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.

The vertex locations of this honeycomb are known as the face-centered cubic or FCC lattice, which has the important property that placing spheres at each of the points that touch each other results in a maximally dense packing of equal spheres. (There are infinitely many cubic close packings, but the FCC lattice has the highest symmetry.) This geometric property makes the FCC lattice ubiquitous in chemistry, such as in the structure of sodium chloride crystals (as found in table salt).

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:


 * (full symmetry)
 * (P4 symmetry, cyclotetrahedral honeycomb)
 * (as alternated cubic honeycomb)
 * (as alternated square prismatic honeycomb)
 * (as alternated product of three diapeirogons)
 * (as alternated square prismatic honeycomb)
 * (as alternated product of three diapeirogons)
 * (as alternated product of three diapeirogons)