Metabigyrate rhombicosidodecahedron

The metabigyrate rhombicosidodecahedron, or mabgyrid, is one of the 92 Johnson solids (J74). It consists of 2+2+2+2+4+4+4 triangles, 1+1+2+2+4+4+4+4+4+4 squares, and 2+2+2+2+4 pentagons. It can be constructed by rotating two non-opposite pentagonal cupolaic caps of the small rhombicosidodecahedron by 36°.

Vertex coordinates
A metabigyrate rhombicosidodecahedron of edge length 1 has vertices given by:
 * $$\left(±\frac{5+\sqrt5}{4},\,0,\,±\frac{3+\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac{3+\sqrt5}{4},\,-\frac{5+\sqrt5}{4}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,0\right),$$
 * $$\left(±\frac12,\,±\frac12,\,±\frac{2+\sqrt5}{2}\right),$$
 * $$\left(±\frac{2+\sqrt5}{2},\,±\frac12,\,±\frac12\right),$$
 * $$\left(±\frac12,\,±\frac{2+\sqrt5}{2},\,-\frac12\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{2}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,-\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{5+4\sqrt5}{10},\,\frac{10+3\sqrt5}{10}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac{5+2\sqrt5}{5},\,\frac{15+\sqrt5}{20}\right),$$
 * $$\left(0,\,±\frac{15+13\sqrt5}{20},\,\frac{5+\sqrt5}{20}\right).$$