Triangular hebesphenorotunda

92nd Johnson solid (20 faces)

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	C_3v
Dual polyhedron	-
Properties	convex
Net

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 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 $J_{92}$ .^[3] It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids.^[4]

The surface area of a triangular hebesphenorotunda of edge length $a$ as:^[2]

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]

V={\frac {1}{6}}\left(15+7{\sqrt {5}}\right)a^{3}\approx 5.10875a^{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:

{\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,

\tau

is denoted as the golden ratio.^[5]

References

^ Johnson, N. W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, S2CID 122006114, Zbl 0132.14603.
^ ^a ^b ^c Berman, M. (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute, 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8, MR 0290245.
^ Francis, D. (August 2013), "Johnson solids & their acronyms", Word Ways, 46 (3): 177.
^ Cromwell, P. R. (1997), Polyhedra, Cambridge University Press, p. 87, ISBN 978-0-521-66405-9.
^ Timofeenko, A. V. (2009), "The non-Platonic and non-Archimedean noncomposite polyhedra", Journal of Mathematical Science, 162 (5): 717, doi:10.1007/s10958-009-9655-0, S2CID 120114341.