View source for Trigonal trapezohedron

{{Short description|Polyhedron with 6 congruent rhombus faces}}
{| class=wikitable align="right" width=200
!bgcolor=#e7dcc3 colspan=2|Trigonal trapezohedron
|-
|align=center colspan=2|[[Image:TrigonalTrapezohedron.svg|120px|Trigonal trapezohedron]]
|-
|bgcolor=#e7dcc3|Type||[[Trapezohedron|trapezohedron]]
|-
|bgcolor=#e7dcc3|[[Conway polyhedron notation|Conway notation]]||{{math|dA3}}
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|bgcolor=#e7dcc3|Coxeter diagram||{{CDD||node_fh|2x|node_fh|6|node}}<BR>{{CDD||node_fh|2x|node_fh|3|node_fh}}
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|bgcolor=#e7dcc3|Faces||6 [[Rhombus|rhombi]]
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|bgcolor=#e7dcc3|Edges||12
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|bgcolor=#e7dcc3|Vertices||8
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|bgcolor=#e7dcc3|Face configuration||{{math|3,3,3,3}}
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|bgcolor=#e7dcc3|[[List of spherical symmetry groups|Symmetry group]]||{{math|''D''<sub>''3d''</sub>, [2<sup>+</sup>,6], (2*3),}} order 12
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|bgcolor=#e7dcc3|[[Point groups in three dimensions#Rotation groups|Rotation group]]||{{math|''D''<sub>''3''</sub>, [2,3]<sup>+</sup>, (223),}} order 6
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|bgcolor=#e7dcc3|[[Dual polyhedron]]||trigonal [[Antiprism|antiprism]]
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|bgcolor=#e7dcc3|Properties||convex, [[Equilateral polygon|equilateral polygon]], face-transitive, [[Zonohedron|zonohedron]], [[Parallelohedron|parallelohedron]]
|}

In [[Geometry|geometry]], a '''trigonal trapezohedron''' is a [[Rhombohedron|rhombohedron]] (a [[Polyhedron|polyhedron]] with six [[Rhombus|rhombus]]-shaped [[Face (geometry)|faces]]) in which, additionally, all six faces are [[Congruence (geometry)|congruent]]. Alternative names for the same shape are the ''trigonal deltohedron''<ref name=ffc>{{cite journal
 | last2 = Frantz | first2 = M.
 | doi = 10.1080/17513472.2014.974483
 | issue = 3-4
 | journal = Journal of Mathematics and the Arts
 | mr = 3292158
 | pages = 111–119
 | title = The cross ratio as a shape parameter for Dürer's solid
 | volume = 8
 | year = 2014}}</ref> or ''isohedral rhombohedron''.<ref name=":0">{{Cite book
 |title=Solid geometry: with chapters on space-lattices, sphere-packs and crystals
 |last=Lines
 |first=L
 |publisher=Dover Publications
 |year=1965}}</ref> Some sources just call them ''rhombohedra''.<ref name=grunbaum>{{cite journal
 | doi = 10.1007/s00283-010-9138-7
 | issue = 4
 | journal = [[The Mathematical Intelligencer]]
 | mr = 2747698
 | pages = 5–15
 | title = The Bilinski dodecahedron and assorted parallelohedra, zonohedra, monohedra, isozonohedra, and otherhedra
 | volume = 32
 | year = 2010| hdl = 1773/15593
 | hdl-access = free
 }}</ref>

==Geometry==
Six identical rhombic faces can construct two configurations of trigonal trapezohedra. The ''acute'' or ''prolate'' form has three acute angle corners of the rhombic faces meeting at the two polar axis vertices. The ''obtuse'' or ''oblate'' or ''flat'' form has three obtuse angle corners of the rhombic faces meeting at the two polar axis vertices.

More strongly than having all faces congruent, the trigonal trapezohedra are [[Isohedral figure|isohedral figure]]s, meaning that they have symmetries that take any face to any other face.<ref name=grunbaum/>

=={{anchor|Golden_rhombohedron}} Special cases==
A [[Cube|cube]] can be interpreted as a special case of a trigonal trapezohedron, with square rather than rhombic faces.

A gyroelongated triangular bipyramid constructed with equilateral triangles can also be seen as a trigonal trapezohedron when its coplanar triangles are merged into rhombi.

The two '''golden rhombohedra''' are the acute and obtuse form of the trigonal trapezohedron with [[Golden rhombus|golden rhombus]] faces.
Copies of these can be assembled to form other convex polyhedra with golden rhombus faces, including the [[Bilinski dodecahedron]] and [[Rhombic triacontahedron|rhombic triacontahedron]].<ref>{{cite book
 | contribution = Donald and the golden rhombohedra
 | mr = 2209027
 | pages = 159–177
 | publisher = American Mathematical Society
 | location = Providence, Rhode Island
 | title = The Coxeter Legacy
 | year = 2006}}</ref>

{{multiple image|total_width=480px|align=center
|image1=Acute golden rhombohedron.png|caption1=Acute golden rhombohedron
|image2=Flat golden rhombohedron.png|caption2=Obtuse golden rhombohedron
}}

Four oblate rhombohedra whose ratio of face diagonal lengths are the square root of two can be assembled to form a [[Rhombic dodecahedron|rhombic dodecahedron]]. The same rhombohedra also tile space in the [[Trigonal trapezohedral honeycomb|trigonal trapezohedral honeycomb]].<ref>{{cite book
 | last2 = Burgiel | first2 = Heidi
 | last3 = Goodman-Strauss | first3 = Chaim
 | isbn = 978-1-56881-220-5
 | mr = 2410150
 | page = 294
 | publisher = A K Peters | location = Wellesley, Massachusetts
 | title = The Symmetries of Things
 | url = https://books.google.com/books?id=Drj1CwAAQBAJ&pg=PA294
 | year = 2008}}</ref>

==Related polyhedra==
The trigonal trapezohedra are special cases of [[Trapezohedron|trapezohedra]], polyhedra with an even number of congruent [[Kite (geometry)|kite]]-shaped faces. When this number of faces is six, the kites [[Degeneracy (mathematics)|degenerate]] to rhombi, and the result is a trigonal trapezohedron. As with the rhombohedra more generally, the trigonal trapezohedra are also special cases of [[Parallelepiped|parallelepiped]]s, and are the only parallelepipeds with six congruent faces. Parallelepipeds are [[Zonohedron|zonohedra]], and Evgraf Fedorov proved that the trigonal trapezohedra are the only infinite family of zonohedra whose faces are all congruent rhombi.<ref name=grunbaum/>

Dürer's solid is generally presumed to be a [[Truncated triangular trapezohedron|truncated triangular trapezohedron]], a trigonal trapezohedron with two opposite vertices [[Truncation (geometry)|truncated]], although its precise shape is still a matter for debate.<ref name=ffc/>

== See also ==
* [[Truncated triangular trapezohedron]]

== References ==
{{reflist}}

==External links==
* {{mathworld | urlname = TrigonalTrapezohedron | title = Trigonal Trapezohedron}}


[[Category:Space-filling polyhedra]]
[[Category:Zonohedra]]
[[Category:Polyhedra]]



{{Sourceattribution|Trigonal trapezohedron}}