Editing Regular polytope (section)

==Classification and description==
{{See also|List of regular polytopes}}

Regular polytopes are classified primarily according to their dimension.

Three classes of regular polytopes exist in every number of dimensions:
*[[Simplex|Regular simplex]]
*[[Measure polytope]] (Hypercube)
*[[Cross polytope]] (Orthoplex)
Any other regular polytope is said to be exceptional.

In [[One-dimensional space|one dimension]], the [[line segment]] simultaneously serves as the 1-simplex, the 1-hypercube and the 1-orthoplex.
 
In [[two dimensions]], there are infinitely many [[regular polygon]]s, namely the regular ''n''-sided polygon for ''n'' ≥ 3. The triangle is the 2-simplex. The square is both the 2-hypercube and the 2-orthoplex. The ''n''-sided polygons for ''n'' ≥ 5 are exceptional. 

In [[Three-dimensional space|three]] and [[Four-dimensional space|four dimensions]], there are several more exceptional [[regular polyhedra]] and [[4-polytope]]s respectively.

In [[Five-dimensional space|five dimensions]] and above, the simplex, hypercube and orthoplex are the only regular polytopes. There are no exceptional regular polytopes in these dimensions. 

Regular polytopes can be further classified according to [[symmetry]]. For example, the [[cube]] and the regular [[octahedron]] share the same symmetry, as do the [[regular dodecahedron]] and [[regular icosahedron]]. Two distinct regular polytopes with the same symmetry are [[Duality (mathematics)#Dimension-reversing dualities|dual]] to one another. Indeed, symmetry groups are sometimes named after regular polytopes, for example the [[Tetrahedral symmetry|tetrahedral]] and [[Icosahedral symmetry|icosahedral symmetries]].

The idea of a polytope is sometimes generalised to include related kinds of geometrical object. Some of these have regular examples, as discussed in the section on historical discovery below.

===Schläfli symbols===
{{main|Schläfli symbol}}

A concise symbolic representation for regular polytopes was developed by [[Ludwig Schläfli]] in the 19th century, and a slightly modified form has become standard. The notation is best explained by adding one dimension at a time.

*A [[convex polygon|convex]] [[regular polygon]] having ''n'' sides is denoted by {''n''}. So an equilateral triangle is {3}, a square {4}, and so on indefinitely. A regular ''n''-sided [[star polygon]] which winds ''m'' times around its centre is denoted by the fractional value {''n''/''m''}, where ''n'' and ''m'' are [[co-prime]], so a regular [[pentagram]] is {5/2}.
*A [[regular polyhedron]] having faces {''n''} with ''p'' faces joining around a vertex is denoted by {''n'', ''p''}. The nine [[regular polyhedra]] are [[Tetrahedron|{3, 3}]]; [[Octahedron|{3, 4}]]; [[Cube|{4, 3}]]; [[Regular icosahedron|{3, 5}]]; [[Regular dodecahedron|{5, 3}]]; [[Great icosahedron|{3, 5/2}]]; [[Great stellated dodecahedron|{5/2, 3}]]; [[Great dodecahedron|{5, 5/2}]]; and [[Small stellated dodecahedron|{5/2, 5}]]. {''p''} is the ''[[vertex figure]]'' of the polyhedron.
*A regular 4-polytope having cells {''n'', ''p''} with ''q'' cells joining around an edge is denoted by {''n'', ''p'', ''q''}. The vertex figure of the 4-polytope is a {''p'', ''q''}.
*A regular 5-polytope is an {''n'', ''p'', ''q'', ''r''}. And so on.

===Duality of the regular polytopes===

The [[Dual polytope|dual]] of a regular polytope is also a regular polytope. The Schläfli symbol for the dual polytope is just the original symbol written backwards: {3, 3} is self-dual, {3, 4} is dual to {4, 3}, {4, 3, 3} to {3, 3, 4} and so on.

The [[vertex figure]] of a regular polytope is the dual of the dual polytope's [[Facet (geometry)|facet]]. For example, the vertex figure of {3, 3, 4} is {3, 4}, the dual of which is {4, 3} &mdash; a [[Cell (geometry)|cell]] of {4, 3, 3}.

The [[hypercube|measure]] and [[cross polytope]]s in any dimension are dual to each other.

