and at times adjudicated varies widely among these four bodies of churches. In all three traditions, a canon was originally a rule adopted by a church council; these canons formed the foundation of canon law.
## Etymology
Greek *kanon* / *κανών*, Arabic *\[\[qaanoon\]\]* / *قانون*, Hebrew *kaneh* / *קָנֶה*, \'straight\'; a rule, code, standard, or measure; the root meaning in all these languages is \'reed\'; see also the Romance-language ancestors of the English word *cane*.
In the fourth century, the First Council of Nicaea (325) calls canons the disciplinary measures of the church: the term canon, κανὠν, means in Greek, a rule. There is a very early distinction between the rules enacted by the church and the legislative measures taken by the state called *leges*, Latin for laws.
## Apostolic Canons {#apostolic_canons}
The *Apostolic Canons* or *Ecclesiastical Canons of the Same Holy Apostles* is a collection of ancient ecclesiastical decrees (eighty-five in the Eastern, fifty in the Western Church) concerning the government and discipline of the Early Christian Church, incorporated with the Apostolic Constitutions which are part of the Ante-Nicene Fathers.
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# Canon law
## Catholic Church {#catholic_church}
In the Catholic Church, canon law is the system of laws and legal principles made and enforced by the church\'s hierarchical authorities to regulate its external organization and government and to order and direct the activities of Catholics toward the mission of the church. It was the first modern Western legal system and is the oldest continuously functioning legal system in the West.
In the Latin Church, positive ecclesiastical laws, based directly or indirectly upon immutable divine law or natural law, derive formal authority in the case of universal laws from the supreme legislator (i.e., the Supreme Pontiff), who possesses the totality of legislative, executive, and judicial power in his person, while particular laws derive formal authority from a legislator inferior to the supreme legislator. The actual subject material of the canons is not just doctrinal or moral in nature, but all-encompassing of the human condition, and therefore extending beyond what is taken as revealed truth.
The Catholic Church also includes the main five rites (groups) of churches which are in full union with the Holy See and the Latin Church:
1. Alexandrian Rite Churches which include the Coptic Catholic Church, Eritrean Catholic Church, and Ethiopian Catholic Church.
2. West Syriac Rite which includes the Maronite Church, Syriac Catholic Church and the Syro-Malankara Catholic Church.
3. Armenian Rite Church which includes the Armenian Catholic Church.
4. Byzantine Rite Churches which include the Albanian Greek Catholic Church, Belarusian Greek Catholic Church, Bulgarian Greek Catholic Church, Greek Catholic Church of Croatia and Serbia, Greek Byzantine Catholic Church, Hungarian Greek Catholic Church, Italo-Albanian Catholic Church, Macedonian Greek Catholic Church, Melkite Greek Catholic Church, Romanian Greek Catholic Church, Russian Greek Catholic Church, Ruthenian Greek Catholic Church, Slovak Greek Catholic Church and Ukrainian Greek Catholic Church.
5. East Syriac Rite Churches which includes the Chaldean Catholic Church and Syro-Malabar Church.
All of these church groups are in full communion with the Supreme Pontiff and are subject to the *Code of Canons of the Eastern Churches*.
### History, sources of law, and codifications {#history_sources_of_law_and_codifications}
thumb\|left\|upright=1.35\|Image of pages from the *Decretum* of Burchard of Worms, an 11th-century book of canon law *Main article: Legal history of the Catholic Church, Philosophy, theology, and fundamental theory of Catholic canon law*
The Catholic Church has what is claimed to be the oldest continuously functioning internal legal system in Western Europe, much later than Roman law but predating the evolution of modern European civil law traditions.
The history of Latin canon law can be divided into four periods: the *jus antiquum*, the *jus novum*, the *jus novissimum* and the *Code of Canon Law*. In relation to the Code, history can be divided into the *jus vetus* (all law before the Code) and the *jus novum* (the law of the Code, or *jus codicis*).
