# Truncated octahedron

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 G_{IV}(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.^{[1]}

Its dual polyhedron is the tetrakis hexahedron. If the original truncated octahedron has unit edge length, its dual tetrakis hexahedron has edge lengths 9/8√2 and 3/2√2.

A truncated octahedron is constructed from a regular octahedron with side length 3*a* by the removal of six right square pyramids, one from each point. These pyramids have both base side length (*a*) and lateral side length (*e*) of *a*, to form equilateral triangles. The base area is then *a*^{2}. Note that this shape is exactly similar to half an octahedron or Johnson solid J_{1}.

From the properties of square pyramids, we can now find the slant height, *s*, and the height, *h*, of the pyramid:

Because six pyramids are removed by truncation, there is a total lost volume of √2*a*^{3}.

The *truncated octahedron* has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: Hexagon, and square. The last two correspond to the B_{2} and A_{2} Coxeter planes.

The truncated octahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

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 (± 1/√3, ± 1/√3, ± 1/√3). The dot product between pairs of two face normals is the cosine of the dihedral angle between adjacent faces, either −1/3 or −1/√3. 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 be dissected into a central octahedron, surrounded by 8 triangular cupolae on each face, and 6 square pyramids above the vertices.^{[2]}

Removing the central octahedron and 2 or 4 triangular cupolae creates two Stewart toroids, with dihedral and tetrahedral symmetry:

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 *x* + *y* + *z* + *w* = 10. 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 surface area *S* and the volume *V* of a truncated octahedron of edge length *a* are:

There are two uniform colorings, with tetrahedral symmetry and octahedral symmetry, and two 2-uniform coloring with dihedral symmetry as a *truncated triangular antiprism*. The constructional names are given for each. Their Conway polyhedron notation is given in parentheses.

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.^{[3]}

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

This polyhedron is a member of a sequence of uniform patterns with vertex figure (4.6.2*p*) 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 4.2*n*.2*n*, extending into the hyperbolic plane:

The *truncated octahedron* (bitruncated cube), is first in a sequence of bitruncated hypercubes:

It is possible to slice a tesseract by a hyperplane so that its sliced cross-section is a truncated octahedron.^{[4]}

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

The cell-transitive bitruncated cubic honeycomb can also be seen as the Voronoi tessellation of the body-centered cubic lattice. The truncated octahedron is one of five three-dimensional primary parallelohedra.

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.^{[5]} It has book thickness 3 and queue number 2.^{[6]}

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].^{[7]}