Octahedral symmetry

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] =
Cycle graph
The four hexagonal cycles have the inversion (the black knot on top) in common. The hexagons are symmetric, so e.g. 3 and 4 are in the same cycle, but not 3 and 12.

A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual of an octahedron.

The group of orientation-preserving symmetries is S4, the symmetric group or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four pairs of opposite sides of the octahedron.


Chiral and full (or achiral) octahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. They are among the crystallographic point groups of the cubic crystal system.

Conjugacy classes
Elements of O Inversions of elements of O
identity 0 inversion 0'
3 × rotation by 180° about a 4-fold axis 7, 16, 23 3 × reflection in a plane perpendicular to a 4-fold axis 7', 16', 23'
8 × rotation by 120° about a 3-fold axis 3, 4, 8, 11, 12, 15, 19, 20 8 × rotoreflection by 60° 3', 4', 8', 11', 12', 15', 19', 20'
6 × rotation by 180° about a 2-fold axis 1, 2, 5, 6, 14, 21 6 × reflection in a plane perpendicular to a 2-fold axis 1', 2', 5', 6', 14', 21'
6 × rotation by 90° about a 4-fold axis 9, 10, 13, 17, 18, 22 6 × rotoreflection by 90° 9', 10', 13', 17', 18', 22'

As the hyperoctahedral group of dimension 3 the full octahedral group is the wreath product ,
and a natural way to identify its elements is as pairs with and .
But as it is also the direct product , one can simply identify the elements of as and their inversions as .

So e.g. the identity is represented as and the inversion as .
is represented as and is as .

The pairs can be seen in the six files below. Each file is denoted by the , and the position of each permutation in the file corresponds to the .

A rotoreflection is a combination of rotation and reflection:
: The reflection applied on the 120° rotation gives the 60° rotoreflection .
: The reflection applied on the 90° rotation gives the 90° rotoreflection .

O, 432, or [4,3]+ of order 24, is chiral octahedral symmetry or rotational octahedral symmetry . This group is like chiral tetrahedral symmetry T, but the C2 axes are now C4 axes, and additionally there are 6 C2 axes, through the midpoints of the edges of the cube. Td and O are isomorphic as abstract groups: they both correspond to S4, the symmetric group on 4 objects. Td is the union of T and the set obtained by combining each element of O \ T with inversion. O is the rotation group of the cube and the regular octahedron.

Chiral octahedral symmetry
Orthogonal projection Stereographic projection
2-fold 4-fold 3-fold 2-fold

Oh, *432, [4,3], or m3m of order 48 - achiral octahedral symmetry or full octahedral symmetry. This group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of Td and Th. This group is isomorphic to S4.C4, and is the full symmetry group of the cube and octahedron. It is the hyperoctahedral group for n = 3. See also the isometries of the cube.

A dual cube-octahedron.

Each face of the disdyakis dodecahedron is a fundamental domain

The octahedral group Oh with fundamental domain

With the 4-fold axes as coordinate axes, a fundamental domain of Oh is given by 0 ≤ xyz. An object with this symmetry is characterized by the part of the object in the fundamental domain, for example the cube is given by z = 1, and the octahedron by x + y + z = 1 (or the corresponding inequalities, to get the solid instead of the surface). ax + by + cz = 1 gives a polyhedron with 48 faces, e.g. the disdyakis dodecahedron.

Faces are 8-by-8 combined to larger faces for a = b = 0 (cube) and 6-by-6 for a = b = c (octahedron).

The 9 mirror lines of full octahedral symmetry can be divided into two subgroups of 3 and 6 (drawn in purple and red), representing in two orthogonal subsymmetries: D2h, and Td. D2h symmetry can be doubled to D4h by restoring 2 mirrors from one of three orientations.

