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.

Triangular hebesphenorotunda
Type	Johnson J91 – J92 – J1
Faces	13 triangles 3 squares 3 pentagons 1 hexagon
Edges	36
Vertices	18
Vertex configuration	3(33.5) 6(3.4.3.5) 3(3.5.3.5) 2.3(32.4.6)
Symmetry group	C3v
Dual polyhedron	-
Properties	convex, elementary
Net

3D model of a triangular hebesphenorotunda

Properties

The triangular hebesphenorotunda is named by Johnson (1966), with the prefix hebespheno- referring 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. 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.^[1] Therefore, the triangular hebesphenorotunda has 20 faces: 13 equilateral triangles, 3 squares, 3 regular pentagons, and 1 regular hexagon.^[2] The faces are all regular polygons, categorizing the triangular hebesphenorotunda as a Johnson solid, enumerated the last one ${\displaystyle J_{92}}$.^[3] It is an elementary polyhedron, meaning that it cannot be separated by a plane into two small regular-faced polyhedra.^[4]

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

Cartesian coordinates

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