The canon law of the Eastern Catholic Churches, which had developed some different disciplines and practices, underwent its own process of codification, resulting in the Code of Canons of the Eastern Churches promulgated in 1990 by Pope John Paul II.
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# Canon law
## Catholic Church {#catholic_church}
### Catholic canon law as legal system {#catholic_canon_law_as_legal_system}
Roman Catholic canon law is a fully developed legal system, with all the necessary elements: courts, lawyers, judges, a fully articulated legal code, principles of legal interpretation, and coercive penalties, though it lacks civilly-binding force in most secular jurisdictions. One example where conflict between secular and canon law occurred was in the English legal system, as well as systems, such as the U.S., that derived from it. Here criminals could apply for the benefit of clergy. Being in holy orders, or fraudulently claiming to be, meant that criminals could opt to be tried by ecclesiastical rather than secular courts. The ecclesiastical courts were generally more lenient. Under the Tudors, the scope of clerical benefit was steadily reduced by Henry VII, Henry VIII, and Elizabeth I. The papacy disputed secular authority over priests\' criminal offenses. The benefit of clergy was systematically removed from English legal systems over the next 200 years, although it still occurred in South Carolina in 1855. In English Law, the use of this mechanism, which by that point was a legal fiction used for first offenders, was abolished by the Criminal Law Act 1827.
The academic degrees in Catholic canon law are the J.C.B. (*Juris Canonici Baccalaureatus*, Bachelor of Canon Law, normally taken as a graduate degree), J.C.L. (*Juris Canonici Licentiatus*, Licentiate of Canon Law) and the J.C.D. (*Juris Canonici Doctor*, Doctor of Canon Law). Because of its specialized nature, advanced degrees in civil law or theology are normal prerequisites for the study of canon law.
Much of Catholic canon law\'s legislative style was adapted from the Roman Code of Justinian. As a result, Roman ecclesiastical courts tend to follow the Roman Law style of continental Europe with some variation, featuring collegiate panels of judges and an investigative form of proceeding, called \"inquisitorial\", from the Latin \"inquirere\", to enquire. This is in contrast to the adversarial form of proceeding found in the common law system of English and U.S. law, which features such things as juries and single judges.
The institutions and practices of Catholic canon law paralleled the legal development of much of Europe, and consequently, both modern civil law and common law bear the influences of canon law. As Edson Luiz Sampel, a Brazilian expert in Catholic canon law, says, canon law is contained in the genesis of various institutes of civil law, such as the law in continental Europe and Latin American countries. Indirectly, canon law has significant influence in contemporary society.
Catholic Canonical jurisprudential theory generally follows the principles of Aristotelian-Thomistic legal philosophy. While the term \"law\" is never explicitly defined in the Catholic Code of Canon Law, the *Catechism of the Catholic Church* cites Aquinas in defining law as \"an ordinance of reason for the common good, promulgated by the one who is in charge of the community\" and reformulates it as \"a rule of conduct enacted by competent authority for the sake of the common good\".
### Code for the Eastern Churches {#code_for_the_eastern_churches}
The law of the Eastern Catholic Churches in full communion with the Roman papacy was in much the same state as that of the Latin Church before 1917; much more diversity in legislation existed in the various Eastern Catholic Churches. Each had its own special law, in which custom still played an important part. One major difference in Eastern Europe however, specifically in the Eastern Orthodox Christian churches, was in regards to divorce. Divorce started to slowly be allowed in specific instances such as adultery being committed, abuse, abandonment, impotence, and barrenness being the primary justifications for divorce. Eventually, the church began to allow remarriage to occur (for both spouses) post-divorce. In 1929 Pius XI informed the Eastern Churches of his intention to work out a Code for the whole of the Eastern Church. The publication of these Codes for the Eastern Churches regarding the law of persons was made between 1949 through 1958 but finalized nearly 30 years later.
