Augmented tridiminished icosahedron

The augmented tridiminished icosahedron, or auteddi, is one of the 92 Johnson solids (J64). It consists of 1+3+3 triangles and 3 pentagons. It can be constructed by attaching a tetrahedron, seen as a triangular pyramid, to the triangular face of the tridiminished icosahedron that is connected only to pentagons.

Vertex coordinates
An augmented tridiminished icosahedron of edge length 1 has the following vertices:
 * $$\left(0,\,0,\,\frac{\sqrt3+2\sqrt6+\sqrt{15}}{6}\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,\frac{\sqrt3+\sqrt{15}}{6}\right),$$
 * $$\left(0,\,\frac{\sqrt3}{3},\,\frac{\sqrt3+\sqrt{15}}{6}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,-\frac{\sqrt3+\sqrt{15}}{12},\,0\right),$$
 * $$\left(0,\,\frac{\sqrt3+\sqrt{15}}{6},\,0\right),$$
 * $$\left(±\frac12,\,\frac{\sqrt3}{6},\,-\frac{\sqrt3}{3}\right),$$
 * $$\left(0,\, -\frac{\sqrt3}{3},\,-\frac{\sqrt3}{3}\right).$$

Alternatively, orienting it so that it is derived from the vertices of a regular icosahedron, we obtain:


 * $$\left(0,\,\frac12,\,\frac{1+\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac12,\,-\frac{1+\sqrt5}{4}\right),$$
 * $$\left(\frac12,\,\frac{1+\sqrt5}{4},\,0\right),$$
 * $$\left(±\frac12,\,-\frac{1+\sqrt5}{4},\,0\right),$$
 * $$\left(\frac{1+\sqrt5}{4},\,0,\,\frac12\right),$$
 * $$\left(-\frac{1+\sqrt5}{4},\,0,\,±\frac12\right),$$
 * $$\left(\frac{3+4\sqrt2+\sqrt5}{12},\,\frac{3+4\sqrt2+\sqrt5}{12},\,\frac{3+4\sqrt2+\sqrt5}{12}\right).$$