![]() Involutional symmetry Cs, (*) [ ] = ![]() | ![]() Cyclic symmetry Cnv, (*nn) [n] = ![]() ![]() ![]() | ![]() Dihedral symmetry Dnh, (*n22) [n,2] = ![]() ![]() ![]() ![]() ![]() | |
Polyhedral group, [n,3], (*n32) | |||
---|---|---|---|
![]() Tetrahedral symmetry Td, (*332) [3,3] = ![]() ![]() ![]() ![]() ![]() | ![]() Octahedral symmetry Oh, (*432) [4,3] = ![]() ![]() ![]() ![]() ![]() | ![]() Icosahedral symmetry Ih, (*532) [5,3] = ![]() ![]() ![]() ![]() ![]() |
In geometry, the polyhedral groups are the symmetry groups of the Platonic solids.
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, Td ≅ S4 , 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, Oh ≅ S4 × C2 , are:
The conjugacy classes of full icosahedral symmetry, Ih ≅ A5 × C2 , include also each with inversion:
Name (Orb.) | Coxeter notation | Order | Abstract structure | Rotation points # valence | Diagrams | |||
---|---|---|---|---|---|---|---|---|
Orthogonal | Stereographic | |||||||
T (332) | ![]() ![]() ![]() ![]() ![]() [3,3]+ | 12 | A4 | 43 ![]() ![]() 32 ![]() | ![]() | ![]() | ![]() | ![]() |
Th (3*2) | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [4,3+] | 24 | A4 × C2 | 43 ![]() 3*2 ![]() | ![]() | ![]() | ![]() | ![]() |
O (432) | ![]() ![]() ![]() ![]() ![]() [4,3]+ | 24 | S4 | 34 ![]() 43 ![]() 62 ![]() | ![]() | ![]() | ![]() | ![]() |
I (532) | ![]() ![]() ![]() ![]() ![]() [5,3]+ | 60 | A5 | 65 ![]() 103 ![]() 152 ![]() | ![]() | ![]() | ![]() | ![]() |
Weyl Schoe. (Orb.) | Coxeter notation | Order | Abstract structure | Coxeter number (h) | Mirrors (m) | Mirror diagrams | |||
---|---|---|---|---|---|---|---|---|---|
Orthogonal | Stereographic | ||||||||
A3 Td (*332) | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [3,3] | 24 | S4 | 4 | 6![]() | ![]() | ![]() | ![]() | ![]() |
B3 Oh (*432) | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [4,3] | 48 | S4 × C2 | 8 | 3![]() >6 ![]() | ![]() | ![]() | ![]() | ![]() |
H3 Ih (*532) | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [5,3] | 120 | A5 × C2 | 10 | 15![]() | ![]() | ![]() | ![]() | ![]() |