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 -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 twenty faces: thirteen equilateral triangles, three squares, three regular pentagons, and one regular hexagon. The faces are all regular polygons, categorizing the triangular hebesphenorotunda as a Johnson solid, enumerated the last one. It is an elementary polyhedron, 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 is:
and its volume is:
Cartesian coordinates
The triangular hebesphenorotunda with edge length 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, denotes the golden ratio.