Octahedral symmetry

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Point groups in three dimensions
Sphere symmetry group cs.png
Involutional symmetry
Cs, [1], (*)
Sphere symmetry group c3v.png
Cyclic symmetry
Cnv, [n], (*nn)
Sphere symmetry group d3h.png
Dihedral symmetry
Dnh, [n,2], (*n22)
Polyhedral group, [n,3], (*n32)
Sphere symmetry group td.png
Tetrahedral symmetry
Td, [3,3], (*332)
Sphere symmetry group oh.png
Octahedral symmetry
Oh, [4,3], (*432)
Sphere symmetry group ih.png
Icosahedral symmetry
Ih, [5,3], (*532)
Octahedral symmetry fundamental domains
The cube is the most common shape with octahedral symmetry

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.

Details[edit]

The octahedral rotation group O with fundamental domain

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.

Chiral octahedral symmetry[edit]

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.

Subgroups of chiral octahedral symmetry[edit]

Schoenflies
notation
Coxeter
notation
Orbifold
notation
Hermann
Mauguin
notation
Order Index Subgroup relations
O [4,3]+ 432 432 24 1 Chiral octahedral subgroup tree.png
T [3,3]+ 332 23 12 2
D4 [4,2]+ 422 422 8 3
D3 [3,2]+ 322 322 6 4
D2 [2,2]+ 222 222 4 6
C4 [4]+ 44 4 4 6
C3 [3]+ 33 3 3 8
C2 [2]+ 22 2 2 12
S1 [ ]+ 11 1 1 24

Conjugacy classes[edit]

  • identity
  • 6 × rotation by 90°
  • 8 × rotation by 120°
  • 3 × rotation by 180° about a 4-fold axis
  • 6 × rotation by 180° about a 2-fold axis

Achiral octahedral symmetry[edit]

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.

Dual Cube-Octahedron.svg
A dual cube-octahedron.
Disdyakisdodecahedron.jpg
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).

Reflective subgroups of [4,3]

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
Orthographic
projection
Stereographic projection
4-fold 3-fold 2-fold
Oh, [4,3] octahedral symmetry (3+6 mirrors)
Sphere symmetry group oh.png Disdyakis dodecahedron stereographic D4.png Disdyakis dodecahedron stereographic D3.png Disdyakis dodecahedron stereographic D2.png
Td, [3,3] tetrahedral subsymmetry (6 mirrors)
Sphere symmetry group td.png Tetrakis hexahedron stereographic D4.png Tetrakis hexahedron stereographic D3.png Tetrakis hexahedron stereographic D2.png
C3v, [3] dihedral subsymmetry (3 mirrors)
Spherical hexagonal hosohedron2.png Trigonal hosohedron stereographic D4.png Trigonal hosohedron stereographic D3.png Trigonal hosohedron stereographic D2.png
D4h, [4,2] dihedral subsymmetry (1+2+2 mirrors)
Octagonal bipyramidal orthogonal.png Octagonal bipyramidal stereographic D4.png Octagonal bipyramidal stereographic D3.png Octagonal bipyramidal stereographic D2.png
D2h, [2,2] dihedral subsymmetry (1+1+1 mirrors)
Octahedron orthographic.png Octahedron stereographic D4.png Octahedron stereographic D3.png Octahedron stereographic D2.png

Subgroups of full octahedral symmetry[edit]

Schoenflies
notation
Coxeter
notation
Orbifold
notation
Hermann
Mauguin
notation
Order Index Subgroup relations
Oh [4,3] *432 m3m 48 1 Octahedral subgroups
O [4,3]+ 432 432 24 2
Td [3,3] *332 43m 24 2
Th [4,3+] 3*2 m3 24 2
D4h [4,2] *422 4/mmm 16 3
T [3,3]+ 332 23 12 4
D3d [2+,6] 2*3 3m 12 4
D2d [2+,4] 2*2 42m 8 6
D4 [4,2]+ 422 422 8 6
C4h [4+,2] 4* 4/m 8 6
C4v [4] *44 4mm 8 6
D2h [2,2] *222 mmm 8 6
D3 [3,2]+ 322 32 6 8
C3v [3] *33 3m 6 8
S6 [2+,6+] 3 6 8
D2 [2,2]+ 222 222 4 12
C2h [2,2+] 2* 2/m 4 12
C2v [2] *22 mm2 4 12
C4 [4]+ 44 4 4 12
S4 [2+,4+] 8 4 12
C3 [3]+ 33 3 3 16
C2 [2]+ 22 2 2 24
S2 [2+,2+] × 1 2 24
Cs [ ] * 2 or m 2 24
C1 [ ]+ 11 1 1 48

Conjugacy classes[edit]

  • 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
48 symmetry elements of a cube

The isometries of the cube[edit]

(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:

