Polyhedral group

Last updated
Selected point groups in three dimensions
Sphere symmetry group cs.png
Involutional symmetry
Cs, (*)
[ ] = CDel node c2.png
Sphere symmetry group c3v.png
Cyclic symmetry
Cnv, (*nn)
[n] = CDel node c1.pngCDel n.pngCDel node c1.png
Sphere symmetry group d3h.png
Dihedral symmetry
Dnh, (*n22)
[n,2] = CDel node c1.pngCDel n.pngCDel node c1.pngCDel 2.pngCDel node c1.png
Polyhedral group, [n,3], (*n32)
Sphere symmetry group td.png
Tetrahedral symmetry
Td, (*332)
[3,3] = CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.png
Sphere symmetry group oh.png
Octahedral symmetry
Oh, (*432)
[4,3] = CDel node c2.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.png
Sphere symmetry group ih.png
Icosahedral symmetry
Ih, (*532)
[5,3] = CDel node c2.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c2.png

In geometry, the polyhedral groups are the symmetry groups of the Platonic solids.

Contents

Groups

There are three polyhedral groups:

These symmetries double to 24, 48, 120 respectively for the full reflectional groups. The reflection symmetries have 6, 9, and 15 mirrors respectively. The octahedral symmetry, [4,3] can be seen as the union of 6 tetrahedral symmetry [3,3] mirrors, and 3 mirrors of dihedral symmetry Dih2, [2,2]. Pyritohedral symmetry is another doubling of tetrahedral symmetry.

The conjugacy classes of full tetrahedral symmetry, TdS4 , are:

The conjugacy classes of pyritohedral symmetry, Th, include those of T, with the two classes of 4 combined, and each with inversion:

The conjugacy classes of the full octahedral group, OhS4 × C2 , are:

The conjugacy classes of full icosahedral symmetry, IhA5 × C2 , include also each with inversion:

Chiral polyhedral groups

Chiral polyhedral groups
Name
(Orb.)
Coxeter
notation
Order Abstract
structure
Rotation
points
# valence
Diagrams
OrthogonalStereographic
T
(332)
CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png
[3,3]+
12 A4 43 3-fold rotation axis.svg Purple Fire.svg
32 Rhomb.svg
Sphere symmetry group t.png Tetrakis hexahedron stereographic D4 gyrations.png Tetrakis hexahedron stereographic D3 gyrations.png Tetrakis hexahedron stereographic D2 gyrations.png
Th
(3*2)
CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png
CDel node c2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png
[4,3+]
24 A4 × C2 43 3-fold rotation axis.svg
3*2CDel node c2.png
Sphere symmetry group th.png Disdyakis dodecahedron stereographic D4 pyritohedral.png Disdyakis dodecahedron stereographic D3 pyritohedral.png Disdyakis dodecahedron stereographic D2 pyritohedral.png
O
(432)
CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png
[4,3]+
24 S4 34 Monomino.png
43 3-fold rotation axis.svg
62 Rhomb.svg
Sphere symmetry group o.png Disdyakis dodecahedron stereographic D4 gyrations.png Disdyakis dodecahedron stereographic D3 gyrations.png Disdyakis dodecahedron stereographic D2 gyrations.png
I
(532)
CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 3.pngCDel node h2.png
[5,3]+
60 A5 65 Patka piechota.png
103 3-fold rotation axis.svg
152 Rhomb.svg
Sphere symmetry group i.png Disdyakis triacontahedron stereographic d5 gyrations.png Disdyakis triacontahedron stereographic d3 gyrations.png Disdyakis triacontahedron stereographic d2 gyrations.png

Full polyhedral groups

Full polyhedral groups
Weyl
Schoe.
(Orb.)
Coxeter
notation
Order Abstract
structure
Coxeter
number

(h)
Mirrors
(m)
Mirror diagrams
OrthogonalStereographic
A3
Td
(*332)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.png
[3,3]
24 S4 46CDel node c1.png Spherical tetrakis hexahedron.svg Tetrakis hexahedron stereographic D4.png Tetrakis hexahedron stereographic D3.png Tetrakis hexahedron stereographic D2.png
B3
Oh
(*432)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node c2.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.png
[4,3]
48S4 × C283CDel node c2.png
>6CDel node c1.png
Spherical disdyakis dodecahedron.svg Disdyakis dodecahedron stereographic D4.png Disdyakis dodecahedron stereographic D3.png Disdyakis dodecahedron stereographic D2.png
H3
Ih
(*532)
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
CDel node c1.pngCDel 5.pngCDel node c1.pngCDel 3.pngCDel node c1.png
[5,3]
120 A5 × C21015CDel node c1.png Spherical disdyakis triacontahedron.svg Disdyakis triacontahedron stereographic d5.svg Disdyakis triacontahedron stereographic d3.svg Disdyakis triacontahedron stereographic d2.svg

See also

References