View source for Small cubicuboctahedron

{{Uniform polyhedra db|Uniform polyhedron stat table|lCCO}}
[[File:Small cubicuboctahedron.stl|thumb|3D model of a small cubicuboctahedron]]
In [[geometry]], the '''small cubicuboctahedron''' is a [[uniform star polyhedron]], indexed as U<sub>13</sub>. It has 20 faces (8 [[Triangle|triangles]], 6 [[Square|squares]], and 6 [[Octagon|octagons]]), 48 edges, and 24 vertices.<ref>{{Cite web|url=https://www.mathconsult.ch/static/unipoly/13.html|title=13: small cubicuboctahedron|last=Maeder|first=Roman|date=|website=MathConsult|archive-url=|archive-date=|access-date=}}</ref> Its [[vertex figure]] is a [[quadrilateral#More quadrilaterals|crossed quadrilateral]].

The small cubicuboctahedron is a [[faceting]] of the [[rhombicuboctahedron]].  Its square faces and its octagonal faces are parallel to those of a [[cube]], while its triangular faces are parallel to those of an [[octahedron]]: hence the name ''cubicuboctahedron''. The ''small'' suffix serves to distinguish it from the [[great cubicuboctahedron]], which also has faces in the aforementioned directions.<ref>{{Cite web|url=https://www.software3d.com/SmallCubicubocta.php|title=Small Cubicuboctahedron|last=Webb|first=Robert|date=|website=Stella: Polyhedron Navigator|archive-url=|archive-date=|access-date=}}</ref>

== Related polyhedra ==
It shares its [[vertex arrangement]] with the [[stellated truncated hexahedron]]. It additionally shares its [[edge arrangement]] with the rhombicuboctahedron (having the triangular faces and 6 square faces in common), and with the [[small rhombihexahedron]] (having the octagonal faces in common).

{| class="wikitable" width="400" style="vertical-align:top;text-align:center"
| [[Image:Small rhombicuboctahedron.png|100px]]<BR>[[Rhombicuboctahedron]]
| [[Image:Small cubicuboctahedron.png|100px]]<BR>'''Small cubicuboctahedron'''
| [[Image:Small rhombihexahedron.png|100px]]<BR>[[Small rhombihexahedron]]
| [[Image:Stellated truncated hexahedron.png|100px]]<BR>[[Stellated truncated hexahedron]]
|}

== Related tilings ==
[[File:Uniform tiling 443-t01.png|thumb|The 3 4 {{pipe}} 4 tiling is the tiling on the universal cover of the small cubicuboctahedron.<br/>
<small>(Yellow and red reversed in this tiling, compared to polyhedron.)</small>]]
As the Euler characteristic suggests, the small cubicuboctahedron is a [[toroidal polyhedron]] of genus 3 (topologically it is a surface of genus 3), and thus can be interpreted as a (polyhedral) [[Immersion (mathematics)|immersion]] of a genus 3 polyhedral surface, in the complement of its 24 vertices, into 3-space. (A neighborhood of any vertex is topologically a cone on a figure-8, which cannot occur in an immersion. Note that the Richter reference overlooks this fact.) The underlying polyhedron (ignoring self-intersections) defines a uniform tiling of this surface, and so the small cubicuboctahedron is a uniform polyhedron. In the language of [[abstract polytope]]s, the small cubicuboctahedron is a ''faithful realization'' of this abstract toroidal polyhedron, meaning that it is a nondegenerate polyhedron and that they have the same symmetry group. In fact, every automorphism of the abstract genus 3 surface with this tiling is realized by an isometry of Euclidean space.

Higher genus surfaces (genus 2 or greater) admit a metric of negative [[constant curvature]] (by the [[uniformization theorem]]), and the [[universal cover]] of the resulting [[Riemann surface]] is the [[models of the hyperbolic plane|hyperbolic plane]]. The corresponding [[Uniform tilings in hyperbolic plane|tiling of the hyperbolic plane]] has vertex figure 3.8.4.8 (triangle, octagon, square, octagon). If the surface is given the appropriate metric of curvature = −1, the covering map is a [[local isometry]] and thus the [[abstract vertex figure|''abstract'' vertex figure]] is the same. This tiling may be denoted by the [[Wythoff symbol]] 3 4 | 4, and is depicted on the right.

[[File:Order-7 triangular tiling.svg|thumb|The small cubicuboctahedron can also be interpreted as a polyhedral immersion (a coloring of) the [[Klein quartic]],<ref name="tiling" /> which is a quotient of the [[order-7 triangular tiling]].]]
Alternatively and more subtly, by chopping up each square face into 2 triangles and each octagonal face into 6 triangles, the small cubicuboctahedron can be interpreted as a non-regular ''coloring'' of the combinatorially ''regular'' (not just ''uniform'') tiling of the genus 3 surface by 56 equilateral triangles, meeting at 24 vertices, each with degree 7.<ref name="tiling">{{Harv|Richter}} Note each face in the polyhedron consist of multiple faces in the tiling, hence the description as a "coloring" – two triangular faces constitute a square face and so forth, as per [http://homepages.wmich.edu/~drichter/images/mathieu/hypercolors.jpg this explanatory image].</ref> This regular tiling is significant as it is a tiling of the [[Klein quartic]], the genus 3 surface with the most symmetric metric (automorphisms of this tiling equal isometries of the surface), and the orientation-preseserving automorphism group of this surface is isomorphic to the [[projective special linear group]] PSL(2,7), equivalently GL(3,2) (the order 168 group of all orientation-preserving isometries). Note that the small cubicuboctahedron is ''not'' a realization of this abstract polyhedron, as it only has 24 orientation-preserving symmetries (not every abstract automorphism is realized by a Euclidean isometry) – the isometries of the small cubicuboctahedron preserve not only the triangular tiling, but also the coloring, and hence are a proper subgroup of the full isometry group.

The corresponding tiling of the hyperbolic plane (the universal covering) is the [[order-7 triangular tiling]]. The automorphism group of the Klein quartic can be augmented (by a symmetry which is not realized by a symmetry of the polyhedron, namely "exchanging the two endpoints of the edges that bisect the squares and octahedra) to yield the [[Mathieu group]] M<sub>24</sub>.<ref>{{Harv|Richter}}</ref>

== See also ==
* [[Compound of five small cubicuboctahedra]]
* [[List of uniform polyhedra]]

== References ==
{{reflist}}
{{refbegin}}
* {{citation | first = David A. | last = Richter | url = http://homepages.wmich.edu/~drichter/mathieu.htm | title = How to Make the Mathieu Group M<sub>24</sub> | accessdate = 2010-04-15 }}
{{refend}}

== External links ==
* {{mathworld2 | urlname = SmallCubicuboctahedron| title = Small cubicuboctahedron | urlname2 = UniformPolyhedron| title2 = Uniform polyhedron}}

[[Category:Toroidal polyhedra]]