Disnub icosahedron

The disnub icosahedron, dasi, or compound of twenty octahedra is a uniform polyhedron compound. It consists of 40+120 triangles. The vertices coincide in pairs, and thus eight triangles join at each vertex.

This compound is a special case of the more general altered disnub icosahedron, with θ = $$\arccos\left(\sqrt{\frac{-1+3\sqrt5+3\sqrt{22+10\sqrt5}}{8}\right) ≈ 14.33033°$$. It has the same edges as the uniform great dirhombicosidodecahedron.

Its quotient prismatic equivalent is the triangular antiprismatic icosayodakoorthowedge, which is 22-dimensional.

Vertex coordinates
The vertices of a disnub icosahedron of edge length 1 are given by all even permutations of:
 * $$\left(±\sqrt{\frac{\sqrt5-1-2\sqrt{\sqrt5-2}}{2}},\,±\sqrt{\frac{3-\sqrt5-\sqrt{10\sqrt5-22}}{8}},\,±\sqrt{\frac{2+\sqrt{2\sqrt5-2}}{8}}\right),$$
 * $$\left(0,\,±\frac{\sqrt{3-\sqrt5}}{2},\,±\frac{\sqrt{\sqrt5-1}}{2}\right),$$
 * $$\left(±\sqrt{\frac{3-\sqrt5+\sqrt{10\sqrt5-22}}{8}},\,±\sqrt{\frac{2-\sqrt{2\sqrt5-2}}{8}},\,±\sqrt{\frac{\sqrt5-1+2\sqrt{\sqrt5-2}}{8}}\right).$$