Tetrahedral-octahedral honeycomb

The tetrahedral-octahedral honeycomb, or octet, also known as the alternated cubic honeycomb, is a convex uniform honeycomb. 6 octahedra and 8 tetrahedra join at each vertex of this honeycomb, with a cuboctahedron as the vertex figure. As one of its names suggests, it can be formed by alternation of the cubic honeycomb. It is also the 3D simplectic honeycomb.

Vertex coordinates
The vertices of a tetrahedral-octahedral honeycomb of edge length 1 are given by
 * $$\left(i\frac{\sqrt2}{2},\,j\frac{\sqrt2}{2},\,k\frac{\sqrt2}{2}\right),$$

where i, j, and k are integers, and i+j+k is even.

Representations
A tetrahedral-octahedral honeycomb has the following Coxeter diagrams:


 * x3o3o *b4o (full symmetry)
 * x3o3o3o3*a (P4 symmetry, cyclotetrahedral honeycomb)
 * s4o3o4o (as alternated cubic honeycomb)
 * s4o3o4s
 * o3o3o *b4s
 * s∞o2s4o4o (as alternated square prismatic honeycomb)
 * s∞o2o4s4o