Involutional symmetry C_{s}, (*) [ ] = |
Cyclic symmetry C_{nv}, (*nn) [n] = |
Dihedral symmetry D_{nh}, (*n22) [n,2] = | |||
Polyhedral group, [n,3], (*n32)
Tetrahedral symmetry T_{d}, (*332) [3,3] = | Octahedral symmetry O_{h}, (*432) [4,3] = |
Icosahedral symmetry I_{h}, (*532) [5,3] = |
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In geometry, the polyhedral group is any of 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 Dih_{2}, [2,2]. Pyritohedral symmetry is another doubling of tetrahedral symmetry.
The conjugacy classes of full tetrahedral symmetry, T_{d} ≅ S_{4}, are:
The conjugacy classes of pyritohedral symmetry, T_{h}, include those of T, with the two classes of 4 combined, and each with inversion:
The conjugacy classes of the full octahedral group, O_{h} ≅ S_{4} × C_{2}, are:
The conjugacy classes of full icosahedral symmetry, I_{h} ≅ A_{5} × C_{2}, include also each with inversion:
Name (Orb.) |
Coxeter notation |
Order | Abstract structure |
Rotation points #_{valence} |
Diagrams | |||
---|---|---|---|---|---|---|---|---|
Orthogonal | Stereographic | |||||||
T (332) |
[3,3]^{+} |
12 | A_{4} | 4_{3} 3_{2} |
||||
T_{h} (3*2) |
[4,3^{+}] |
24 | A_{4} × C_{2} | 4_{3} 3_{*2} |
||||
O (432) |
[4,3]^{+} |
24 | S_{4} | 3_{4} 4_{3} 6_{2} |
||||
I (532) |
[5,3]^{+} |
60 | A_{5} | 6_{5} 10_{3} 15_{2} |
Weyl Schoe. (Orb.) |
Coxeter notation |
Order | Abstract structure |
Coxeter number (h) |
Mirrors (m) |
Mirror diagrams | |||
---|---|---|---|---|---|---|---|---|---|
Orthogonal | Stereographic | ||||||||
A_{3} T_{d} (*332) |
[3,3] |
24 | S_{4} | 4 | 6 | ||||
B_{3} O_{h} (*432) |
[4,3] |
48 | S_{4} × C_{2} | 8 | 3 >6 |
||||
H_{3} I_{h} (*532) |
[5,3] |
120 | A_{5} × C_{2} | 10 | 15 |