Octahedral symmetry

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.
The group of orientation-preserving symmetries is "S"4, 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 faces of the octahedron.
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.
As the hyperoctahedral group of dimension 3 the full octahedral group is the wreath product formula_1,<br>and a natural way to identify its elements is as pairs formula_2 with formula_3 and formula_4.<br>But as it is also the direct product formula_5, one can simply identify the elements of tetrahedral subgroup "Td" as formula_6 and their inversions as formula_7.
So e.g. the identity formula_8 is represented as formula_9 and the inversion formula_10 as formula_11.<br>
formula_12 is represented as formula_13 and formula_14 as formula_15.
A rotoreflection is a combination of rotation and reflection.
Chiral octahedral symmetry.
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 "S"4, 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.
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 "S"4."C"2, 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.
With the 4-fold axes as coordinate axes, a fundamental domain of Oh is given by 0 ≤ "x" ≤ "y" ≤ "z". 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 3x3 permutation matrices and assign a + sign or a - sign to each of the three 1s. There are 6 permutations x 8 sign combinations = 48 matrices in total giving the full octahedral group. There are exactly 24 matrices with determinant = +1 and these are the rotation matrices of the chiral octahedral group. The other 24 matrices correspond to a reflection or inversion.
Three reflectional generator matices are needed for octahedral symmetry, which represent the three mirrors of a Coxeter-Dynkin diagram. The product of the reflections produce 3 rotational generators.
The isometries of the cube.
The cube has 48 isometries (symmetry elements), forming the symmetry group "O""h", isomorphic to "S"4 × "C"2. 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, 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.