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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.

Abstract symmetry[edit | edit source]

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.)