The tetrahedral-octahedral honeycomb, also called the alternated cubic honeycomb, is the quasiregular honeycomb of three-dimensional Euclidean space whose cells are regular tetrahedra and regular octahedra (Coxeter 1940, 1973).
Eight tetrahedra and six octahedra meet at every polyhedron vertex, so the vertex
figure is a cuboctahedron. Double-counting
vertex-cell incidences in a periodic repeat unit shows that the numbers of tetrahedral
and octahedral cells occur in the ratio 
. For common edge length 
,
the volumes of these cells are 
and 
, respectively, in the ratio 
. The tetrahedra therefore
occupy one-third of space and the octahedra occupy
two-thirds. At every polyhedron edge, two tetrahedra
and two octahedra meet. This is possible because their
dihedral angles are 
and 
, whose sum is 
.
Its vertex set is the face-centered cubic lattice. Equal spheres centered at these vertices
and tangent along the edges form cubic
close packing. Subtracting these spheres from the
cells gives the tetrahedral and octahedral interstitial regions. The Voronoi
cells of the sphere centers
are rhombic dodecahedra. One regular octahedron
together with the eight regular tetrahedra
attached to its faces forms a stella
octangula. Finite slabs of its 
-skeleton are used as octet
trusses.