The first Code of Canon Law (1917) was exclusively for the Latin Church, with application to the Eastern Churches only \"in cases which pertain to their very nature\". After the Second Vatican Council (1962 - 1965), the Vatican produced the *Code of Canons of the Eastern Churches* which became the first code of Eastern Catholic Canon Law.
## Eastern Orthodox Church {#eastern_orthodox_church}
The Eastern Orthodox Church, principally through the work of 18th-century Athonite monastic scholar Nicodemus the Hagiorite, has compiled canons and commentaries upon them in a work known as the *Pēdálion* (*Πηδάλιον*, \'Rudder\'), so named because it is meant to \"steer\" the church in her discipline. The dogmatic determinations of the Councils are to be applied rigorously since they are considered to be essential for the church\'s unity and the faithful preservation of the Gospel.
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# Canon law
## Anglican Communion {#anglican_communion}
In the Church of England, the ecclesiastical courts that formerly decided many matters such as disputes relating to marriage, divorce, wills, and defamation, still have jurisdiction of certain church-related matters (e.g. discipline of clergy, alteration of church property, and issues related to churchyards). Their separate status dates back to the 12th century when the Normans split them off from the mixed secular/religious county and local courts used by the Saxons. In contrast to the other courts of England, the law used in ecclesiastical matters is at least partially a civil law system, not common law, although heavily governed by parliamentary statutes. Since the Reformation, ecclesiastical courts in England have been royal courts. The teaching of canon law at the Universities of Oxford and Cambridge was abrogated by Henry VIII; thereafter practitioners in the ecclesiastical courts were trained in civil law, receiving a Doctor of Civil Law (D.C.L.) degree from Oxford, or a Doctor of Laws (LL.D.) degree from Cambridge. Such lawyers (called \"doctors\" and \"civilians\") were centered at \"Doctors Commons\", a few streets south of St Paul\'s Cathedral in London, where they monopolized probate, matrimonial, and admiralty cases until their jurisdiction was removed to the common law courts in the mid-19th century.
Other churches in the Anglican Communion around the world (e.g., the Episcopal Church in the United States and the Anglican Church of Canada) still function under their own private systems of canon law.
In 2002 a Legal Advisors Consultation meeting at Canterbury concluded:
> \(1\) There are principles of canon law common to the churches within the Anglican Communion; (2) Their existence can be factually established; (3) Each province or church contributes through its own legal system to the principles of canon law common within the Communion; (4) these principles have strong persuasive authority and are fundamental to the self-understanding of each of the member churches; (5) These principles have a living force, and contain within themselves the possibility for further development; and (6) The existence of the principles both demonstrates and promotes unity in the Communion.
## Presbyterian and Reformed churches {#presbyterian_and_reformed_churches}
In Presbyterian and Reformed churches, canon law is known as \"practice and procedure\" or \"church order\", and includes the church\'s laws respecting its government, discipline, legal practice, and worship.
Roman canon law had been criticized by the Presbyterians as early as 1572 in the Admonition to Parliament. The protest centered on the standard defense that canon law could be retained so long as it did not contradict the civil law. According to Polly Ha, the Reformed church government refuted this, claiming that the bishops had been enforcing canon law for 1500 years.
## Lutheranism
The Book of Concord is the historic doctrinal statement of the Lutheran Church, consisting of ten credal documents recognized as authoritative in Lutheranism since the 16th century. However, the Book of Concord is a confessional document (stating orthodox belief) rather than a book of ecclesiastical rules or discipline, like canon law. Each Lutheran national church establishes its own system of church order and discipline, though these are referred to as \"canons\".
## United Methodist Church {#united_methodist_church}
The Book of Discipline contains the laws, rules, policies, and guidelines for The United Methodist Church. Its latest edition was published in 2024
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# Computational complexity
In computer science, the **computational complexity** or simply **complexity** of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) and memory storage requirements. The complexity of a problem is the complexity of the best algorithms that allow solving the problem.
