Symmetry

A symmetry of a polytope (or polytope-like object) is a distance-preserving transformation of its containing space 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.

The symmetry group of a polytope is the 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 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 (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.

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 cube lying on a table has 48 symmetries structured as the group B3, but only eight of those symmetries preserve the top face of the cube (imagine painting the top face a different color and transforming the cube so that its appearance does not change).

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.

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.