If the Schläfli symbol is [[palindromic]] (i.e. reads the same forwards and backwards), then the polytope is self-dual. The self-dual regular polytopes are:
* All [[regular polygon]]s - {a}.
* All regular ''n''-[[simplex]]es - {3,3,...,3}.
* The regular [[24-cell]] - {3,4,3} - in 4&nbsp;dimensions.
* The [[great 120-cell]] - {5,5/2,5} - and [[grand stellated 120-cell]] - {5/2,5,5/2} - in 4&nbsp;dimensions.
* All regular ''n''-dimensional [[Hypercubic honeycomb|hypercubic]] [[Honeycomb (geometry)|honeycombs]] - {4,3,...,3,4}. These may be treated as [[#Apeirotopes — infinite polytopes|infinite polytope]]s.
* Hyperbolic tilings and honeycombs (tilings {p,p} with p>4 in 2 dimensions; [[Order-4 square tiling honeycomb|{4,4,4}]], [[order-5 dodecahedral honeycomb|{5,3,5}]], [[Icosahedral honeycomb|{3,5,3}]], [[order-6 hexagonal tiling honeycomb|{6,3,6}]], and [[Hexagonal tiling honeycomb|{3,6,3}]] in 3 dimensions; [[order-5 120-cell honeycomb|{5,3,3,5}]] in 4 dimensions; and [[16-cell honeycomb honeycomb|{3,3,4,3,3}]] in 5 dimensions).

===Regular simplices===
{|class="wikitable" align="right" style="border-width:30%;"
|+ Graphs of the 1-simplex to 4-simplex.
|align=center|[[Image:1-simplex t0.svg|80px]]
|align=center|[[Image:2-simplex t0.svg|80px]]
|align=center|[[Image:3-simplex t0.svg|80px]]
|align=center|[[Image:4-simplex t0.svg|80px]]
|-
| [[Line segment]]
| [[Equilateral triangle|Triangle]]
| [[Tetrahedron]]
| [[Pentachoron]]
|-
| &nbsp;
| [[Image:Regular triangle.svg|80px]]
| [[Image:Tetrahedron.svg|80px]]
| [[Image:Schlegel wireframe 5-cell.png|80px]]
|}
{{main|Simplex}}

These are the '''regular simplices''' or '''simplexes'''. Their names are, in order of dimension:

:0. [[Point (geometry)|Point]]
:1. [[Line segment]]
:2. [[Equilateral triangle]] (regular trigon)
:3. Regular [[tetrahedron]] (triangular pyramid)
:4. Regular [[pentachoron]] ''or'' 4-simplex
:5. Regular [[hexateron]] ''or'' 5-simplex
:... An ''n''-simplex has ''n''+1 vertices.

The process of making each simplex can be visualised on a graph: Begin with a point ''A''. Mark point ''B'' at a distance ''r'' from it, and join to form a [[line segment]]. Mark point ''C'' in a second, [[orthogonal]], dimension at a distance ''r'' from both, and join to ''A'' and ''B'' to form an [[equilateral triangle]]. Mark point ''D'' in a third, orthogonal, dimension a distance ''r'' from all three, and join to form a regular [[tetrahedron]]. This process is repeated further using new points to form higher-dimensional simplices.

===Measure polytopes (hypercubes)===
{|class="wikitable" align="right" style="border-width:30%;"
|+ Graphs of the 2-cube to 4-cube.
|align=center|[[Image:Cross graph 2.svg|80px]]
|align=center|[[Image:Cube graph ortho vcenter.png|80px]]
|align=center|[[Image:Hypercubestar.svg|80px]]
|-
| [[Square (geometry)|Square]]
| [[Cube]]
| [[Tesseract]]
|-
| [[Image:Kvadrato.svg|80px]]
| [[Image:Hexahedron.svg|80px]]
| [[Image:Schlegel wireframe 8-cell.png|80px]]
|}
{{main|Hypercube}}

These are the '''measure polytopes''' or '''hypercubes'''. Their names are, in order of dimension:

:0. Point
:1. Line segment
:2. [[Square (geometry)|Square]] (regular tetragon)
:3. [[Cube]] (regular hexahedron)
:4. [[Tesseract]] (regular octachoron) ''or'' 4-cube
:5. [[Penteract]] (regular decateron) ''or'' 5-cube
:... An ''n''-cube has ''2<sup>n</sup>'' vertices.

The process of making each hypercube can be visualised on a graph: Begin with a point ''A''. Extend a line to point ''B'' at distance ''r'', and join to form a line segment. Extend a second line of length ''r'', orthogonal to ''AB'', from ''B'' to ''C'', and likewise from ''A'' to ''D'', to form a [[Square (geometry)|square]] ''ABCD''. Extend lines of length ''r'' respectively from each corner, orthogonal to both ''AB'' and ''BC'' (i.e. upwards). Mark new points ''E'',''F'',''G'',''H'' to form the [[cube]] ''ABCDEFGH''. This process is repeated further using new lines to form higher-dimensional hypercubes.

