Polyhedral Symmetry
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Polyhedral Symmetry
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

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, 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, 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, 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

Chiral polyhedral groups
Name
(Orb.)
Coxeter
notation
Order Abstract
structure
Rotation
points
#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
CDel node c2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png
[4,3+]
24 A4×2 43Armed forces red triangle.svg
3*2CDel node c2.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

Full polyhedral groups
Weyl
Schoe.
(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.svg Disdyakis triacontahedron stereographic d3.svg Disdyakis triacontahedron stereographic d2.svg

See also

References

External links

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

  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Polyhedral_symmetry
 



 



 
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