Polyhedron group

The group $ \mathop{\rm Sym} P $ of symmetries of a polytope (cf. Polyhedron) $ P $ in an $ n $- dimensional Euclidean space $ E ^ {n} $, that is, the group of all motions of $ E ^ {n} $ which send $ P $ to itself. A polytope $ P $ is called regular if $ \mathop{\rm Sym} P $ acts transitively on the set of its "flag set of a polytopeflags" , that is, collections

$$ F = \{ \Gamma _ {0} \dots \Gamma _ {n-} 1 \} $$

where $ \Gamma _ {k} $ is a $ k $- dimensional closed face and $ \Gamma _ {k-} 1 \subset \Gamma _ {k} $. The group of symmetries of a regular polytope is generated by reflections (see Reflection group). Its fundamental domain is a simplicial cone $ K $ whose vertex is the centre of the polytope $ P $, and whose edges pass through the centres of the faces constituting some flag $ F $. By the same token the generating reflections $ r _ {1} \dots r _ {n} $ of the group $ \mathop{\rm Sym} P $ have a natural enumeration: $ r _ {k} $ is the reflection relative to the hyperplane bounding $ K $ which does not pass through the centre of the face $ \Gamma _ {k-} 1 $. The generators $ r _ {k} $ and $ r _ {l} $ commute for $ | k - l | \geq 2 $, and the order of $ r _ {k} r _ {k+} 1 $ is equal to $ p _ {k} $— the number of $ k $- dimensional (or $ ( k- 1 ) $- dimensional) faces of the polytope $ \Gamma _ {k+} 1 $ containing the face $ \Gamma _ {k-} 2 $( if it is assumed that $ \Gamma _ {n} = P $ and $ \Gamma _ {-} 1 = \emptyset $). The sequence $ \{ p _ {1} \dots p _ {n-} 1 \} $ is called the Schläfli symbol of the polytopes. The three-dimensional regular polytopes (Platonic solids) have the following Schläfli symbols: the tetrahedron — $ \{ 3 , 3 \} $, the cube — $ \{ 4 , 3 \} $, the octahedron — $ \{ 3 , 4 \} $, the dodecahedron — $ \{ 5 , 3 \} $, and the icosahedron — $ \{ 3 , 5 \} $.

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 $ ( n- 1 ) $- dimensional faces of $ P $. 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 $ n\geq 5 $ there are only 3 regular polytopes in $ E ^ {n} $: the simplex, the cube and the polytope reciprocal to the cube (the analogue of the octahedron). Their Schläfli symbols are $ \{ 3 \dots 3 \} $, $ \{ 4 , 3 \dots 3 \} $ and $ \{ 3 \dots 3 , 4 \} $. In $ 4 $- dimensional space there are 6 regular polytopes: $ \{ 3 , 3 , 3 \} $, $ \{ 4 , 3 , 3 \} $, $ \{ 3 , 3 , 4 \} $, $ \{ 3 , 4 , 3 \} $, $ \{ 5 , 3 , 3 \} $, and $ \{ 3 , 3 , 5 \} $.

Each face of a regular polytope $ P $ is also a regular polytope, the Schläfli symbol of which is the initial segment of the Schläfli symbol of $ P $. For example, a $ 3 $- dimensional face of the polytope $ \{ 5 , 3 , 3 \} $ has the Schläfli symbol $ \{ 5 , 3 \} $, 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

$$ < r _ {1} \dots r _ {n} \mid ( r _ {k} r _ {l} ) ^ {2} = 1 \ \ \textrm{ for } | k- l | \geq 2 ; $$

$$ ( r _ {k} r _ {k+} 1 ) ^ {2} = p _ {k} \ \textrm{ for } k = 1 \dots n- 1 > . $$

This shows that this group is a Coxeter group.

References

[a1]	H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1990)