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File:J92 triangular hebesphenorotunda.stl
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.

PropertiesEdit

The triangular hebesphenorotunda is named by Template:Harvtxt, 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.Template:R Therefore, the triangular hebesphenorotunda has 20 faces: 13 equilateral triangles, 3 squares, 3 regular pentagons, and 1 regular hexagon.Template:R The faces are all regular polygons, categorizing the triangular hebesphenorotunda as a Johnson solid, enumerated the last one <math> J_{92} </math>.Template:R It is an elementary polyhedron, meaning that it cannot be separated by a plane into two small regular-faced polyhedra.Template:R

The surface area of a triangular hebesphenorotunda of edge length <math> a </math> as:Template:R <math display="block"> A = \left(3+\frac{1}{4}\sqrt{1308+90\sqrt{5}+114\sqrt{75+30\sqrt{5}}}\right)a^2 \approx 16.389a^2, </math> and its volume as:Template:R <math display="block"> V = \frac{1}{6}\left(15+7\sqrt{5}\right)a^3\approx5.10875a^3. </math>

Cartesian coordinatesEdit

The triangular hebesphenorotunda with edge length <math> \sqrt{5} - 1 </math> can be constructed by the union of the orbits of the Cartesian coordinates: <math display="block"> \begin{align}

\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{align} </math> under the action of the group generated by rotation by 120° around the z-axis and the reflection about the yz-plane. Here, <math> \tau </math> denotes the golden ratio.Template:R

ReferencesEdit

Template:Reflist

External linksEdit

Template:Johnson solids navigator