The binary tetrahedral and octahedral group
The tetrahedral group and the octahedral group are subgroups of the continuous group of 3-dimensional rotations SO(3). The elements are all rotations that leave an tetrahedron (c.q. octahedron) invariant. There are 12 elements of the tetrahedral group and 24 of the octahedral group. The vertices of a 24-cell and its dual correspond to the 24 elements of the octahedral group. In the stereographic projection of the 24-cell and its dual the rotational axis of a rotational symmetry is given by the line of the origin to a vertex. The angle of rotation ß is given by the length of the line segment i.e. the distance of the vertex to the origin. The relation is:
distance = tan(ß/4) = sin (ß/2)/(1+cos(ß/2))
Thus, if ß = 90° then distance = SQR(2) - 1 = 0.41421 ;
if ß = 180° distance = 1;
if ß = 270° distance = SQR(2) +1 = 2.41421 ;
if ß = 120° distance = SQR(3)/3 = 0.57735 ;
if ß = 240° distance = SQR(3) = 1.73205 .
The hypercube (with edge-length = 1, which implies that its vertices lie on the unit hypersphere of radius 1) on the right gives the threefold symmetry axes with angles of 120° and 240° . One line through the the origin contains 4 vertices. Two opposite ones on the small inner cube correspond to rotations of 120°, having distance SQR(3)/3 to the origin. Two opposites on the outer cube correspond to 240° , having distance SQR(3) to the origin. This results in 8 rotations about threefold axes.
Adding the 8 midpoints of the cubic cells results in a 24-cell:
The vertex in the centre corresponds to the identity and so does the vertex at infinity. The other 6 new vertices correspond in pairs of opposites to an angle of 180°, having distance = 1 to the origin.
This results in 3 rotations about twofold symmetry axes and the identity.
Together with the 8 threefold rotations we get the 12 elements of the tetrahedral group (or the 24 elements of the binary tetrahedral group).
Now we can add the 2x12 = 24 vertices of a dual 24-cell to find the octahedral group elements.
First, we can add rotations of 90° and 270° to the 3 twofold axes of the tetrahedral group.
Two opposite vertices of the small octahedron correspond to rotations of 90°, having distance SQR(2)-1 to the origin.
Two opposite vertices of the big octahedron correspond to rotations of 270° , having distance SQR(2) +1 to the origin.
This results in 6 new rotations about fourfold axes and together with the 3 rotations of tetrahedral group around the same axes completes to 9 rotations about fourfold symmetry axes.
Secondly, let's
add the 6 rotations about twofold symmetry axes.
These axes are no symmetry axes of the tetrahedron as they transform a tetrahedron into its dual. (A tetrahedron and its dual form a stella octangula.)
The corresponding 12 vertices of the 24-cell all lie on a sphere of radius 1. They are the vertices of a cuboctahedron whose vertices lie on the midpoints of the 12 edges of cube.
Two opposite vertices of the cuboctahedron corresponsd to a rotation of 180° , having distance = 1 to the centre.
Resuming results we have:
16 vertices for 16/2 = 8 rotations about a threefold axis,
6+12 = 18 vertices for 18/2 = 9 rotations about a fourfold axis,
12 vertices for 12/2 = 6 rotations about a twofold axis and
2 vertices for 2/2 = 1 identity rotation. The two 24-cells that are dual to each other form a compound that can be compared to the 3-dimensinonal stella octangula consisting of 2 dual tetrahedra. A 24-cell is self-dual like a tetrahedron but does not coincide with it, again like the tetrahedron.
If we let one of two opposite vertices correspond to a negative angle we get a negative distance (= tan(ß/4)). We can identify this with a rotation-reflection. We thus get the 48 elements of the full symmetry group of the octahedron.
