View source for Cubohemioctahedron

{{Short description|Polyhedron with 10 faces}}
{{Uniform polyhedra db|Uniform polyhedron stat table|ChO}}
[[File:Cubohemioctahedron.stl|thumb|3D model of a cubohemioctahedron]]
In [[geometry]], the '''cubohemioctahedron''' is a [[nonconvex uniform polyhedron]], indexed as U<sub>15</sub>. It has 10 faces (6 [[Square|squares]] and 4 regular [[Hexagon|hexagons]]), 24 edges and 12 vertices.<ref>{{Cite web|url=https://www.mathconsult.ch/enwiki/static/unipoly/15.html|title=15: cubohemioctahedron|last=Maeder|first=Roman|website=MathConsult}}</ref> Its [[vertex figure]] is a [[antiparallelogram|crossed quadrilateral]].

It is given [[Wythoff symbol]] '''{{frac|4|3}} 4 | 3''', although that is a double-covering of this figure.

A nonconvex polyhedron has intersecting faces which do not represent new edges or faces. In the picture vertices are marked by golden spheres, and edges by silver cylinders.

It is a [[hemipolyhedron]] with 4 [[hexagon]]al faces passing through the model center. The hexagons intersect each other and so only triangular portions of each are visible.

== Related polyhedra ==
It shares the [[vertex arrangement]] and [[edge arrangement]] with the [[cuboctahedron]] (having the square faces in common), and with the [[octahemioctahedron]] (having the hexagonal faces in common).

{| class="wikitable" width="300"
|- align=center
| [[Image:cuboctahedron.png|100px]]<BR>[[Cuboctahedron]]
| [[Image:cubohemioctahedron.png|100px]]<BR>'''Cubohemioctahedron'''
| [[Image:octahemioctahedron.png|100px]]<BR>[[Octahemioctahedron]]
|}

=== Tetrahexagonal tiling ===
The ''cubohemioctahedron'' can be seen as a [[Net (polyhedron)|net]] on the hyperbolic [[tetrahexagonal tiling]] with vertex figure 4.6.4.6.

[[File:Uniform tiling 64-t1.png|240px]]
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=== Hexahemioctacron===
{{Uniform polyhedra db|Uniform dual polyhedron stat table|ChO|No-Face-Image=True}}
The '''hexahemioctacron''' is the dual of the cubohemioctahedron, and is one of nine [[dual hemipolyhedra]]. It appears visually indistinct from the [[octahemioctacron]].

Since the cubohemioctahedron has four hexagonal [[Face (geometry)|faces]] passing through the model center, thus it is [[degeneracy (mathematics)|degenerate]], and can be seen as having four [[Vertex (geometry)|vertices]] at infinity.

In [[Magnus Wenninger]]'s ''Dual Models'', they are represented with intersecting infinite [[prism (geometry)|prisms]] passing through the model center, cut off at a certain point that is convenient for the maker.

== See also ==
* [[Hemicube (geometry)|Hemi-cube]] - The four vertices at infinity correspond directionally to the four vertices of this abstract polyhedron.

==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}} (Page 101, Duals of the (nine) hemipolyhedra)

== External links ==
* {{mathworld | urlname = Hexahemioctacron| title =Hexahemioctacron}}
* {{mathworld2 | urlname = Cubohemioctahedron| title = Cubohemioctahedron | urlname2 = UniformPolyhedron| title2 = Uniform polyhedron}}
* [https://web.archive.org/web/20171110075259/http://gratrix.net/polyhedra/uniform/summary/ Uniform polyhedra and duals]

{{Nonconvex polyhedron navigator}}

[[Category:Uniform polyhedra]]


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