  • O (the identity and 23 proper rotations) with the following conjugacy classes (in parentheses are given the permutations of the body diagonals and the unit quaternion representation):
    • identity (identity; 1)
    • rotation about an axis from the center of a face to the center of the opposite face by an angle of 90°: 3 axes, 2 per axis, together 6 ((1 2 3 4), etc.; ((1±i)/√2, etc.)
    • ditto by an angle of 180°: 3 axes, 1 per axis, together 3 ((1 2) (3 4), etc.; i,j,k)
    • rotation about an axis from the center of an edge to the center of the opposite edge by an angle of 180°: 6 axes, 1 per axis, together 6 ((1 2), etc.; ((i±j)/√2, etc.)
    • rotation about a body diagonal by an angle of 120°: 4 axes, 2 per axis, together 8 ((1 2 3), etc.; (1±i±j±k)/2)
  • The same with inversion (x is mapped to −x) (also 24 isometries). Note that rotation by an angle of 180° about an axis combined with inversion is just reflection in the perpendicular plane. The combination of inversion and rotation about a body diagonal by an angle of 120° is rotation about the body diagonal by an angle of 60°, combined with reflection in the perpendicular plane (the rotation itself does not map the cube to itself; the intersection of the reflection plane with the cube is a regular hexagon).

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

  • by the faces three given adjacent faces (say 1, 2, and 3 on a die) are mapped to
  • by the image of a cube with on one face a non-symmetric marking: the face with the marking, whether it is normal or a mirror image, and the orientation
  • by a permutation of the four body diagonals (each of the 24 permutations is possible), combined with a toggle for inversion of the cube, or not

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

Examples:

  • C4v, [4], (*422): if one face has a different color (or two opposite faces have colors different from each other and from the other four), the cube has 8 isometries, like a square has in 2D.
  • D2h, [2,2], (*222): if opposite faces have the same colors, different for each set of two, the cube has 8 isometries, like a cuboid.
  • D4h, [4,2], (*422): if two opposite faces have the same color, and all other faces have one different color, the cube has 16 isometries, like a square prism (square box).
  • C2v, [2], (*22):
    • if two adjacent faces have the same color, and all other faces have one different color, the cube has 4 isometries.
    • if three faces, of which two opposite to each other, have one color and the other three one other color, the cube has 4 isometries.
    • if two opposite faces have the same color, and two other opposite faces also, and the last two have different colors, the cube has 4 isometries, like a piece of blank paper with a shape with a mirror symmetry.
  • Cs, [ ], (*):
    • if two adjacent faces have colors different from each other, and the other four have a third color, the cube has 2 isometries.
    • if two opposite faces have the same color, and all other faces have different colors, the cube has 2 isometries, like an asymmetric piece of blank paper.
  • C3v, [3], (*33): if three faces, of which none opposite to each other, have one color and the other three one other color, the cube has 6 isometries.

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:

  • D2d, [2+,4], (2*2): if one face has a line segment dividing the face into two equal rectangles, and the opposite has the same in perpendicular direction, the cube has 8 isometries; there is a symmetry plane and 2-fold rotational symmetry with an axis at an angle of 45° to that plane, and, as a result, there is also another symmetry plane perpendicular to the first, and another axis of 2-fold rotational symmetry perpendicular to the first.
  • Th, [3+,4], (3*2): if each face has a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do not meet at the edge, the cube has 24 isometries: the even permutations of the body diagonals and the same combined with inversion (x is mapped to −x).
  • Td, [3,3], (*332): if the cube consists of eight smaller cubes, four white and four black, put together alternatingly in all three standard directions, the cube has again 24 isometries: this time the even permutations of the body diagonals and the inverses of the other proper rotations.
  • T, [3,3]+, (332): if each face has the same pattern with 2-fold rotational symmetry, say the letter S, such that at all edges a top of one S meets a side of the other S, the cube has 12 isometries: the even permutations of the body diagonals.

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[edit]

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[edit]

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

Solids with full octahedral symmetry[edit]

Class Name Picture Faces Edges Vertices Dual name Picture
Platonic solid Cube Hexahedron (cube) 6 12 8 Octahedron Octahedron
Archimedean solid
(dual Catalan solid)
Cuboctahedron Cuboctahedron.png 14 24 12 Rhombic dodecahedron Rhombicdodecahedron.jpg
Truncated cube Truncated hexahedron.png 14 36 24 Triakis octahedron Triakisoctahedron.jpg
Truncated octahedron Truncated octahedron.png 14 36 24 Tetrakis hexahedron Tetrakishexahedron.jpg
Rhombicuboctahedron Small rhombicuboctahedron.png 26 48 24 Deltoidal icositetrahedron Deltoidalicositetrahedron.jpg
Truncated cuboctahedron Great rhombicuboctahedron.png 26 72 48 Disdyakis dodecahedron Disdyakisdodecahedron.jpg
Regular
compound
polyhedron
Stella octangula Stella octangula.png 8 12 8 Self-dual
cube and octahedron Compound of cube and octahedron.png 14 24 14 Self-dual

See also[edit]

References[edit]

  • Peter R. Cromwell, Polyhedra (1997), p.295
  • The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
  • N.W. Johnson: Geometries and Transformations, (2015) Chapter 11: Finite symmetry groups

External links[edit]