Parabigyrate rhombicosidodecahedron

The parabigyrate rhombicosidodecahedron, or pabgyrid, is one of the 92 Johnson solids (J73). It consists of 10+10 triangles, 10+10+10 squares, and 2+10 pentagons. It can be constructed by rotating two opposite pentagonal cupolaic caps of the small rhombicosidodecahedron by 36°.

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