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Archimedean solid

In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. 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. Like the cube, it can tessellate (or "pack") 3 dimensional space, as a permutohedron. The truncated octahedron was called the "mecon" by Buckminster Fuller. Its dual polyhedron is the tetrakis hexahedron. If the original truncated octahedron has unit edge length, its dual tetrakis hexahedron has edge lengths and .

Orthogonal projection in bounding box(±2,±2,±2)Truncated octahedron with hexagons replaced by 6 coplanar triangles. There are 8 new vertices at: (±1,±1,±1). Truncated octahedron subdivided into as a topological rhombic triacontahedron
All permutations of (0, ±1, ±2) are Cartesian coordinates of the vertices of a truncated octahedron of edge length a = √2 centered at the origin. The vertices are thus also the corners of 12 rectangles whose long edges are parallel to the coordinate axes. The edge vectors have Cartesian coordinates (0, ±1, ±1) and permutations of these. The face normals (normalized cross products of edges that share a common vertex) of the 6 square faces are (0, 0, ±1), (0, ±1, 0) and (±1, 0, 0). The face normals of the 8 hexagonal faces are The dot product between pairs of two face normals is the cosine of the dihedral angle between adjacent faces, either − or −. The dihedral angle is approximately 1.910633 radians (109.471° ) at edges shared by two hexagons or 2.186276 radians (125.263° ) at edges shared by a hexagon and a square.

The truncated octahedron can also be represented by even more symmetric coordinates in four dimensions: all permutations of (1, 2, 3, 4) form the vertices of a truncated octahedron in the three dimensional subspace Therefore, the truncated octahedron is the permutohedron of order 4: each vertex corresponds to a permutation of (1, 2, 3, 4) and each edge represents a single pairwise swap of two elements.

The truncated octahedron exists in the structure of the faujasite crystals.

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

This polyhedron is a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter–Dynkin diagram For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedra), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling. The truncated octahedron is topologically related as a part of sequence of uniform polyhedra and tilings with vertex figures n.6.6, extending into the hyperbolic plane:

The truncated octahedron is topologically related as a part of sequence of uniform polyhedra and tilings with vertex figures 4.2n.2n, extending into the hyperbolic plane:

In the mathematical field of graph theory, a truncated octahedral graph is the graph of vertices and edges of the truncated octahedron. It has 24 vertices and 36 edges, and is a cubic Archimedean graph. It has book thickness 3 and queue number 2. 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].