Octahedral symmetry

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Selected 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
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. Full octahedral group; cycle graph.svg
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.

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

Contents

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 diagonals of the cube.

Details

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 OInversions of elements of O
identity0inversion0′
3 × rotation by 180° about a 4-fold axis7, 16, 233 × reflection in a plane perpendicular to a 4-fold axis7′, 16′, 23′
8 × rotation by 120° about a 3-fold axis3, 4, 8, 11, 12, 15, 19, 208 × rotoreflection by 60°3′, 4′, 8′, 11′, 12′, 15′, 19′, 20′
6 × rotation by 180° about a 2-fold axis1′, 2′, 5′, 6′, 14′, 21′6 × reflection in a plane perpendicular to a 2-fold axis1, 2, 5, 6, 14, 21
6 × rotation by 90° about a 4-fold axis9′, 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 (m, n) with and .
But as it is also the direct product S4 × S2, one can simply identify the elements of tetrahedral subgroup Td as and their inversions as .

So e.g. the identity (0, 0) is represented as 0 and the inversion (7, 0) as 0′.
(3, 1) is represented as 6 and (4, 1) as 6′.

A rotoreflection is a combination of rotation and reflection.

Chiral octahedral symmetry

Gyration axes
C4
Monomino.png
C3
> Armed forces red triangle.svg
C2
Rhomb.svg
346

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 projectionStereographic projection
2-fold4-fold3-fold2-fold
Sphere symmetry group o.png Disdyakis dodecahedron stereographic D4 gyrations.png Disdyakis dodecahedron stereographic D3 gyrations.png Disdyakis dodecahedron stereographic D2 gyrations.png

Full octahedral symmetry

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.C2, 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.

Polyhedron great rhombi 6-8 dual max.png
Each face of the disdyakis dodecahedron is a fundamental domain.
Sphere symmetry group oh.png
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.

Rotation matrices

Take the set of all 3×3 permutation matrices and assign a + or − sign to each of the three 1s. There are permutations and sign combinations for a total of 48 matrices, giving the full octahedral group. 24 of these matrices have a determinant of +1; these are the rotation matrices of the chiral octahedral group. The other 24 matrices have a determinant of −1 and correspond to a reflection or inversion.

Three reflectional generator matrices are needed for octahedral symmetry, which represent the three mirrors of a Coxeter–Dynkin diagram. The product of the reflections produce 3 rotational generators.

[4,3], CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 3.pngCDel node n2.png
ReflectionsRotations Rotoreflection
GeneratorsR0R1R2R0R1R1R2R0R2R0R1R2
GroupCDel node n0.pngCDel node n1.pngCDel node n2.pngCDel node h2.pngCDel 4.pngCDel node h2.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel node h2.pngCDel 6.pngCDel node h4.pngCDel 2x.pngCDel node h2.png
Order2224326
Matrix

