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 = 0°, all eight octahedra coincide. We rotate these octahedra around their 3-fold axes of symmetry (2 each, in opposite directions). At θ = $$\arccos\left(\frac{3\sqrt5-1}{8}\right) ≈ 44.47751º$$, lateral triangle planes also coincide in pairs, forming the hexagrammatic disnub octahedron. At θ = 60° octahedra coincide in pairs forming a double cover of the snub octahedron.

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

Its quotient prismatic equivalent is the triangular antiprismatic octaexoorthowedge, which is ten-dimensional.

Vertex coordinates
The vertices of a disnub octahedron of edge length 1 and rotation angle θ are given by all permutations of:
 * $$\left(±\frac{\sqrt2-\sqrt2\cos(\theta)+\sqrt6\sin(\theta)}{6},\,±\frac{\sqrt2-\sqrt2\cos(\theta)-\sqrt6\sin(\theta)}{6},\,±\frac{\sqrt2+2\sqrt2\cos(\theta)}{6}\right).$$