# Symmetry

A graphical representation of H3. Any two triangular regions are related by one of its 120 symmetries.

A symmetry of a polytope is a distance-preserving transformation of its containing space that maps each of its elements to another element of the same type. Such transformations are called isometries. For example, by rotating a square 90° around its center, each vertex is mapped to a vertex, and each edge is mapped to an edge. Thus, this rotation is one of the square’s symmetries.

The symmetry group of a polytope is the group formed by the symmetries of a polytope, with composition as its operation.

The study of symmetries is strongly linked to the study of polytopes. Many classes of polytopes, such as regular and uniform polytopes, are explicitly defined in terms of symmetries. Even when investigating categories as the CRFs that don’t directly involve the subject, symmetry can be useful in finding new shapes or simplifying calculations.

Some families of symmetry groups and example polytopes
rank Simplex Orthoplex/Hypercube Demicube Gosset Icositetrachoric Pentagonal Polygonal
n An Bn Dn En F4 Hn I2(p)
2 triangle
square
pentagon
heptagon (p = 7)
3 tetrahedron
octahedron
tetrahedron
icosahedron
4 pentachoron
rectified pentachoron
icositetrachoron
hexacosichoron
5 5-simplex
5-orthoplex
5-demicube
5-demicube
6 6-simplex
6-orthoplex
6-demicube
27-72-peton
7 7-simplex
7-orthoplex
7-demicube
126-576-exon
8 8-simplex
8-orthoplex
8-demicube
2160-17280-zetton
9 9-simplex
9-orthoplex
9-demicube
n n-simplex n-hypercube n-demicube (n-4)2,1 polytope

## Reflection groups

An important subclass of the polytope symmetry groups is the class of reflection groups, which are symmetry groups generated by reflections. Reflection groups can be represented using Coxeter diagrams.