A common alternative that requires less volume is the truncated octahedron.

Therefore, the truncated octahedron is the permutohedron of order 4.

The truncated octahedron, with 6 square and 8 hexagonal faces.

The truncated octahedron is one of five three-dimensional primary parallelohedra.

The truncated octahedron, which also tessellates 3-dimensional space, is the only permutohedron.

Since each of its faces has point symmetry the truncated octahedron is a zonohedron.

For example, a bitruncated cube is a truncated octahedron.

Its dual is the truncated octahedron, an Archimedean solid.

These coordinates represent the truncated octahedron with alternated vertices deleted.

It can also be constructed as an alternated truncated octahedron, having pyritohedral symmetry.

This polyhedron is similar to the uniform truncated octahedron:

The projection envelope is a truncated octahedron.

The pseudoicosahedron is an alternated truncated octahedron.

This was followed by a DNA truncated octahedron.

The truncated cube and the truncated octahedron are Archimedean solids with 36 edges.

The cuboctahedron, the truncated cube, and the truncated octahedron each have fourteen faces.

Hence, the truncated 16-cell may be thought of as the 4-dimensional analogue of the truncated octahedron.

The area A and the volume V of a truncated octahedron of edge length a are:

The parallel projection of the truncated 24-cell into 3-dimensional space, truncated octahedron first, has the following layout:

Two of the truncated octahedra project onto a truncated octahedron lying in the center of the envelope.

Truncated octahedron (6 squares, 8 regular hexagons)

It has two symmetry constructions, one from the truncated octahedron, and one as an omnitruncation of the tetrahedron.

It shares the same vertex arrangement as a nonuniform truncated octahedron, having irregular hexagons alternating with long and short edges.

The truncated octahedron is one of a family of uniform polyhedra related to the cube and regular octahedron.

The truncated octahedron exists in three different convex uniform honeycombs (space-filling tessellations):