View source for Nonconvex great rhombicosidodecahedron

{{Short description|Polyhedron with 62 faces}}
{{Uniform polyhedra db|Uniform polyhedron stat table|ugrID}}
[[File:Nonconvex great rhombicosidodecahedron.stl|thumb|3D model of a nonconvex great rhombicosidodecahedron]]
In [[geometry]], the nonconvex '''great rhombicosidodecahedron''' is a [[nonconvex uniform polyhedron]], indexed as U<sub>67</sub>. It has 62 faces (20 [[triangle]]s, 30 [[square]]s and 12 [[pentagram]]s), 120 edges, and 60 vertices.<ref>{{Cite web|url=https://www.mathconsult.ch/static/unipoly/67.html|title=67: great rhombicosidodecahedron|last=Maeder|first=Roman|website=MathConsult}}</ref> It is also called the '''quasirhombicosidodecahedron'''. It is given a [[Schläfli symbol]] {{math|rr{{mset|{{Frac|5|3}},3}}.}}  Its [[vertex figure]] is a [[quadrilateral#More quadrilaterals|crossed quadrilateral]].

This model shares the name with the convex ''great rhombicosidodecahedron'', also known as the [[truncated icosidodecahedron]].

== Cartesian coordinates ==
[[Cartesian coordinates]] for the vertices of a nonconvex great rhombicosidodecahedron are all the even permutations of
<math display=block>\begin{array}{ccclc}
  \Bigl(& \pm\,\frac{1}{\varphi^2},& 0,& \pm \bigl[2-\frac{1}{\varphi}\bigr] &\Bigr) \\
  \Bigl(& \pm\,1,& \pm\,\frac{1}{\varphi^3},& \pm\,1 &\Bigr) \\
  \Bigl(& \pm\,\frac{1}{\varphi},& \pm\,\frac{1}{\varphi^2},& \pm\,\frac{2}{\varphi} &\Bigr)
\end{array}</math>

where <math>\varphi = \tfrac{1+\sqrt 5}{2}</math> is the [[golden ratio]].

== Related polyhedra ==

It shares its [[vertex arrangement]] with the [[truncated great dodecahedron]], and with the [[Polyhedron compound#Uniform compounds|uniform compounds]] of [[compound of six pentagonal prisms|6]] or [[compound of twelve pentagonal prisms|12 pentagonal prisms]]. It additionally shares its [[edge arrangement]] with the [[great dodecicosidodecahedron]] (having the triangular and pentagrammic faces in common), and the [[great rhombidodecahedron]] (having the square faces in common).

{| class="wikitable" width="400" style="vertical-align:top;text-align:center"
| [[Image:Uniform great rhombicosidodecahedron.png|100px]]<BR>'''Nonconvex great rhombicosidodecahedron'''
| [[Image:Great dodecicosidodecahedron.png|100px]]<BR>[[Great dodecicosidodecahedron]]
| [[Image:Great rhombidodecahedron.png|100px]]<BR>[[Great rhombidodecahedron]]
|-
| [[Image:Great truncated dodecahedron.png|100px]]<BR>[[Truncated great dodecahedron]]
| [[Image:UC34-6 pentagonal prisms.png|100px]]<BR>[[Compound of six pentagonal prisms]]
| [[Image:UC35-12 pentagonal prisms.png|100px]]<BR>[[Compound of twelve pentagonal prisms]]
|}

{{-}}

=== Great deltoidal hexecontahedron ===
{{Uniform polyhedra db|Uniform dual polyhedron stat table|ugrID}}
[[File:Great deltoidal hexecontahedron.stl|thumb|3D model of a great deltoidal hexecontahedron]]
The '''great deltoidal hexecontahedron''' is a nonconvex [[Isohedral figure|isohedral]] [[polyhedron]]. It is the [[Dual polyhedron|dual]] of the nonconvex great rhombicosidodecahedron. It is visually identical to the [[great rhombidodecacron]]. It has 60 intersecting [[cross quadrilateral]] faces, 120 edges, and 62 vertices.

It is also called a ''great strombic hexecontahedron.''

== See also ==
* [[List of uniform polyhedra]]

==References==
{{Reflist}}
*{{Citation | last1=Wenninger | first1=Magnus | author1-link=Magnus Wenninger | title=Dual Models | publisher=[[Cambridge University Press]] | isbn=978-0-521-54325-5 |mr=730208 | year=1983 | doi=10.1017/CBO9780511569371}}

== External links ==
* {{mathworld | urlname = UniformGreatRhombicosidodecahedron| title = Great rhombicosidodecahedron}}
* {{mathworld | urlname = GreatDeltoidalHexecontahedron| title =Great deltoidal hexecontahedron}}
* [https://web.archive.org/web/20171110075259/http://gratrix.net/polyhedra/uniform/summary/ Uniform polyhedra and duals]

{{Nonconvex polyhedron navigator}}

[[Category:Uniform polyhedra]]