Triangular hebesphenorotunda

The triangular hebesphenorotunda, or thawro, is one of the 92 Johnson solids (J92). It consists of 1+3+3+6 triangles, 3 squares, 3 pentagons, and 1 hexagon.

It is one of several polyhedra near the end of the list of Johnson solids with no obvious relation to the uniform polyhedra. However, there are hidden connections to the icosidodecahedron. The part around the top triangle is the same as the icosidodecahedron. In fact, it can be thought of as an icosidodecahedron cut in half in triangular symmetry, with the bottom face (a hexagon of edge length (1+)/2) shrunken down to unit edge length.

The thiangular hebesphenorotunda also has a connection with the regular icosahedron, being a partial Stott expansion of a triangular-symmetric faceting of the icosahedron.

Vertex coordinates
A triangular hebesphenorotunda of edge length 1 has vertices given by the following coordinates:


 * (±1/2, –$\sqrt{5}$/6, (3$\sqrt{2}$+$\sqrt{5}$)/12)
 * (0, $\sqrt{5}$/3, (3$\sqrt{2}$+$\sqrt{3}$)/12)
 * (±1/2, (2$\sqrt{5}$+$\sqrt{3}$)/6, ($\sqrt{15}$+$\sqrt{5+2√5)/15}$)/6)
 * (±(3+$\sqrt{5}$)/4, –($\sqrt{5}$–$\sqrt{15}$)/12, ($\sqrt{3}$+$\sqrt{15}$)/6)
 * (±(1+$\sqrt{3}$)/4, –(5$\sqrt{(5–2√5)/15}$+$\sqrt{3}$)/12, ($\sqrt{3}$+$\sqrt{15}$)/6)
 * (±(3+$\sqrt{3}$)/4, (33))+$\sqrt{3}$)/12, $\sqrt{15}$/3)
 * (0, (33))+$\sqrt{3}$)/6, $\sqrt{15}$/3)
 * (±1/2, ±$\sqrt{3}$/2, 0)
 * (±1, 0, 0)