![]() Involutional symmetry Cs, (*) [ ] = ![]() |
![]() Cyclic symmetry Cnv, (*nn) [n] = ![]() ![]() ![]() |
![]() Dihedral symmetry Dnh, (*n22) [n,2] = ![]() ![]() ![]() ![]() ![]() | |
Polyhedral group, [n,3], (*n32) | |||
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![]() Tetrahedral symmetry Td, (*332) [3,3] = ![]() ![]() ![]() ![]() ![]() |
![]() Octahedral symmetry Oh, (*432) [4,3] = ![]() ![]() ![]() ![]() ![]() |
![]() Icosahedral symmetry Ih, (*532) [5,3] = ![]() ![]() ![]() ![]() ![]() |
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 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 ![]() |
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Th (3*2) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [4,3+ |
24 | A4 × C2 | 43
![]() 3*2 ![]() |
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O (432) |
![]() ![]() ![]() ![]() ![]() [4,3]+ |
24 | S4 | 34
![]() 43 ![]() 62 ![]() |
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I (532) |
![]() ![]() ![]() ![]() ![]() [5,3]+ |
60 | A5 | 65
![]() 103 ![]() 152 ![]() |
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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![]() |
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B3 Oh (*432) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [4,3] |
48 | S4 × C2 | 8 | 3![]() >6 ![]() |
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H3 Ih (*532) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [5,3] |
120 | A5 × C2 | 10 | 15![]() |
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![]() 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 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 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 ![]() |
![]() |
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Th (3*2) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [4,3+ |
24 | A4 × C2 | 43
![]() 3*2 ![]() |
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O (432) |
![]() ![]() ![]() ![]() ![]() [4,3]+ |
24 | S4 | 34
![]() 43 ![]() 62 ![]() |
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I (532) |
![]() ![]() ![]() ![]() ![]() [5,3]+ |
60 | A5 | 65
![]() 103 ![]() 152 ![]() |
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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![]() |
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B3 Oh (*432) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [4,3] |
48 | S4 × C2 | 8 | 3![]() >6 ![]() |
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H3 Ih (*532) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [5,3] |
120 | A5 × C2 | 10 | 15![]() |
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