Truncated octahedron

The truncated octahedron or toe is one of the 13 Archimedean solids. It consists of 6 squares and 8 ditrigons. Each vertex joins one square and two hexagons. As the name suggests, it can be obtained by the truncation of the octahedron.

It is the only Archimedean solid that can tile 3D space by itself. This results in the bitruncated cubic honeycomb.

It can be alternated into the icosahedron after all edge lengths are made equal.

Vertex coordinates
A truncated octahedron of edge length 1 has vertex coordinates given by all permutations of
 * (±$\sqrt{2}$, ±$\sqrt{3}$/2, 0).

Representations
A truncated octahedron has the following Coxeter diagrams:


 * o4x3x (full symmetry)
 * x3x3x (A3 symmetry, as great rhombitetratetrahedron)
 * s4x3x (as hexagon-alternated great rhombicuboctahedron)
 * xuxux4ooqoo&#xt (BC2 axial, square-first)
 * xxux3xuxx&#xt (A2 axial, hexagon-first)
 * Qqo xux4ooq&#zx (BC2×A1 symmetry)
 * xu(wx)(wx)ux oq(oQ)(oQ)qo&#xt (A1×A1 axial, edge-first)
 * xu(xd)ux xu(dx)ux&#xt (square-first when seen as rectangle)

Great rhombitetratetrahedron
The truncated octahedron is also the omnitruncate of the A3 family, where it is known as the great rhombitetratetrahedron or the truncated tetratetrahedron. In this subsymmetry, the 8 hexagons split into 2 sets of 4, with the squares connecting to 2 in each set.

This is one of three Wythoffian non-prismatic polyhedra whose Coxeter diagrams are all ringed, the other two being the great rhombicuboctahedron and the great rhombicosidodecahedron. It can be represented as x3x3x.