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92nd Johnson solid (20 faces) From Wikipedia, the free encyclopedia
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.
Triangular hebesphenorotunda | |
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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 | |
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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 the Johnson solid, enumerated the last one .[3] It is elementary polyhedra, 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 as:[2] and its volume as:[2]
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, is denoted as the golden ratio.[5]
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