Polyhedral group

In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids.

Point groups in three dimensions

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] =

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, TdS4, 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, OhS4 × 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, IhA5 × 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

Chiral polyhedral groups
Name
(Orb.)
Coxeter
notation
Order Abstract
structure
Rotation
points
#valence
Diagrams
Orthogonal Stereographic
T
(332)

[3,3]+
12A443
32
Th
(3*2)


[4,3+]
24A4×243
3*2
O
(432)

[4,3]+
24S434
43
62
I
(532)

[5,3]+
60A565
103
152

Full polyhedral groups

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

(h)
Mirrors
(m)
Mirror diagrams
Orthogonal Stereographic
A3
Td
(*332)


[3,3]
24S446
B3
Oh
(*432)


[4,3]
48S4×283
6
H3
Ih
(*532)


[5,3]
120A5×21015

See also

References

  • Weisstein, Eric W. "PolyhedralGroup". MathWorld.
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