# Symmetry

A symmetry of a polytope (or polytope-like object) is a distance-preserving transformation of its containing space (isometry) that maps each of its elements to another element of the same type. 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.

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

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

## Distinction of symmetry groups

In contrast to typical treatment of groups in abstract algebra, there are symmetry groups that are isomorphic but are considered distinct because they describe different sets of polytopes. Some distinct symmetry groups from different dimensions of Euclidean space are isomorphic, such as chiral icosahedral symmetry (3D) and chiral pentachoric symmetry (4D). There are even isomorphic but distinct symmetry groups in the same dimension, such as those of the pentagonal prism and pentagonal antiprism. To formally distinguish these groups, geometers consider symmetry groups identical iff they belong to the same conjugacy class in the broader group of isometries in a given metric space (which implies isomorphism). Intuitively, conjugacy classes contain symmetry groups that are related to each other by a change of basis.

## Abstract symmetry

A different but related definition of symmetry concerns abstract polytopes. A symmetry of an abstract polytope is defined as an automorphism (a bijection mapping each element to an element of the same rank such that the structure of the polytope is preserved), and its symmetry group is defined as its automorphism group.

Isomorphic symmetry groups concerning polytopes of the same rank are conventionally considered identical. (Distinctions based on conjugacy classes don't work here, as there is no containing space and no notion of an isometry.)

## Symmetry group of an element

The symmetries of an individual element E "relative to" a polytope P are defined as the symmetries of P that transform that E to itself. Those symmetries form a group known as the symmetry group of E relative to P.

For example, a snub cube has 24 symmetries forming the chiral cubic symmetry group. The symmetry group of a single square relative to the entire polyhedron is chiral square symmetry, even though a square alone in 2D Euclidean space is not chiral. Furthermore, the 24 "snub triangles" have no symmetries relative to the snub cube except for the identity transformation.

If E is a facet of P, then the symmetry group of E relative to P is not only a subgroup of the symmetry group of P, but also a subgroup of the symmetry group of E alone in its affine subspace. (This assumes that all elements are in fact in affine subspaces; i.e. P is not skew.) This fact does not necessarily hold true if E is not a facet. For example, a point is asymmetrical, but a vertex of a cube possesses nontrivial symmetries relative to that cube (imagine rotating the cube about the space diagonal connecting the opposite vertex of the cube).

## Notable symmetry groups of polytopes

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        hexadecachoron        hexadecachoron      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.