Disnub cuboctahedron

The disnub cuboctahedron, disco, or compound of two snub cubes is a uniform polyhedron compound. It consists of 48 snub triangles, 16 further triangles, and 12 squares (the latter two can combine in pairs due to faces in the same plane). Four triangles and one square join at each vertex.

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

Its volume V ≈ 15.77896 is given by twice 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 has coordinates given by all permutations of:
 * (±c1, ±c2, ±c3),

where


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