View source for Tetrahemihexahedron

{{Short description|Polyhedron with 7 faces}}
{{Uniform polyhedra db|Uniform polyhedron stat table|ThH}}
[[File:Tetrahemihexahedron.stl|thumb|3D model of a tetrahemihexahedron]]
[[File:Tetrahemihexahedron rotation.gif|thumb|Rotating model]]
[[File:Net of tetrahemihexahedron.svg|thumb|Net of tetrahemihexahedron]]
In [[geometry]], the '''tetrahemihexahedron''' or '''hemicuboctahedron''' is a [[uniform star polyhedron]], indexed as U<sub>4</sub>. It has 7 faces (4 [[Triangle|triangles]] and 3 [[Square|squares]]), 12 edges, and 6 vertices.<ref>{{Cite web|url=https://www.mathconsult.ch/static/unipoly/04.html|title=04: tetrahemihexahedron|last=Maeder|first=Roman|website=MathConsult}}</ref> Its [[vertex figure]] is a [[antiparallelogram|crossed quadrilateral]]. Its [[Coxeter–Dynkin diagram]] is {{CDD|node_1|3x|rat|d2|node|3|node_1}} (although this is a double covering of the tetrahemihexahedron).

The tetrahemihexahedron is the only non-[[prismatic uniform polyhedron]] with an odd number of faces. Its [[Wythoff symbol]] is '''3/2 3 | 2''', but that represents a double covering of the tetrahemihexahedron with eight triangles and six squares, paired and coinciding in space. (It can more intuitively be seen as two coinciding tetrahemihexahedra.)

The tetrahemihexahedron is a [[hemipolyhedron]]. The "hemi" part of the name means some of the faces form a group with half as many members as some regular polyhedron—here, three square faces form a group with half as many faces as the regular hexahedron, better known as the cube—hence ''hemihexahedron''. Hemi faces are also oriented in the same direction as the regular polyhedron's faces. The three square faces of the tetrahemihexahedron are, like the three facial orientations of the cube, mutually [[perpendicular]].

The "half-as-many" characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. Visually, each square is divided into four [[right triangle]]s, with two visible from each side.

<!--The tetrahemihexahedron is the three-dimensional [[demicross polytope]]. (Seems to be original research, but nevertheless an interesting one to interpret).-->

==Related surfaces==
The tetrahemihexahedron is a [[orientability|non-orientable]] surface. It is unique as the only [[uniform polyhedron]] with an [[Euler characteristic]] of 1 and is hence a [[projective polyhedron]], yielding a representation of the [[real projective plane]]<ref name="richter" /> very similar to the ''[[Roman surface]]''.

{| class=wikitable
|- align=center
|[[File:RomanSurfaceTopView.PNG|320px]]<BR>[[Roman surface]]
|}

==Related polyhedra==
The tetrahemihexahedron has the same vertices and edges as the regular [[octahedron]]. It also shares 4 of the 8 triangular faces of the octahedron, but has three additional square faces passing through the centre of the polyhedron.

{| class="wikitable" width="400" style="vertical-align:top;text-align:center"
| [[Image:octahedron.png|100px]]<BR>[[Octahedron]]
| [[Image:tetrahemihexahedron.png|100px]]<BR>'''Tetrahemihexahedron'''
|}

The dual figure of the tetrahemihexahedron is the [[tetrahemihexacron]].

The tetrahemihexahedron is [[covering space|2-covered]] by the [[cuboctahedron]],<ref name="richter">{{Harv|Richter}}</ref> which accordingly has the same abstract [[vertex figure]] (2 triangles and two squares: 3.4.3.4) and twice the vertices, edges, and faces. It has the same topology as the [[abstract polytope|abstract polyhedron]] [[hemi-cuboctahedron]].

{| class="wikitable" width="400" style="vertical-align:top;text-align:center"
| [[Image:cuboctahedron.png|100px]]<br>[[Cuboctahedron]]
| [[Image:tetrahemihexahedron.png|100px]]<BR>Tetrahemihexahedron
|}

The tetrahemihexahedron may also be constructed as a crossed triangular [[cupola (geometry)|cuploid]]. All cuploids and their duals are topologically projective planes.<ref>[https://archive.bridgesmathart.org/2018/bridges2018-543.pdf Polyhedral Models of the Projective Plane], Paul Gailiunas, Bridges 2018 Conference Proceedings</ref>
{{Star-cuploids}}

=== Tetrahemihexacron===
{{Uniform polyhedra db|Uniform dual polyhedron stat table|ThH|No-Face-Image=True}}
The '''tetrahemihexacron''' is the [[dual polyhedron|dual]] of the tetrahemihexahedron, and is one of nine [[dual hemipolyhedra]].

Since the hemipolyhedra have [[Face (geometry)|faces]] passing through the center, the [[Dual polyhedron|dual figures]]  have corresponding [[Vertex (geometry)|vertices]] at infinity; properly, on the [[real projective plane]] at infinity.<ref>{{Harv|Wenninger|2003|loc = [https://books.google.com/books?id=mfmzUjhs-_8C&pg=PA101 p. 101]}}</ref> In [[Magnus Wenninger]]'s ''Dual Models'', they are represented with intersecting [[prism (geometry)|prisms]], each extending in both directions to the same vertex at infinity, in order to maintain symmetry. In practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of [[stellation]] figures, called ''stellation to infinity''. However, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions.

Topologically, the tetrahemihexacron is considered to contain seven vertices. The three vertices considered at infinity (the [[real projective plane]] at infinity) correspond directionally to the three vertices of the [[hemi-octahedron]], an abstract polyhedron. The other four vertices exist at alternate corners of a central cube (a [[demicube]], in this case a [[tetrahedron]]).

==References==
{{reflist}}
{{refbegin}}
* {{citation | first = David A. | last = Richter | url = http://homepages.wmich.edu/~drichter/rptwo.htm | title = Two Models of the Real Projective Plane }}
*{{citation|last1=Wenninger|first1=Magnus|author1-link=Magnus Wenninger|title=Dual Models|publisher=[[Cambridge University Press]]|isbn=978-0-521-54325-5|mr=730208|orig-year=1983|year=2003|doi=10.1017/CBO9780511569371}} (Page 101, Duals of the (nine) hemipolyhedra)
{{refend}}

==External links==
*{{mathworld2|urlname=Tetrahemihexahedron|title=Tetrahemihexahedron|urlname2=UniformPolyhedron|title2=Uniform polyhedron}}
*[http://gratrix.net/polyhedra/uniform/summary Uniform polyhedra and duals]
*[http://www.software3D.com/Tetrahemi.php Paper model]

{{Nonconvex polyhedron navigator}}

[[Category:Uniform polyhedra]]
[[Category:Projective polyhedra]]