Editing Triangular hebesphenorotunda

{{Short description|92nd Johnson solid (20 faces)}}
{{Infobox polyhedron
|image=Triangular hebesphenorotunda.png
|type=[[Johnson solid|Johnson]]<br>{{math|[[bilunabirotunda|''J''{{sub|91}}]] – '''''J''{{sub|92}}''' – [[square pyramid|''J''{{sub|1}}]]}}
|faces=13 [[triangle]]s <br> 3 [[Square (geometry)|square]]s <br> 3 [[pentagon]]s <br> 1 [[hexagon]]
|edges=36
|vertices=18
|symmetry={{math|C{{sub|3v}}}}
|vertex_config= {{math|3(3{{sup|3}}.5) <br> 6(3.4.3.5) <br> 3(3.5.3.5) <br> 2.3(3{{sup|2}}.4.6)}}
|dual=-
|properties=[[convex set|convex]], [[Elementary polyhedron|elementary]]
|net=Johnson solid 92 net.png
}}
[[File:J92 triangular hebesphenorotunda.stl|thumb|3D model of a triangular hebesphenorotunda]]

In [[geometry]], the '''triangular hebesphenorotunda''' is a [[Johnson solid]] with 13 [[equilateral triangle]]s, 3 [[Square (geometry)|square]]s, 3 [[regular pentagon]]s, and 1 [[regular hexagon]], meaning the total of its faces is 20.

== Properties ==
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''&mdash;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 triangle]]s, 3 [[Square (geometry)|square]]s, 3 [[regular pentagon]]s, and 1 [[regular hexagon]].{{r|berman}} The faces are all [[regular polygon]]s, categorizing the triangular hebesphenorotunda as a [[Johnson solid]], enumerated the last one <math> J_{92} </math>.{{r|francis}} It is an [[elementary polyhedra|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 <math> a </math> as:{{r|berman}}
<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:{{r|berman}}
<math display="block"> V = \frac{1}{6}\left(15+7\sqrt{5}\right)a^3\approx5.10875a^3. </math>

== Cartesian coordinates ==
The triangular hebesphenorotunda with edge length <math> \sqrt{5} - 1 </math> can be constructed by the union of the orbits of the [[Cartesian coordinate]]s:
<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 [[Symmetry group|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]].{{r|timofeenko}}

== References ==
{{reflist|refs=

<ref name="berman">{{citation
 | last = Berman | first = M.
 | doi = 10.1016/0016-0032(71)90071-8
 | journal = Journal of the Franklin Institute
 | mr = 290245
 | pages = 329–352
 | title = Regular-faced convex polyhedra
 | volume = 291
 | year = 1971| issue = 5
}}.</ref>

<ref name="cromwell">{{citation
 | last = Cromwell | first = P. R.
 | title = Polyhedra
 | year = 1997
 | url = https://archive.org/details/polyhedra0000crom/page/87/mode/1up
 | publisher = [[Cambridge University Press]]
 | isbn = 978-0-521-66405-9
 | page = 86&ndash;87, 89
}}.</ref>

<ref name="francis">{{citation
 | last = Francis | first = D.
 | title = Johnson solids & their acronyms
 | journal = Word Ways
 | date = August 2013
 | volume = 46 | issue = 3 | page = 177
 | url = https://go.gale.com/ps/i.do?id=GALE%7CA340298118
}}.</ref>

<ref name="johnson">{{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
}}.</ref>

<ref name="timofeenko">{{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
}}.</ref>

}}

==External links==
* {{Mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid | urlname =TriangularHebesphenorotunda| title = Triangular hebesphenorotunda}}

{{Johnson solids navigator}}

[[Category:Elementary polyhedron]]
[[Category:Johnson solids]]