The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 edges, and 24 vertices.

Since each of its faces has point symmetry the truncated octahedron is a 6-zonohedron.

It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces.

The truncated octahedron was called the "mecon" by Buckminster Fuller.

Setting the edge length of the regular octahedron equal to

(the square pyramid is has four equilateral triangles as faces, the first Johnson solid).

The truncated octahedron is one of the thirteen Archimedean solids.

In other words, it has a highly symmetric and semi-regular polyhedron with two or more different regular polygonal faces that meet in a vertex.

[3] The dual polyhedron of a truncated octahedron is the tetrakis hexahedron.

[5] The dihedral angle of a truncated octahedron between square-to-hexagon is

[6] The Cartesian coordinates of the vertices of a truncated octahedron with edge length 1 are all permutations of[citation needed]

The truncated octahedron can be described as a permutohedron of order 4 or 4-permutohedron, meaning it can be represented with even more symmetric coordinates in four dimensions: all permutations of

form the vertices of a truncated octahedron in the three-dimensional subspace

and each edge represents a single pairwise swap of two elements.

If, instead, the truncated octahedron is labeled by the inverse permutations, the edges correspond to swaps of elements whose positions differ by one.

With this alternative labeling, the edges and vertices of the truncated octahedron form the Cayley graph of the symmetric group

, the group of four-element permtutations, as generated by swaps of consecutive positions.

It is classified as plesiohedron, meaning it can be defined as the Voronoi cell of a symmetric Delone set.

[10] Plesiohedra, translated without rotating, can be repeated to fill space.

There are five three-dimensional primary parallelohedrons, one of which is the truncated octahedron.

[11] More generally, every permutohedron and parallelohedron is a zonohedron, a polyhedron that is centrally symmetric and can be defined by a Minkowski sum.

[12] In chemistry, the truncated octahedron is the sodalite cage structure in the framework of a faujasite-type of zeolite crystals.

[13] In solid-state physics, the first Brillouin zone of the face-centered cubic lattice is a truncated octahedron.

[14] The truncated octahedron (in fact, the generalized truncated octahedron) appears in the error analysis of quantization index modulation (QIM) in conjunction with repetition coding.

[16] Removing the central octahedron and 2 or 4 triangular cupolae creates two Stewart toroids, with dihedral and tetrahedral symmetry: It is possible to slice a tesseract by a hyperplane so that its sliced cross-section is a truncated octahedron.

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

[19] As a Hamiltonian cubic graph, it can be represented by LCF notation in multiple ways: [3, −7, 7, −3]6, [5, −11, 11, 7, 5, −5, −7, −11, 11, −5, −7, 7]2, and [−11, 5, −3, −7, −9, 3, −5, 5, −3, 9, 7, 3, −5, 11, −3, 7, 5, −7, −9, 9, 7, −5, −7, 3].