Snub Cube

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The snub cube, also called the cubus simus (Kepler 1619, Weissbach and Martini 2002) or snub cuboctahedron, is an Archimedean solid having 38 faces (32 triangular and 6 square), 60 edges, and 24 vertices. It is a chiral solid, and hence has two enantiomorphous forms known as laevo (left-handed) and dextro (right-handed). A laevo snub dodecahedron is illustrated above together with a wireframe version and a net that can be used for its construction.

It is also the uniform polyhedron with Maeder index 12 (Maeder 1997), Wenninger index 17 (Wenninger 1989), Coxeter index 24 (Coxeter et al. 1954), and Har'El index 17 (Har'El 1993). It has Schläfli symbol and Wythoff symbol .

Some symmetric projections of the snub cube are illustrated above.

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

The tribonacci constant is intimately related to the metric properties of the snub cube.

It can be constructed by snubification of a unit cube with outward offset

			(1)
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and twist angle

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Here, the notation indicates a polynomial root and is the tribonacci constant.

An attractive dual of the two enantiomers superposed on one another is illustrated above.

Its dual polyhedron is the pentagonal icositetrahedron, with which it is illustrated above.

Its skeleton is the snub cubical graph, several illustrations of which are illustrated above.

The midradius of the dual and solid and circumradius for a snub cube of unit edge length are given by

			(7)
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The distances from the center to the centroids of the triangular and square faces are given by the unique positive roots to the equations

			(14)
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			(20)
			(21)

The surface area of the snub cube of side length 1 is

(22)

and the volume by

			(23)
			(24)
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The dihedral angles are

			(28)
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			(37)

The angle subtended by an edge from the center is

			(38)
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			(40)
			(41)