Paragyrate diminished rhombicosidodecahedron

The paragyrate diminished rhombicosidodecahedron, or pagydrid, is one of the 92 Johnson solids (J77). It consists of 5+5+5 triangles, 5+5+5+10 squares, 1+5+5 pentagons, and 1 decagon. It can be constructed by removing one of the pentagonal cupolaic caps of the small rhombicosidodecahedron, and rotating the opposite cap by 36°.

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