Disnub tetrahedron

The disnub tetrahedron, dissit, or compound of four octahedra with rotational freedom is a uniform polyhedron compound. It consists of 8+24 triangles, with 4 triangles joining at each vertex.

This compound has rotational freedom, represented by an angle θ. At θ = 0°, all four octahedra coincide. We rotate these octahedra around their 3-fold axes of symmetry. At θ = $$\arccos\left(\frac{3\sqrt5-1}{8}\right) ≈ 44.47751°$$, lateral triangle planes coincide in pairs, forming the hexagrammatic disnub tetrahedron. At θ = 60° the compound gains double symmetry and forms the snub octahedron.

Variations with θ < 44.47751° are sometimes referred to as the inner disnub tetrahedron or idsit, while cases with 44.47751° < θ < 60° are called the outer disnub tetrahedron or odsit.

Its quotient prismatic equivalent is the triangular antiprismatic pyritotetrahedroorthowedge, which is six-dimensional.

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