The study of the complexity of explicitly given algorithms is called analysis of algorithms, while the study of the complexity of problems is called computational complexity theory. Both areas are highly related, as the complexity of an algorithm is always an upper bound on the complexity of the problem solved by this algorithm. Moreover, for designing efficient algorithms, it is often fundamental to compare the complexity of a specific algorithm to the complexity of the problem to be solved. Also, in most cases, the only thing that is known about the complexity of a problem is that it is lower than the complexity of the most efficient known algorithms. Therefore, there is a large overlap between analysis of algorithms and complexity theory.
As the amount of resources required to run an algorithm generally varies with the size of the input, the complexity is typically expressed as a function `{{math|''n'' → ''f''(''n'')}}`{=mediawiki}, where `{{math|''n''}}`{=mediawiki} is the size of the input and `{{math|''f''(''n'')}}`{=mediawiki} is either the worst-case complexity (the maximum of the amount of resources that are needed over all inputs of size `{{math|''n''}}`{=mediawiki}) or the average-case complexity (the average of the amount of resources over all inputs of size `{{math|''n''}}`{=mediawiki}). Time complexity is generally expressed as the number of required elementary operations on an input of size `{{math|''n''}}`{=mediawiki}, where elementary operations are assumed to take a constant amount of time on a given computer and change only by a constant factor when run on a different computer. Space complexity is generally expressed as the amount of memory required by an algorithm on an input of size `{{math|''n''}}`{=mediawiki}.
## Complexity as a function of input size {#complexity_as_a_function_of_input_size}
It is impossible to count the number of steps of an algorithm on all possible inputs. As the complexity generally increases with the size of the input, the complexity is typically expressed as a function of the size `{{math|''n''}}`{=mediawiki} (in bits) of the input, and therefore, the complexity is a function of `{{math|''n''}}`{=mediawiki}. However, the complexity of an algorithm may vary dramatically for different inputs of the same size. Therefore, several complexity functions are commonly used.
The worst-case complexity is the maximum of the complexity over all inputs of size `{{mvar|n}}`{=mediawiki}, and the average-case complexity is the average of the complexity over all inputs of size `{{mvar|n}}`{=mediawiki} (this makes sense, as the number of possible inputs of a given size is finite). Generally, when \"complexity\" is used without being further specified, this is the worst-case time complexity that is considered.
## Asymptotic complexity {#asymptotic_complexity}
It is generally difficult to compute precisely the worst-case and the average-case complexity. In addition, these exact values provide little practical application, as any change of computer or of model of computation would change the complexity somewhat. Moreover, the resource use is not critical for small values of `{{mvar|n}}`{=mediawiki}, and this makes that, for small `{{mvar|n}}`{=mediawiki}, the ease of implementation is generally more interesting than a low complexity.
For these reasons, one generally focuses on the behavior of the complexity for large `{{mvar|n}}`{=mediawiki}, that is on its asymptotic behavior when `{{mvar|n}}`{=mediawiki} tends to the infinity. Therefore, the complexity is generally expressed by using big O notation.
For example, the usual algorithm for integer multiplication has a complexity of $O(n^2),$ this means that there is a constant $c_u$ such that the multiplication of two integers of at most `{{mvar|n}}`{=mediawiki} digits may be done in a time less than $c_un^2.$ This bound is *sharp* in the sense that the worst-case complexity and the average-case complexity are $\Omega(n^2),$ which means that there is a constant $c_l$ such that these complexities are larger than $c_ln^2.$ The radix does not appear in these complexity, as changing of radix changes only the constants $c_u$ and $c_l.$
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# Computational complexity
## Models of computation {#models_of_computation}
The evaluation of the complexity relies on the choice of a model of computation, which consists in defining the basic operations that are done in a unit of time. When the model of computation is not explicitly specified, it is generally implicitely assumed as being a multitape Turing machine, since several more realistic models of computation, such as random-access machines are asymptotically equivalent for most problems. It is only for very specific and difficult problems, such as integer multiplication in time $O(n\log n),$ that the explicit definition of the model of computation is required for proofs.
