Disnub octahedron

The disnub octahedron, daso, or compound of eight octahedra with rotational freedom is a uniform polyhedron compound. It consists of 16+48 triangles (the set of 16 form coplanar pairs, forming 8 hexagrams), with 4 triangles joining at each vertex.

This compound has rotational freedom, represented by an angle θ. At θ = 0º, all eight octahedra coincide. We rotate these octahedra around their 3-fold axes of symmetry (2 each, in opposite directions). At θ = acos((3√5–1)/8) ≈ 44.47751º, lateral triangle planes also coincide in pairs, forming the hexagrammatic disnub octahedron. At θ = 60º octahedra coincide in pairs forming a double cover of the snub octahedron.

Variations with θ < 44.47751º are sometimes referred to as the inner disnub octahedron or idso, while cases with 44.47751º < θ < 60º are called the outer disnub octahedron or odso.

Vertex coordinates
The vertices of a disnub octahedron of edge length 1 and rotation angle θ are given by all permutations of:
 * (±($\sqrt{2}$–$\sqrt{6}$cos(θ)+$\sqrt{2}$sin(θ))/6, ±($\sqrt{2}$–$\sqrt{2}$cos(θ)–$\sqrt{6}$sin(θ))/6, ±($\sqrt{2}$+2$\sqrt{2}$cos(θ))/6)