Triangular hebesphenorotunda

The triangular hebesphenorotunda is named by {{harvtxt|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.{{r|johnson}} Therefore, the triangular hebesphenorotunda has 20 faces: 13 equilateral triangles, 3 squares, 3 regular pentagons, and 1 regular hexagon.{{r|berman}} The faces are all regular polygons, categorizing the triangular hebesphenorotunda as a Johnson solid, enumerated the last one $J\_\{92\}$.{{r|francis}} It is an elementary polyhedron, meaning that it cannot be separated by a plane into two small regular-faced polyhedra.{{r|cromwell}}

The surface area of a triangular hebesphenorotunda of edge length $a$ as:{{r|berman}}

$$A\; =\; \backslash left(3+\backslash frac\{1\}\{4\}\backslash sqrt\{1308+90\backslash sqrt\{5\}+114\backslash sqrt\{75+30\backslash sqrt\{5\}\}\}\backslash right)a^2\; \backslash approx\; 16.389a^2,$$

and its volume as:{{r|berman}}

$$V\; =\; \backslash frac\{1\}\{6\}\backslash left(15+7\backslash sqrt\{5\}\backslash right)a^3\backslash approx5.10875a^3.$$

The triangular hebesphenorotunda with edge length $\backslash sqrt\{5\}\; -\; 1$ can be constructed by the union of the orbits of the Cartesian coordinates:

$$\backslash begin\{align\}$$

\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}

under the action of the group generated by rotation by 120° around the z-axis and the reflection about the yz-plane. Here, $\backslash tau$ denotes the golden ratio.{{r|timofeenko}}

{{reflist|refs=

{{citation

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| journal = Journal of the Franklin Institute

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| volume = 291

| year = 1971| issue = 5

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{{citation

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| title = Polyhedra

| year = 1997

| url = https://archive.org/details/polyhedra0000crom/page/87/mode/1up

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{{citation

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{{citation

| last = Johnson | first = N. W. | author-link = Norman Johnson (mathematician)

| title = Convex polyhedra with regular faces

| journal = Canadian Journal of Mathematics

| volume = 18 | pages = 169–200

| year = 1966

| doi = 10.4153/cjm-1966-021-8|mr=0185507

| zbl = 0132.14603

| s2cid = 122006114

| doi-access = free

}}.

{{citation

| last = Timofeenko | first = A. V.

| year = 2009

| title = The non-Platonic and non-Archimedean noncomposite polyhedra

| journal = Journal of Mathematical Science

| volume = 162 | issue = 5 | pages = 717

| doi = 10.1007/s10958-009-9655-0

| s2cid = 120114341

}}.

}}

• {{Mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid | urlname =TriangularHebesphenorotunda| title = Triangular hebesphenorotunda}}

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Category:Elementary polyhedron

Category:Johnson solids