Tridiminished icosahedron

The tridiminished icosahedron is one of the 92 Johnson solids (J63). It consists of 1+1+3 triangles and 3 pentagons. It can be constructed by removing 3 mutually non-adjacent vertices from a regular icosahedron.

Vertex coordinates
A tridiminished icosahedron of edge length 1 has the following vertices:
 * $$\left(0,\,\frac12,\,\frac{1+\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac12,\,-\frac{1+\sqrt5}{4}\right),$$
 * $$\left(\frac12,\,\frac{1+\sqrt5}{4},\,0\right),$$
 * $$\left(±\frac12,\,-\frac{1+\sqrt5}{4},\,0\right),$$
 * $$\left(\frac{1+\sqrt5}{4},\,0,\,\frac12\right),$$
 * $$\left(-\frac{1+\sqrt5}{4},\,0,\,±\frac12\right).$$

These are the vertices of an icosahedron, but with three missing.

An alternate set of coordinates can be given in a way that positions the tridiminished icosahedron within the symmetry axis:
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,\frac{\sqrt3+\sqrt{15}}{6}\right),$$
 * $$\left(0,\,\frac{\sqrt3}{3},\,\frac{\sqrt3+\sqrt{15}}{6}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,-\frac{\sqrt3+\sqrt{15}}{12},\,0\right),$$
 * $$\left(0,\,\frac{\sqrt3+\sqrt{15}}{6},\,0\right),$$
 * $$\left(±\frac12,\,\frac{\sqrt3}{6},\,-\frac{\sqrt3}{3}\right),$$
 * $$\left(0,\, -\frac{\sqrt3}{3},\,-\frac{\sqrt3}{3}\right).$$

Related polyhedra
A tetrahedron can be attached to the tridiminished icosahedron at the triangular face surrounded by pentagons to form the augmented tridiminished icosahedron.

In vertex figures
The tridiminished icosahedron is the vertex figure of the uniform snub disicositetrachoron.