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.

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.

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.