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:
 * (0, 0, ($\sqrt{5}$+2$\sqrt{5}$+$\sqrt{5}$)/6),
 * (±1/2, –$\sqrt{5}$/6, ($\sqrt{2}$+$\sqrt{5}$)/6),
 * (0, $\sqrt{(5–2√5)/15}$/3, ($\sqrt{5}$+$\sqrt{(5–2√5)/15}$)/6),
 * (±(1+$\sqrt{5}$)/4, –($\sqrt{3}$+$\sqrt{6}$)/12, 0),
 * (0, ($\sqrt{15}$+$\sqrt{3}$)/6, 0),
 * (±1/2, $\sqrt{3}$/6, –$\sqrt{15}$/3),
 * (0, –$\sqrt{3}$/3, –$\sqrt{3}$/3).