Truncated octahedron

Truncated octahedral graph
3-fold symmetric schlegel diagram
Vertices	24
Edges	36
Automorphisms	48
Chromatic number	2
Properties	Cubic, Hamiltonian, regular, zero-symmetric

In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces (8 regular hexagonal and 6 square), 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a 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.

Its dual polyhedron is the tetrakis hexahedron.

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

A truncated octahedron is constructed from a regular octahedron with side length 3a 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 a². Note that this shape is exactly similar to half an octahedron or Johnson solid J₁.