Snub cube

The snub cube or snic, also called the snub cuboctahedron, is one of the 13 Archimedean solids. Its surface consists of 24 snub or scalene triangles, 8 triangles, and 6 squares, with four triangles and one square meeting at each vertex. It can be obtained by alternation of the great rhombicuboctahedron, followed by adjustment of edge lengths to be all equal.

This is one of nine uniform snub polyhedra generated with one set of digonal faces.

Measures

 * The circumradius of the snub cube with unit edge length is the largest real root of 32x6–80x4+44x2–7.
 * Its volume is the largest real root of 729x6–45684x4+19386x2–12482.
 * The dihedral angles between two triangular faces equal acos(ξ1), where ξ1 ≈ –0.89286 equals the real root of 27x3–9x2–15x+13.
 * The dihedral angles between a square face and a triangular face equal acos(ξ2), where ξ2 ≈ –0.79846 equals the negative real root of 27x6–99x4+129x2-49.

Vertex coordinates
A snub cube of edge length 1, centered at the origin, has coordinates given by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes, of
 * (c1, c2, c3),

where


 * $$c_1=\sqrt{\frac{1}{12}\left(4-\sqrt[3]{17+3\sqrt{33}}+\sqrt[3]{-17+3\sqrt{33}}\right)},$$
 * $$c_2=\sqrt{\frac{1}{12}\left(2+\sqrt[3]{17+3\sqrt{33}}-\sqrt[3]{-17+3\sqrt{33}}\right)},$$
 * $$c_3=\sqrt{\frac{1}{12}\left(4+\sqrt[3]{199+3\sqrt{3}}+\sqrt[3]{199-3\sqrt{3}}\right)}.$$