Compound of two snub cubes

Compound of two snub cubes
Type	Uniform compound
Index	UC68
Schläfli symbol	βr{4,3}
Coxeter diagram
Polyhedra	2 snub cubes
Faces	16+48 triangles 12 squares
Edges	120
Vertices	48
Symmetry group	octahedral (Oh)
Subgroup restricting to one constituent	chiral octahedral (O)

This uniform polyhedron compound is a composition of the 2 enantiomers of the snub cube. As a holosnub, it is represented by Schläfli symbol βr{4,3} and Coxeter diagram .

The vertex arrangement of this compound is shared by a convex nonuniform truncated cuboctahedron, having rectangular faces, alongside irregular hexagons and octagons, each alternating with two edge lengths.

Cartesian coordinates

Cartesian coordinates for the vertices are all the permutations of

(±1, ±ξ, ±1/ξ)

where ξ is the real solution to

$\xi^3+\xi^2+\xi=1, \,$

which can be written

$\xi = \frac{1}{3}\left(\sqrt[3]{17+3\sqrt{33}} - \sqrt[3]{-17+3\sqrt{33}} - 1\right)$

or approximately 0.543689. ξ is the reciprocal of the tribonacci constant.

Truncated cuboctahedron

This compound can be seen as the union of the two chiral alternations of a truncated cuboctahedron:

References

• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society 79: 447–457, doi:10.1017/S0305004100052440, MR 0397554.