Triangular hebesphenorotunda

File:J92 triangular hebesphenorotunda.stl

3D model of a triangular hebesphenorotunda

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

Properties

The original name is attributed to , with two affixes. The prefix hebespheno- refers to a blunt wedge-like complex formed by three adjacent lunes, a figure where two equilateral triangles are attached at the opposite sides of a square, whereas 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.Template:R Therefore, the triangular hebesphenorotunda has twenty faces: thirteen equilateral triangles, three squares, three regular pentagons, and one regular hexagon.Template:R The faces are all regular polygons, categorizing the triangular hebesphenorotunda as a Johnson solid, enumerated the last one $J_{92}$.Template:R It is an elementary polyhedron, meaning that it cannot be separated by a plane into two small regular-faced polyhedra.Template:R

The surface area of a triangular hebesphenorotunda of edge length $a$ is:Template:R $$A=(3+\frac{1}{4}\sqrt{1308+90\sqrt{5}+114\sqrt{75+30\sqrt{5}}})a^2\approx 16.389a^2,$$ and its volume is:Template:R $$V=\frac{1}{6}(15+7\sqrt{5})a^3\approx 5.10875a^3.$$

Cartesian coordinates

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

External links

• Template:Mathworld2