===Cross polytopes (orthoplexes)===
{| class="wikitable" align="right" style="border-width:30%;"
|+ Graphs of the 2-orthoplex to 4-orthoplex.
|align=center|[[Image:2-orthoplex.svg|80px]]
|align=center|[[Image:3-orthoplex.svg|80px]]
|align=center|[[Image:4-orthoplex.svg|80px]]
|-
| [[Square (geometry)|Square]]
| [[Octahedron]]
| [[16-cell]]
|-
| [[Image:Kvadrato.svg|80px]]
| [[Image:Octahedron.svg|80px]]
| [[Image:Schlegel wireframe 16-cell.png|80px]]
|}
{{main|Orthoplex}}

These are the '''cross polytopes''' or '''orthoplexes'''. Their names are, in order of dimensionality:

:0. Point
:1. Line segment
:2. Square (regular tetragon)
:3. Regular [[octahedron]]
:4. Regular hexadecachoron ([[16-cell]]) ''or'' 4-orthoplex
:5. Regular triacontakaiditeron ([[pentacross]]) ''or'' 5-orthoplex
:... An ''n''-orthoplex has ''2n'' vertices.

The process of making each orthoplex can be visualised on a graph: Begin with a point ''O''. Extend a line in opposite directions to points ''A'' and ''B'' a distance ''r'' from ''O'' and 2''r'' apart. Draw a line ''COD'' of length 2''r'', centred on ''O'' and orthogonal to ''AB''. Join the ends to form a [[Square (geometry)|square]] ''ACBD''. Draw a line ''EOF'' of the same length and centered on 'O', orthogonal to ''AB'' and ''CD'' (i.e. upwards and downwards). Join the ends to the square to form a regular [[octahedron]]. This process is repeated further using new lines to form higher-dimensional orthoplices.

===Classification by Coxeter groups===

Regular polytopes can be classified by their [[isometry group]]. These are finite [[Coxeter group]]s, but not every finite Coxeter group may be realised as the isometry group of a regular polytope. Regular polytopes are in [[bijection]] with Coxeter groups with linear [[Coxeter-Dynkin diagram]] (without branch point) and an increasing numbering of the nodes. Reversing the numbering gives the [[dual polytope]].

The classification of finite Coxeter groups, which goes back to {{Harv|Coxeter|1935}}, therefore implies the classification of regular polytopes:
* Type <math>A_n</math>, the symmetric group, gives the regular [[simplex]],
*Type <math>B_n</math>, gives the  [[measure polytope]] and the [[cross polytope]] (both can be distinguished by the increasing numbering of the nodes of the Coxeter-Dynkin diagram),
* Exceptional types <math>I_2(n)</math> give the [[regular polygon]]s (with <math>n = 3, 4, ...</math>),
* Exceptional type <math>H_3</math> gives the regular [[dodecahedron]] and [[icosahedron]] (again the numbering allows to distinguish them),
* Exceptional type <math>H_4</math> gives the [[120-cell]] and the [[600-cell]],
* Exceptional type <math>F_4</math> gives the [[24-cell]], which is self-dual.

The bijection between regular polytopes and Coxeter groups with linear Coxeter-Dynkin diagram can be understood as follows. Consider a regular polytope <math>P</math> of dimension <math>n</math> and take its [[barycentric subdivision]]. The [[fundamental domain]] of the isometry group action on <math>P</math> is given by any simplex <math>\Delta</math> in the barycentric subdivision. The simplex <math>\Delta</math> has <math>n+1</math> vertices which can be numbered from 0 to <math>n</math> by the dimension of the corresponding face of <math>P</math> (the face they are the barycenter of). The isometry group of <math>P</math> is generated by the <math>n</math> reflections around the hyperplanes of <math>\Delta</math> containing the vertex number <math>n</math> (since the barycenter of the whole polytope <math>P</math> is fixed by any isometry). These <math>n</math> hyperplanes can be numbered by the vertex of <math>\Delta</math> they do not contain. The remaining thing to check is that any two hyperplanes with adjacent numbers cannot be orthogonal, whereas hyperplanes with non-adjacent numbers are orthogonal. This can be done using induction (since all facets of <math>P</math> are again regular polytopes). Therefore, the Coxeter-Dynkin diagram of the isometry group of <math>P</math> has <math>n</math> vertices numbered from 0 to <math>n-1</math> such that adjacent numbers are linked by at least one edge and non-adjacent numbers are not linked.