Truncated Cube

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The truncated cube is the 14-faced Archimedean solid with faces . It is also the uniform polyhedron with Maeder index 9 (Maeder 1997), Wenninger index 8 (Wenninger 1989), Coxeter index 21 (Coxeter et al. 1954), and Har'El index 14 (Har'El 1993). It has Schläfli symbol t and Wythoff symbol . It is illustrated above together with a wireframe version and a net that can be used for its construction.

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

The truncated cube is the convex hull of the great cubicuboctahedron, great rhombihexahedron, and quasirhombicuboctahedron uniform polyhedra.

The dual polyhedron of the truncated cube is the small triakis octahedron, 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 unit edge length () are

			(1)
			(2)
			(3)

The distances from the center of the solid to the centroids of the triangular and octagonal faces are

			(4)
			(5)

The surface area and volume are

			(6)
			(7)

The unit truncated cube has Dehn invariant

			(8)
			(9)
			(10)

(OEIS A377296), where the first expression uses the basis of Conway et al. (1999, p. 331).