Truncated Octahedron

Make Your Own Truncated Octahedron

Print & fold

Print in 3D

The truncated octahedron is the 14-faced Archimedean solid with faces . It is also the uniform polyhedron with Maeder index 8 (Maeder 1997), Wenninger index 7 (Wenninger 1989), Coxeter index 20 (Coxeter et al. 1954), and Har'El index 13 (Har'El 1993). It has Schläfli symbol t and Wythoff symbol . It was called the "mecon" by Buckminster Fuller (Rawles 1997). It is illustrated above together with a wireframe version and a net that can be used for its construction.

Some symmetric projections of the truncated octahedron are illustrated above.

The truncated octahedron has the octahedral group of symmetries. The form of the fluorite () resembles the truncated octahedron (Steinhaus 1999, pp. 207-208).

The truncated octahedron is a space-filling polyhedron (Steinhaus 1999, pp. 187-190 and 207) and therefore has a Dehn invariant of 0.

It is implemented in the Wolfram Language as PolyhedronData["TruncatedOctahedron"] or UniformPolyhedron["TruncatedTetrahedron"]. Precomputed properties are available as PolyhedronData["TruncatedTetrahedron", prop].

The truncated octahedron is the convex hull of the tetragonal disphenoid 6-compound.

The solid of edge length can be formed from an octahedron of edge length via truncation by removing six square pyramids, each with edge slant height , base on a side, and height . The height and base area of the square pyramid are then

			(1)
			(2)
			(3)

and its volume is

			(4)
			(5)

The volume of the truncated octahedron is then given by the volume of the octahedron

			(6)
			(7)

minus six times the volume of the square pyramid,

			(8)
			(9)

The surface area of the truncated octahedron is

(10)

The dual polyhedron of the truncated octahedron is the tetrakis hexahedron, both of which are illustrated above together with their common midsphere.

The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are

			(11)
			(12)
			(13)

The distances from the center of the solid to the centroids of the square and hexagonal faces are given by

			(14)
			(15)