From Polytope Wiki
Jump to navigation Jump to search
A graphical representation of H3. Any two triangular regions are related by one of its 120 symmetries.

A symmetry of a polytope is a distance-preserving transformation of its containing space that maps each of its elements to another element of the same type. Such transformations are called isometries. 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 group formed by the symmetries of a polytope, with composition as its operation.

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
dimension Simplex Hypercube

(measure polytope)


(cross polytope)


(alternated hypercube)

"Pentagonal" polytope Regular polygon
n An BCn Dn En Fn Gn Hn I2(n)
2 triangle
square (self-dual)
(dyad) hexagon (self-dual)
Regular hexagon.svg
pentagon (self-dual)
heptagon (n=7)
3 tetrahedron
(tetrahedron) dodecahedron
4 pentachoron (4-simplex)
Schlegel wireframe 5-cell.png
tesseract (4-cube)
Schlegel wireframe 8-cell.png
hexadecachoron (4-orthoplex)
Schlegel wireframe 16-cell.png
(hexadecachoron) (pentachoron) icositetrachoron
Schlegel wireframe 24-cell.png
Schlegel wireframe 120-cell.png
Schlegel wireframe 600-cell vertex-centered.png
5 hexateron (5-simplex)
penteract (5-cube)
triacontaditeron (5-orthoplex)
demipenteract (5-demicube)
5-demicube graph.png
6 heptapeton (6-simplex)
hexeract (6-cube)
hexacontatetrapeton (6-orthoplex)
demihexeract (6-demicube)
6-demicube graph.png
mo (122 polytope)
Gosset 1 22 polytope.svg
7 octaexon (7-simplex)
7-simplex graph.png
hepteract (7-cube)
7-cube graph.svg
hecatonicosoctaexon (7-orthoplex)
demihepteract (7-demicube)
7-demicube graph.png
lin (132 polytope)
Gosset 1 32 petrie.svg
8 enneazetton (8-simplex)
8-simplex graph.png
octeract (8-cube)
diacosipentacontahexazetton (8-orthoplex)
demiocteract (8-demicube)
8-demicube graph.png
bif (142 polytope)
Gosset 1 42 polytope petrie.svg
9 decayotton (9-simplex)
9-simplex graph.png
enneract (9-cube)
pentacosidodecayotton (9-orthoplex)
demienneract (9-demicube)
9-demicube graph.png
n n-simplex n-cube n-orthoplex n-demicube

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]