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Polyhedron group

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The group of symmetries of a polytope (cf. Polyhedron) in an -dimensional Euclidean space , that is, the group of all motions of which send to itself. A polytope is called regular if acts transitively on the set of its "flag set of a polytopeflags" , that is, collections

where is a -dimensional closed face and . The group of symmetries of a regular polytope is generated by reflections (see Reflection group). Its fundamental domain is a simplicial cone whose vertex is the centre of the polytope , and whose edges pass through the centres of the faces constituting some flag . By the same token the generating reflections of the group have a natural enumeration: is the reflection relative to the hyperplane bounding which does not pass through the centre of the face . The generators and commute for , and the order of is equal to — the number of -dimensional (or -dimensional) faces of the polytope containing the face (if it is assumed that and ). The sequence is called the Schläfli symbol of the polytopes. The three-dimensional regular polytopes (Platonic solids) have the following Schläfli symbols: the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron.

The Schläfli symbol determines a regular polytope up to a similarity. Reversal of a Schläfli symbol corresponds to transition to the reciprocal polytope, whose vertices ly at the centres of the -dimensional faces of . Reciprocal polytopes have the same symmetry group.

All possible Schläfli symbols of regular polytopes can be obtained from the classification of finite reflection groups, by selecting those with a linear Coxeter graph. For there are only 3 regular polytopes in : the simplex, the cube and the polytope reciprocal to the cube (the analogue of the octahedron). Their Schläfli symbols are , and . In -dimensional space there are 6 regular polytopes: , , , , , and .

Each face of a regular polytope is also a regular polytope, the Schläfli symbol of which is the initial segment of the Schläfli symbol of . For example, a -dimensional face of the polytope has the Schläfli symbol , that is, it is a dodecahedron.

References

[1] H.S.M. Coxeter, "Regular polytopes" , Dover, reprint (1973)
[2] B.A. Rozenfel'd, "Multi-dimensional spaces" , Moscow (1966) (In Russian)


Comments

A presentation of the polyhedron group is given by

This shows that this group is a Coxeter group.

References

[a1] H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1990)
How to Cite This Entry:
Polyhedron group. E.B. Vinberg (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polyhedron_group&oldid=12575
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098