Subgroups of full octahedral symmetry

Subgroup of Oh; S4 blue red; cycle graph.svg
O
Subgroup of Oh; S4 green orange; cycle graph.svg
Td
Subgroup of Oh; A4xC2; cycle graph.svg
Th
Cycle graphs of subgroups of order 24
Subgroups ordered in a Hasse diagram Full octahedral group; subgroups Hasse diagram.svg
Subgroups ordered in a Hasse diagram
Full octahedral group; subgroups Hasse diagram; rotational.svg
Rotational subgroups
Full octahedral group; subgroups Hasse diagram; reflective.svg
Reflective subgroups
Full octahedral group; subgroups Hasse diagram; inversion.svg
Subgroups containing inversion
Schoenflies notation Coxeter Orb. H-M Structure Cyc. Order Index
Oh[4,3]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png*432m3m S4×S2481
Td[3,3]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png*33243mS4 Subgroup of Oh; S4 green orange; cycle graph.svg 242
D4h[2,4]CDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png*2244/mmmD2×D8 GroupDiagramMiniC2D8.svg 163
D2h[2,2]CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png*222mmmD3
2
= D2×D4
GroupDiagramMiniC2x3.svg 86
C4v[4]CDel node.pngCDel 4.pngCDel node.png*444mm D8 GroupDiagramMiniD8.svg 86
C3v[3]CDel node.pngCDel 3.pngCDel node.png*333mD6 = S3 GroupDiagramMiniD6.svg 68
C2v[2]CDel node.pngCDel 2.pngCDel node.png*22mm2D2
2
= D4
GroupDiagramMiniD4.svg 412
Cs = C1v[ ]CDel node.png*2 or mD2 GroupDiagramMiniC2.svg 224
Th[3+,4]CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 4.pngCDel node.png3*2m3 A4×S2 Subgroup of Oh; A4xC2; cycle graph.svg 242
C4h[4+,2]CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 2.pngCDel node.png4*4/m Z4×D2 GroupDiagramMiniC2C4.svg 86
D3d[2+,6]CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 6.pngCDel node.png2*33mD12 = Z2×D6 GroupDiagramMiniD12.svg 124
D2d[2+,4]CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 4.pngCDel node.png2*242mD8 GroupDiagramMiniD8.svg 86
C2h = D1d[2+,2]CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2.pngCDel node.png2*2/mZ2×D2 GroupDiagramMiniD4.svg 412
S6[2+,6+]CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 6.pngCDel node h2.png3Z6 = Z2×Z3 GroupDiagramMiniC6.svg 68
S4[2+,4+]CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 4.pngCDel node h2.png4Z4 GroupDiagramMiniC4.svg 412
S2[2+,2+]CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png×1S2 GroupDiagramMiniC2.svg 224
O[4,3]+CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png432432S4 Subgroup of Oh; S4 blue red; cycle graph.svg 242
T[3,3]+CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png33223A4 GroupDiagramMiniA4.svg 124
D4[2,4]+CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 4.pngCDel node h2.png224422D8 GroupDiagramMiniD8.svg 86
D3[2,3]+CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 3.pngCDel node h2.png223322D6 = S3 GroupDiagramMiniD6.svg 68
D2[2,2]+CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.png222222D4 = Z2
2
GroupDiagramMiniD4.svg 412
C4[4]+CDel node h2.pngCDel 4.pngCDel node h2.png444Z4 GroupDiagramMiniC4.svg 412
C3[3]+CDel node h2.pngCDel 3.pngCDel node h2.png333Z3 = A3 GroupDiagramMiniC3.svg 316
C2[2]+CDel node h2.pngCDel 2x.pngCDel node h2.png222Z2 GroupDiagramMiniC2.svg 224
C1[ ]+CDel node h2.png111Z1 GroupDiagramMiniC1.svg 148
Octahedral symmetry tree conway.png
Octahedral subgroups in Coxeter notation [1]

The isometries of the cube

48 symmetry elements of a cube 48 elementov simmetrii kuba.gif
48 symmetry elements of a cube

The cube has 48 isometries (symmetry elements), forming the symmetry group Oh, isomorphic to S4 ×Z2. 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.

Examples:

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.

Examples:

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, Z4, [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

ClassNamePictureFacesEdgesVerticesDual namePicture
Archimedean solid
(Catalan solid)
snub cube Polyhedron snub 6-8 right max.png 386024 pentagonal icositetrahedron Polyhedron snub 6-8 right dual max.png

Solids with full octahedral symmetry

ClassNamePictureFacesEdgesVerticesDual namePicture
Platonic solid Cube Polyhedron 6 max.png 6128 Octahedron Polyhedron 8 max.png
Archimedean solid
(dual Catalan solid)
Cuboctahedron Polyhedron 6-8 max.png 142412 Rhombic dodecahedron Polyhedron 6-8 dual max.png
Truncated cube Polyhedron truncated 6 max.png 143624 Triakis octahedron Polyhedron truncated 6 dual max.png
Truncated octahedron Polyhedron truncated 8 max.png 143624 Tetrakis hexahedron Polyhedron truncated 8 dual max.png
Rhombicuboctahedron Polyhedron small rhombi 6-8 max.png 264824 Deltoidal icositetrahedron Polyhedron small rhombi 6-8 dual max.png
Truncated cuboctahedron Polyhedron great rhombi 6-8 max.png 267248 Disdyakis dodecahedron Polyhedron great rhombi 6-8 dual max.png
Regular
compound
polyhedron
Stellated octahedron Polyhedron stellated 8 max.png 8128Self-dual
Cube and octahedron Polyhedron pair 6-8 max.png 142414Self-dual

See also

References

  1. John Conway, The Symmetries of Things, Fig 20.8, p. 280