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.

The triangles come in two types. One group of 8 has full triangle symmetry and joins only to other triangles. The other set of 24 has no local symmetry and connects to the square faces, and are the snub faces of the polyhedron.

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

Measures
The circumradius R ≈ 1.34371 of the snub cube with unit edge length is the largest real root of
 * $$32x^6-80x^4+44x^2-7.$$

Its volume V ≈ 7.88948 is given by the largest real root of
 * $$729x^6-45684x^4+19386x^2-12482$$.

Its dihedral angles can be given as acos(α) for the angle between two triangular faces, and acos(β) for the angle between a square face and a triangular face, where α ≈ –0.89286 equals the unique real root of
 * $$27x^3-9x^2-15x+13,$$

and β ≈ –0.79846 equals the unique negative real root of
 * $$27x^6-99x^4+129x^2-49.$$

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


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

Variations
The snub cube has a general variant that maintains the chiral cubic symmetry. It general has 6 squares of one edge size, 8 triangles of a second size, and 24 scalene triangles using the first two edge sizes along with a third edge size, that joins two scalene triangle faces.

The most notable of these variants is the one obtained as the direct alternation of the uniform great rhombicuboctahedron. This variant uses 6 squares of size $$\sqrt{2+\sqrt2}$$, 8 triangles of size $$\sqrt3$$, and 24 scalene triangles, with one of each of these edges along with edges joining them of length $$\sqrt2$$, as faces.

Related polyhedra
The disnub cuboctahedron is a uniform polyhedron compound that consists of the two opposite chiral forms of the snub cube.

It is also related to the cuboctahedron and small rhombicuboctahedron through a twisting operation. Twisting the faces of the small rhombicuboctahedron so the edge-squares become pairs of triangles results in the snub cube. Continuing the twisting until those triangles become edges results in a cuboctahedron.