Symmetry

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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
rank Simplex Orthoplex/Hypercube Demicube Gosset Icositetrachoric Pentagonal Polygonal
n An Bn Dn En F4 Hn I2(p)
2 triangle
2-simplex t0.svg
CDel node 1.pngCDel 3.pngCDel node.png
square
2-cube.svg
CDel node 1.pngCDel 4.pngCDel node.png
pentagon
Regular polygon 5.svg
CDel node 1.pngCDel 5.pngCDel node.png
heptagon (p = 7)
Regular polygon 7.svg
CDel node 1.pngCDel p.pngCDel node.png
3 tetrahedron
3-simplex t0.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
octahedron
3-cube t2.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
tetrahedron
3-simplex t0.svg
CDel nodes 10ru.pngCDel split2.pngCDel node.png
icosahedron
Icosahedron H3 projection.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
4 pentachoron
4-simplex t0.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
hexadecachoron
4-cube t3.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
hexadecachoron
4-demicube t0 D4.svg
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
rectified pentachoron
4-simplex t1.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3b.pngCDel nodeb.png
icositetrachoron
24-cell t0 F4.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
hexacosichoron
600-cell graph H4.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
5 5-simplex
5-simplex t0.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-orthoplex
5-cube t4.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
5-demicube
5-demicube t0 D5.svg
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-demicube
5-demicube t0 D5.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3b.pngCDel nodeb.png
6 6-simplex
6-simplex t0.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-orthoplex
6-cube t5.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-demicube
6-demicube t0 D6.svg
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
27-72-peton
Up 2 21 t0 E6.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3b.pngCDel nodeb.png
7 7-simplex
7-simplex t0.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-orthoplex
7-cube t6.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
7-demicube
7-demicube t0 D7.svg
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
126-576-exon
Up2 3 21 t0 E7.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3b.pngCDel nodeb.png
8 8-simplex
8-simplex t0.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-orthoplex
8-cube t7.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
8-demicube
8-demicube t0 D8.svg
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
2160-17280-zetton
4 21 t0 E8.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3b.pngCDel nodeb.png
9 9-simplex
9-simplex t0.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
9-orthoplex
9-cube t8.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
9-demicube
9-demicube t0 D9.svg
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
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]