Great dirhombicosidodecahedron

The great dirhombicosidodecahedron, or gidrid, is the most complex uniform hemipolyhedron. It consists of 40 triangles, 60 snub squares, and 24 pentagrams, all of which fall into coinciding planes in pairs. Two triangles, four squares, and two pentagrams meet at each vertex.

Vertex coordinates
A great dirhombicosidodecahedron of edge length 1 has vertex coordinates given by all even permutations of:
 * $$\left(±\sqrt{\frac{\sqrt5-1-2\sqrt{\sqrt5-2}}{8}},\,±\sqrt{\frac{3-\sqrt5-\sqrt{10\sqrt5-22}}{8}},\,±\sqrt{\frac{2+\sqrt{2\sqrt5-2}}{8}}\right),$$
 * $$\left(0,\,±\frac{\sqrt{3-\sqrt5}}{2},\,±\frac{\sqrt{\sqrt5-1}}{2}\right),$$
 * $$\left(±\sqrt{\frac{3-\sqrt5+\sqrt{10\sqrt5-22}}{8}},\,±\sqrt{\frac{2-\sqrt{2\sqrt5-2}}{8}},\,±\sqrt{\frac{\sqrt5-1+2\sqrt{\sqrt5-2}}{8}}\right).$$

Related polyhedra
The great dirhombicosidodecahedron has the same edges as two uniform polyhedron compounds: the disnub icosahedron, the compound of 20 octahedra, and the snub pseudosnub rhombicosahedron, the compound of 20 tetrahemihexahedra. It also has the same vertices of the great snub dodecicosidodecahedron, which uses a subset of its edges.

The degenerate great disnub dirhombidodecahedron can be constructed by blending this polyhedron with the disnub icosahedron.