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.

It is the only Johnson solid that is constructed using both diminishing and augmenting, assuming that no diminishing and augmenting cancel each other out.

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:


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