Octahedral symmetry and subgroups
Stereographic projection
4-fold 3-fold 2-fold
Oh, [4,3], , full octahedral symmetry (3+6 mirrors)
Td, [3,3]=[1+,4,3], = , tetrahedral subgroup (6 mirrors)
C3v, [3], , dihedral subgroup (3 mirrors)
Stereographic projection
4-fold 3-fold 2-fold
D4h, [4,2], dihedral subgroup (1+2+2 mirrors)
D2h, [2,2]=[4,3*], = dihedral subgroup (1+1+1 mirrors)
C4v, [4], , dihedral subgroup (2+2 mirrors)

Subgroups of full octahedral symmetry

Cycle graphs of subgroups of order 24
Subgroups ordered in a Hasse diagram
Rotational subgroups
Reflective subgroups
Schoe. Coxeter Orb. H-M Structure Cyc. Order Index
Oh [4,3] *432m3m S4×S2481
Td [3,3] *332 43m S424 2
D4h [2,4] *224 4/mmm Dih1×Dih416 3
D2h [2,2] *222 mmm Dih13=Dih1×Dih28 6
C4v [4] *44 4mm Dih48 6
C3v [3] *33 3m Dih3=S36 8
C2v [2] *22 mm2 Dih24 12
Cs=C1v [ ] * 2 or m Dih12 24
Th [3+,4] 3*2 m3 A4×S224 2
C4h [4+,2] 4* 4/m Z4×Dih18 6
D3d [2+,6] 2*3 3m Dih6=Z2×Dih312 4
D2d [2+,4] 2*2 42m Dih48 6
C2h = D1d [2+,2] 2* 2/m Z2×Dih14 12
S6 [2+,6+] 3 Z6=Z2×Z36 8
S4 [2+,4+] 8 Z44 12
S2 [2+,2+] × 1 S22 24
O [4,3]+ 432 432 S424 2
T [3,3]+ 332 23 A412 4
D4 [2,4]+ 224 422 Dih48 6
D3 [2,3]+ 223 322 Dih3=S36 8
D2 [2,2]+ 222 222 Dih2=Z224 12
C4 [4]+ 44 4 Z44 12
C3 [3]+ 33 3 Z3=A33 16
C2 [2]+ 22 2 Z22 24
C1 [ ]+ 11 1 Z11 48
Octahedral subgroups in Coxeter notation[1]

The isometries of the cube

48 symmetry elements of a cube

(To be integrated in the rest of the text.)

The cube has 48 isometries (symmetry elements), forming the symmetry group Oh, isomorphic to S4 × C2. They can be categorized as follows:

An isometry of the cube can be identified in various ways:

For cubes with colors or markings (like dice have), the symmetry group is a subgroup of Oh.


For some larger subgroups a cube with that group as symmetry group is not possible with just coloring whole faces. One has to draw some pattern on the faces.


The full symmetry of the cube, Oh, [4,3], (*432), is preserved if and only if all faces have the same pattern such that the full symmetry of the square is preserved, with for the square a symmetry group, Dih4, [4], of order 8.

The full symmetry of the cube under proper rotations, O, [4,3]+, (432), is preserved if and only if all faces have the same pattern with 4-fold rotational symmetry, C4, [4]+.

Octahedral symmetry of the Bolza surface

In Riemann surface theory, the Bolza surface, sometimes called the Bolza curve, is obtained as the ramified double cover of the Riemann sphere, with ramification locus at the set of vertices of the regular inscribed octahedron. Its automorphism group includes the hyperelliptic involution which flips the two sheets of the cover. The quotient by the order 2 subgroup generated by the hyperelliptic involution yields precisely the group of symmetries of the octahedron. Among the many remarkable properties of the Bolza surface is the fact that it maximizes the systole among all genus 2 hyperbolic surfaces.

Solids with octahedral chiral symmetry

Class Name Picture Faces Edges Vertices Dual name Picture
Archimedean solid
(Catalan solid)
snub cube 38 60 24 pentagonal icositetrahedron

Solids with full octahedral symmetry

Class Name Picture Faces Edges Vertices Dual name Picture
Platonic solid Cube6128 Octahedron
Archimedean solid
(dual Catalan solid)
Cuboctahedron 14 24 12 Rhombic dodecahedron
Truncated cube 14 36 24 Triakis octahedron
Truncated octahedron 14 36 24 Tetrakis hexahedron
Rhombicuboctahedron 26 48 24 Deltoidal icositetrahedron
Truncated cuboctahedron 26 72 48 Disdyakis dodecahedron
Stella octangula8 12 8Self-dual
cube and octahedron 142414Self-dual

See also


  1. John Conway, The Symmetries of Things, Fig 20.8, p280

External links

This article is issued from Wikipedia - version of the 11/1/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.