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 = 0º, all four octahedra coincide. We rotate these octahedra around their 3-fold axes of symmetry. At θ = acos((3√5–1)/8) ≈ 44.47751º, lateral triangle planes coincide in pairs, forming the hexagrammatic disnub tetrahedron. At θ = 60º the compound gains double symmetry and forms the snub octahedron.

Variations with θ < 44.47751º are sometimes referred to as the inner disnub tetrahedron or idsit, while cases with 44.47751º < θ < 60º 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:
 * (±($\sqrt{2}$–$\sqrt{6}$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)