### Deterministic models {#deterministic_models}
A deterministic model of computation is a model of computation such that the successive states of the machine and the operations to be performed are completely determined by the preceding state. Historically, the first deterministic models were recursive functions, lambda calculus, and Turing machines. The model of random-access machines (also called RAM-machines) is also widely used, as a closer counterpart to real computers.
When the model of computation is not specified, it is generally assumed to be a multitape Turing machine. For most algorithms, the time complexity is the same on multitape Turing machines as on RAM-machines, although some care may be needed in how data is stored in memory to get this equivalence.
### Non-deterministic computation {#non_deterministic_computation}
In a non-deterministic model of computation, such as non-deterministic Turing machines, some choices may be done at some steps of the computation. In complexity theory, one considers all possible choices simultaneously, and the non-deterministic time complexity is the time needed, when the best choices are always done. In other words, one considers that the computation is done simultaneously on as many (identical) processors as needed, and the non-deterministic computation time is the time spent by the first processor that finishes the computation. This parallelism is partly amenable to quantum computing via superposed entangled states in running specific quantum algorithms, like e.g. Shor\'s factorization of yet only small integers (`{{as of|2018|03|lc=yes}}`{=mediawiki}: 21 = 3 × 7).
Even when such a computation model is not realistic yet, it has theoretical importance, mostly related to the P = NP problem, which questions the identity of the complexity classes formed by taking \"polynomial time\" and \"non-deterministic polynomial time\" as least upper bounds. Simulating an NP-algorithm on a deterministic computer usually takes \"exponential time\". A problem is in the complexity class NP, if it may be solved in polynomial time on a non-deterministic machine. A problem is NP-complete if, roughly speaking, it is in NP and is not easier than any other NP problem. Many combinatorial problems, such as the Knapsack problem, the travelling salesman problem, and the Boolean satisfiability problem are NP-complete. For all these problems, the best known algorithm has exponential complexity. If any one of these problems could be solved in polynomial time on a deterministic machine, then all NP problems could also be solved in polynomial time, and one would have P = NP. `{{As of|2017}}`{=mediawiki} it is generally conjectured that `{{nowrap|P ≠ NP,}}`{=mediawiki} with the practical implication that the worst cases of NP problems are intrinsically difficult to solve, i.e., take longer than any reasonable time span (decades!) for interesting lengths of input.
### Parallel and distributed computation {#parallel_and_distributed_computation}
Parallel and distributed computing consist of splitting computation on several processors, which work simultaneously. The difference between the different model lies mainly in the way of transmitting information between processors. Typically, in parallel computing the data transmission between processors is very fast, while, in distributed computing, the data transmission is done through a network and is therefore much slower.
The time needed for a computation on `{{mvar|N}}`{=mediawiki} processors is at least the quotient by `{{mvar|N}}`{=mediawiki} of the time needed by a single processor. In fact this theoretically optimal bound can never be reached, because some subtasks cannot be parallelized, and some processors may have to wait a result from another processor.
The main complexity problem is thus to design algorithms such that the product of the computation time by the number of processors is as close as possible to the time needed for the same computation on a single processor.
### Quantum computing {#quantum_computing}
A quantum computer is a computer whose model of computation is based on quantum mechanics. The Church--Turing thesis applies to quantum computers; that is, every problem that can be solved by a quantum computer can also be solved by a Turing machine. However, some problems may theoretically be solved with a much lower time complexity using a quantum computer rather than a classical computer. This is, for the moment, purely theoretical, as no one knows how to build an efficient quantum computer.
Quantum complexity theory has been developed to study the complexity classes of problems solved using quantum computers. It is used in post-quantum cryptography, which consists of designing cryptographic protocols that are resistant to attacks by quantum computers.
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