Triangular hebesphenorotunda



In geometry, the triangular hebesphenorotunda is a Johnson solid with 13 equilateral triangles, 3 squares, 3 regular pentagons, and 1 regular hexagon, making the total of its faces is 20.

Properties
The triangular hebesphenorotunda is named by, with the prefix hebespheno- referring to a blunt wedge-like complex formed by three adjacent lunes&mdash;a figure where two equilateral triangles are attached at the opposite sides of a square. The suffix (triangular) -rotunda refers to the complex of three equilateral triangles and three regular pentagons surrounding another equilateral triangle, which bears a structural resemblance to the pentagonal rotunda. Therefore, the triangular hebesphenorotunda has 20 faces: 13 equilateral triangles, 3 squares, 3 regular pentagons, and 1 regular hexagon. The faces are all regular polygons, categorizing the triangular hebesphenorotunda as the Johnson solid, enumerated the last one $$ J_{92} $$. It is elementary, meaning that it cannot be separated by a plane into two small regular-faced polyhedra.

The surface area of a triangular hebesphenorotunda of edge length $$ a $$ as: $$ A = \left(3+\frac{1}{4}\sqrt{1308+90\sqrt{5}+114\sqrt{75+30\sqrt{5}}}\right)a^2 \approx 16.389a^2, $$ and its volume as: $$ V = \frac{1}{6}\left(15+7\sqrt{5}\right)a^3\approx5.10875a^3. $$

Cartesian coordinates
The triangular hebesphenorotunda with edge length $$ \sqrt{5} - 1 $$ can be constructed by the union of the orbits of the Cartesian coordinates: $$ \begin{align} \left( 0,-\frac{2}{\tau\sqrt{3}},\frac{2\tau}{\sqrt{3}} \right), \qquad &\left( \tau,\frac{1}{\sqrt{3}\tau^2},\frac{2}{\sqrt{3}} \right) \\ \left( \tau,-\frac{\tau}{\sqrt{3}},\frac{2}{\sqrt{3}\tau} \right), \qquad &\left(\frac{2}{\tau},0,0\right), \end{align} $$ under the action of the group generated by rotation by 120° around the z-axis and the reflection about the yz-plane. Here, $$ \tau $$ is denoted as the golden ratio.