Editing Triangular orthobicupola

{{Short description|27th Johnson solid; 2 triangular cupolae joined base-to-base}}
{{Infobox polyhedron
|image=triangular_orthobicupola.png
|type=[[Johnson solid|Johnson]]<br>{{math|[[gyrobifastigium|''J''{{sub|26}}]] – '''''J''{{sub|27}}''' – [[square orthobicupola|''J''{{sub|28}}]]}}
|faces=2+6 [[triangle]]s<br>6 [[Square (geometry)|square]]s
|edges=24
|vertices=12
|symmetry={{math|D{{sub|3h}}}}
|vertex_config={{math|6(3{{sup|2}}.4{{sup|2}})<br>6(3.4.3.4)}}
|dual=[[Trapezo-rhombic dodecahedron]]
|properties=[[convex polytope|convex]]
|net=Johnson solid 27 net.png
}}

In [[geometry]], the '''triangular orthobicupola''' is one of the [[Johnson solid]]s ({{math|''J''{{sub|27}}}}). As the name suggests, it can be constructed by attaching two [[triangular cupola]]s ({{math|''J''{{sub|3}}}}) along their bases. It has an equal number of squares and triangles at each vertex; however, it is not [[vertex-transitive]]. It is also called an ''anticuboctahedron'', ''twisted cuboctahedron'' or ''disheptahedron''. It is also a [[Midsphere#Canonical polyhedron|canonical polyhedron]].

{{Johnson solid}}

The ''triangular orthobicupola'' is the first in an infinite set of [[Bicupola (geometry)|orthobicupolae]].

== Construction ==
The ''triangular orthobicupola'' can be constructed by attaching two [[triangular cupola]]s onto their bases. Similar to the [[cuboctahedron]], which would be known as the ''triangular gyrobicupola'', the difference is that the two triangular cupolas that make up the triangular orthobicupola are joined so that pairs of matching sides abut (hence, "ortho"); the cuboctahedron is joined so that triangles abut squares and vice versa. Given a triangular orthobicupola, a 60-degree rotation of one cupola before the joining yields a cuboctahedron.{{r|os}} Hence, another name for the triangular orthobicupola is the ''anticuboctahedron''.{{r|becker}} Because the triangular orthobicupola has the property of [[Convex set|convexity]] and its faces are [[regular polygon]]s&mdash;eight [[equilateral triangle]]s and six [[square]]s&mdash;it is categorized as a [[Johnson solid]]. It is enumerated as the twenty-seventh Johnson solid <math> J_{27} </math>{{r|berman|francis}}

== Properties ==
The [[surface area]] <math> A </math> and the [[volume]] <math> V </math> of a triangular orthobicupola are the same as those with cuboctahedron. Its surface area can be obtained by summing all of its polygonal faces, and its volume is by slicing it off into two triangular cupolas and adding their volume. With edge length <math> a </math>, they are:{{r|berman}}
<math display="block"> \begin{align}
 A &= 2\left(3+\sqrt{3}\right)a^2 \approx 9.464a^2, \\
 V &= \frac{5\sqrt{2}}{3}a^3 \approx 2.357a^3.
\end{align} </math>

The dual polyhedron of a triangular orthobicupola is the [[trapezo-rhombic dodecahedron]]. It has 6 rhombic and 6 trapezoidal faces, and is similar to the [[rhombic dodecahedron]].{{r|becker}}

==References==
{{Reflist|refs=

<ref name="becker">{{cite journal
 | last = Becker | first = David A.
 | title = A Peculiarly Cerebroid Convex Zygo-Dodecahedron is an Axiomatically Balanced "House of Blues": The Circle of Fifths to the Circle of Willis to Cadherin Cadenzas
 | journal = Symmetry
 | year = 2012
 | volume = 4 | issue = 4 | pages = 644–666
 | doi = 10.3390/sym4040644
| doi-access = free
 | bibcode = 2012Symm....4..644B
 }}</ref>

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

<ref name="os">{{cite book
 | last1 = Ogievetsky | first1 = O.
 | last2 = Shlosman | first2 = S.
 | editor-last1 = Novikov | editor-first1 = S.
 | editor-last2 = Krichever | editor-first2 = I.
 | editor-last3 = Ogievetsky | editor-first3 = O.
 | editor-last4 = Shlosman | editor-first4 = S.
 | year = 2021
 | title = Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry
 | contribution = Platonic compounds and cylinders
 | url = https://books.google.com/books?id=UsspEAAAQBAJ&pg=PA477
 | page = 477
 | publisher = [[American Mathematical Society]]
 | isbn = 978-1-4704-5592-7
}}</ref>

}}

==External links==
* {{mathworld2 | urlname = JohnsonSolid | title = Johnson solid| urlname2 = TriangularOrthobicupola | title2 = Triangular orthobicupola}}

{{Johnson solids navigator}}
[[Category:Johnson solids]]