View source for Nonconvex great rhombicuboctahedron

{{Short description|Nonconvex uniform polyhedron with 26 faces}}
{{Uniform polyhedra db|Uniform polyhedron stat table|ugrCO}}
[[File:Nonconvex great rhombicuboctahedron.stl|thumb|3D model of a nonconvex great rhombicuboctahedron]]
In [[geometry]], the nonconvex '''great rhombicuboctahedron''' is a [[nonconvex uniform polyhedron]], indexed as U<sub>17</sub>. It has 26 faces (8 [[Triangle|triangles]] and 18 [[Square|squares]]), 48 edges, and 24 vertices.<ref>{{Cite web|url=https://www.mathconsult.ch/static/unipoly/17.html|title=17: great rhombicuboctahedron|last=Maeder|first=Roman|date=|website=MathConsult|archive-url=|archive-date=|access-date=}}</ref> It is represented by the Schläfli symbol rr{4,{{frac|3|2}}}  and [[Coxeter-Dynkin diagram]] of {{CDD|node_1|4|node|d3|rat|d2|node_1}}. Its [[vertex figure]] is a [[quadrilateral#More quadrilaterals|crossed quadrilateral]].

This model shares the name with the convex ''great rhombicuboctahedron'', also called the [[truncated cuboctahedron]].

An alternative name for this figure is '''quasirhombicuboctahedron'''. From that derives its Bowers acronym: '''querco'''.

== Orthographic projections==
[[File:Nonconvex_great_rhombicuboctahedron_ortho_wireframes.png|480px]]

== Cartesian coordinates ==
[[Cartesian coordinates]] for the vertices of a ''nonconvex great rhombicuboctahedron'' centered at the origin with edge length 1 are all the permutations of

<math display=block>\Bigl(\pm \left[\sqrt 2 - 1\right],\ \pm 1,\ \pm 1\Bigr).</math>

== Related polyhedra ==

It shares the [[vertex arrangement]] with the convex [[truncated cube]]. It additionally shares its [[edge arrangement]] with the [[great cubicuboctahedron]] (having the triangular faces and 6 square faces in common), and with the [[great rhombihexahedron]] (having 12 square faces in common). It has the same vertex figure as the [[pseudo great rhombicuboctahedron]], which is not a uniform polyhedron.

{| class="wikitable" style="vertical-align:top;text-align:center"
|- valign=top
|[[Image:Truncated hexahedron.png|100px]]<BR>[[Truncated cube]]
|[[Image:Uniform great rhombicuboctahedron.png|100px]]<BR>'''Great rhombicuboctahedron'''
|[[Image:Great cubicuboctahedron.png|100px]]<BR>[[Great cubicuboctahedron]]
|[[Image:Great rhombihexahedron.png|100px]]<BR>[[Great rhombihexahedron]]
|[[Image:Pseudo-great rhombicuboctahedron.png|100px]]<BR>[[Pseudo great rhombicuboctahedron]]
|}
{{Clear}}

=== Great deltoidal icositetrahedron===
{{Uniform polyhedra db|Uniform dual polyhedron stat table|ugrCO}}
[[File:Great deltoidal icositetrahedron.stl|thumb|3D model of a great deltoidal icositetrahedron]]
The '''great deltoidal icositetrahedron''' is the dual of the nonconvex great rhombicuboctahedron.

== 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= GreatDeltoidalIcositetrahedron|title= Great Deltoidal Icositetrahedron}}
* {{mathworld | urlname = UniformGreatRhombicuboctahedron| title = Uniform great rhombicuboctahedron}}
* [http://www.software3d.com/GRhomCubOct.php Great Rhombicuboctahedron] Paper model

[[Category:Uniform polyhedra]]


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