Triangular hebesphenorotunda explained

Type:	Johnson
Faces:	13 triangles 3 squares 3 pentagons 1 hexagon
Edges:	36
Vertices:	18
Dual:	-
Properties:	convex
Net:	Johnson solid 92 net.png

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 - 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:$$A\; =\; \backslash left(3+\backslash frac\backslash sqrt\backslash right)a^2\; \backslash approx\; 16.389a^2,$$and its volume as:$$V\; =\; \backslash frac\backslash left(15+7\backslash sqrt\backslash right)a^3\backslash approx5.10875a^3.$$

Cartesian coordinates

The triangular hebesphenorotunda with edge length

\sqrt{ 5 } - 1 can be constructed by the union of the orbits of the Cartesian coordinates: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