Symmetry group

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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[edit | edit source]

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 are conjugate subgroups of the broader group of isometries in a given metric space (which implies isomorphism). Intuitively, conjugate subgroups are related to each other by a change of basis.

Symmetry group of an element[edit | edit source]

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[edit | edit source]

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
heptagon (p = 7)
3 tetrahedron
4 pentachoron
rectified pentachoron
5 5-simplex
6 6-simplex
7 7-simplex
8 8-simplex
9 9-simplex
n n-simplex n-hypercube n-demicube (n-4)2,1 polytope

Reflection groups[edit | edit source]

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.

External links[edit | edit source]