Editing Regular polytope (section)

===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.