3D model of a 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 Johnson (1966), 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. 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 J 92 {\displaystyle J_{92}}. 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 a {\displaystyle a} is: A = ( 3 + 1 4 1308 + 90 5 + 114 75 + 30 5 ) a 2 ≈ 16.389 a 2 , {\displaystyle 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 is: V = 1 6 ( 15 + 7 5 ) a 3 ≈ 5.10875 a 3 . {\displaystyle V={\frac {1}{6}}\left(15+7{\sqrt {5}}\right)a^{3}\approx 5.10875a^{3}.}

Cartesian coordinates

The triangular hebesphenorotunda with edge length 5 − 1 {\displaystyle {\sqrt {5}}-1} can be constructed by the union of the orbits of the Cartesian coordinates: ( 0 , − 2 τ 3 , 2 τ 3 ) , ( τ , 1 3 τ 2 , 2 3 ) ( τ , − τ 3 , 2 3 τ ) , ( 2 τ , 0 , 0 ) , {\displaystyle {\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, τ {\displaystyle \tau } denotes the golden ratio.

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