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[[File:Icosahedral reflection domains.png|250px|thumb|A graphical representation of [[H3]]. Any two triangular regions are related by one of its 120 symmetries.]]
A '''symmetry''' of a polytope (or polytope-like object) is a distance-preserving transformation of its containing space ({{w|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.

The '''symmetry group''' of a polytope is the {{w|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 {{w|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 polytope]]s. A symmetry of an abstract polytope is defined as an {{w|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 {{w|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 [[CRF]]s that don’t directly involve the subject, symmetry can be useful in finding new shapes or simplifying calculations.
{{Symmetry groups}}

== 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 diagram]]s.
== External links ==
*{{Wikipedia|Coxeter group#Symmetry groups of regular polytopes|Symmetry groups of regular polytopes}}
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[[Category:Polytope properties]]