Gyrate bidiminished rhombicosidodecahedron

The gyrate bidiminished rhombicosidodecahedron is one of the 92 Johnson solids (J82). It consists of 4×1+3×2 triangles, 4×1+8×2 squares, 4×1+3×2 pentagons, and 2 decagons. It can be constructed by removing two non-opposite pentagonal cupolaic caps of the small rhombicosidodecahedron, and rotating a third cap by 36°.

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