# 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.

How to Cite This Entry:
Polyhedron group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polyhedron_group&oldid=48235
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article