Triangular hebesphenorotunda
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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 Script error: No such module "Footnotes"., 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.[1]Template:R/superscript Therefore, the triangular hebesphenorotunda has twenty faces: thirteen equilateral triangles, three squares, three regular pentagons, and one regular hexagon.[2]Template:R/superscript The faces are all regular polygons, categorizing the triangular hebesphenorotunda as a Johnson solid, enumerated the last one .[3]Template:R/superscript It is an elementary polyhedron, meaning that it cannot be separated by a plane into two small regular-faced polyhedra.[4]Template:R/superscript
The surface area of a triangular hebesphenorotunda of edge length is:[2]Template:R/superscript and its volume is:[2]Template:R/superscript
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.[5]Template:R/superscript