polyhedral group
{{Short description|Geometric polyhedral group}}
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In geometry, the polyhedral groups are the symmetry groups of the Platonic solids.
Groups
There are three polyhedral groups:
- The tetrahedral group of order 12, rotational symmetry group of the regular tetrahedron. It is isomorphic to A4.
- The conjugacy classes of T are:
- identity
- 4 × rotation by 120°, order 3, cw
- 4 × rotation by 120°, order 3, ccw
- 3 × rotation by 180°, order 2
- The octahedral group of order 24, rotational symmetry group of the cube and the regular octahedron. It is isomorphic to S4.
- The conjugacy classes of O are:
- identity
- 6 × rotation by ±90° around vertices, order 4
- 8 × rotation by ±120° around triangle centers, order 3
- 3 × rotation by 180° around vertices, order 2
- 6 × rotation by 180° around midpoints of edges, order 2
- The icosahedral group of order 60, rotational symmetry group of the regular dodecahedron and the regular icosahedron. It is isomorphic to A5.
- The conjugacy classes of I are:
- identity
- 12 × rotation by ±72°, order 5
- 12 × rotation by ±144°, order 5
- 20 × rotation by ±120°, order 3
- 15 × rotation by 180°, order 2
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, {{nowrap|Td ≅ S4}}, are:
- identity
- 8 × rotation by 120°
- 3 × rotation by 180°
- 6 × reflection in a plane through two rotation axes
- 6 × rotoreflection by 90°
The conjugacy classes of pyritohedral symmetry, Th, include those of T, with the two classes of 4 combined, and each with inversion:
- identity
- 8 × rotation by 120°
- 3 × rotation by 180°
- inversion
- 8 × rotoreflection by 60°
- 3 × reflection in a plane
The conjugacy classes of the full octahedral group, {{nowrap|Oh ≅ S4 × C2}}, are:
- inversion
- 6 × rotoreflection by 90°
- 8 × rotoreflection by 60°
- 3 × reflection in a plane perpendicular to a 4-fold axis
- 6 × reflection in a plane perpendicular to a 2-fold axis
The conjugacy classes of full icosahedral symmetry, {{nowrap|Ih ≅ A5 × C2}}, include also each with inversion:
- inversion
- 12 × rotoreflection by 108°, order 10
- 12 × rotoreflection by 36°, order 10
- 20 × rotoreflection by 60°, order 6
- 15 × reflection, order 2
Chiral polyhedral groups
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|+ Chiral polyhedral groups | ||||||||
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!rowspan=2|Coxeter{{br}}notation !rowspan=2|Order !rowspan=2|Abstract{{br}}structure !rowspan=2|Rotation{{br}}points{{br}}#valence !colspan=4|Diagrams | ||||||||
| Orthogonal
!colspan=3|Stereographic | ||||||||
|---|---|---|---|---|---|---|---|---|
| align=center
!T{{br}}(332) | {{CDD|node_h2|3|node_h2|3|node_h2}}{{br}}[3,3]+ | 12 | A4 | 4312px 12px{{br}}3212px | 120px | 120px | 120px | 120px |
| align=center
!Th{{br}}(3*2) | {{CDD|node|4|node_h2|3|node_h2}}{{br}}{{CDD|node_c2|4|node_h2|3|node_h2}}{{br}}[4,3+] | 24 | A4 × C2 | 4312px{{br}}3*2{{CDD|node_c2}} | 120px | |||
| align=center
!O{{br}}(432) | {{CDD|node_h2|4|node_h2|3|node_h2}}{{br}}[4,3]+ | 24 | S4 | 3412px{{br}}4312px{{br}}6212px | 120px | |||
| align=center
!I{{br}}(532) | {{CDD|node_h2|5|node_h2|3|node_h2}}{{br}}[5,3]+ | 60 | A5 | |6512px{{br}}10312px{{br}}15212px | 120px |
Full polyhedral groups
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|+ Full polyhedral groups | |||||||||
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!rowspan=2|Weyl{{br}}Schoe.{{br}}(Orb.) !rowspan=2|Coxeter{{br}}notation !rowspan=2|Order !rowspan=2|Abstract{{br}}structure !rowspan=2|Coxeter{{br}}number{{br}}(h) !rowspan=2|Mirrors{{br}}(m) !colspan=4|Mirror diagrams | |||||||||
| Orthogonal
!colspan=3|Stereographic | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| align=center
!A3{{br}}Td{{br}}(*332) | {{CDD|node|3|node|3|node}}{{br}}{{CDD|node_c1|3|node_c1|3|node_c1}}{{br}}[3,3] | 24 | S4 | 4 | 6{{CDD|node_c1}} | 120px | 120px | 120px | 120px |
| align=center
!B3{{br}}Oh{{br}}(*432) | {{CDD|node|4|node|3|node}}{{br}}{{CDD|node_c2|4|node_c1|3|node_c1}}{{br}}[4,3] | 48 | S4 × C2 | 8 | 3{{CDD|node_c2}}{{br}}>6{{CDD|node_c1}} | 120px | 120px | 120px | 120px |
| align=center
!H3{{br}}Ih{{br}}(*532) | {{CDD|node|5|node|3|node}}{{br}}{{CDD|node_c1|5|node_c1|3|node_c1}}{{br}}[5,3] | 120 | A5 × C2 | 10 | 15{{CDD|node_c1}} | 120px | 120px | 120px | 120px |
See also
References
- Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. (The Polyhedral Groups. §3.5, pp. 46–47)
External links
- {{mathworld | urlname = PolyhedralGroup| title =PolyhedralGroup}}