Polyhedral group

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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 group is any of the symmetry groups of the Platonic solids.

Groups[edit]

There are three polyhedral groups:

  • The tetrahedral group of order 12, rotational symmetry group of the regular tetrahedron.
    • 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.
    • The conjugacy classes of O are:
      • identity
      • 6 × rotation by 90°, order 4
      • 8 × rotation by 120°, order 3
      • 3 × rotation by 180°, order 4
      • 6 × rotation by 180°, order 2
  • The icosahedral group of order 60, rotational symmetry group of the regular dodecahedron and the regular icosahedron.
    • 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 respectly 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, 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, Oh, 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 Ih 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[edit]

Chiral polyhedral groups
Name
(Orb.)
Coxeter
notation
Order Abstract
structure
Gyration
axes
#valence
Diagrams
Orthogonal Stereographic
T
(332)
CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png
[3,3]+
12 A4 43Armed forces red triangle.svg Purple Fire.svg
32Rhomb.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
[4,3+]
24 A4×2 43Armed forces red triangle.svg
3*2Desabilitado para jugar.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 34Monomino.png
43Armed forces red triangle.svg
62Rhomb.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 65Patka piechota.png
103Armed forces red triangle.svg
152Rhomb.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[edit]

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

(h)
Mirrors
(m)
Mirror diagrams
Orthogonal Stereographic
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 4 6CDel node c1.png Spherical tetrakis hexahedron.png 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]
48 S4×2 8 3CDel node c2.png
6CDel node c1.png
Spherical disdyakis dodecahedron.png 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 A5×2 10 15CDel node c1.png Spherical disdyakis triacontahedron.png Disdyakis triacontahedron stereographic d5.png Disdyakis triacontahedron stereographic d3.png Disdyakis triacontahedron stereographic d2.png

See also[edit]

References[edit]

External links[edit]