Trigonal trapezohedron
{{Short description|Polyhedron with 6 congruent rhombus faces}}
{{CS1 config|mode=cs1}}
| class=wikitable align="right" width=200
!bgcolor=#e7dcc3 colspan=2|Trigonal trapezohedron | |||
| align=center colspan=2|Image:TrigonalTrapezohedron.svg | |||
| bgcolor=#e7dcc3|Type | trapezohedron | ||
| bgcolor=#e7dcc3|Conway notation | {{math|dA3}} | ||
| bgcolor=#e7dcc3|Coxeter diagram | {{CDD | node_fh|2x|node_fh|6|node}} {{CDD | node_fh|2x|node_fh|3|node_fh}} |
| bgcolor=#e7dcc3|Faces | 6 rhombi | ||
| bgcolor=#e7dcc3|Edges | 12 | ||
| bgcolor=#e7dcc3|Vertices | 8 | ||
| bgcolor=#e7dcc3|Face configuration | {{math|3,3,3,3}} | ||
| bgcolor=#e7dcc3|Symmetry group | {{math|D3d, [2+,6], (2*3),}} order 12 | ||
| bgcolor=#e7dcc3|Rotation group | {{math|D3, [2,3]+, (223),}} order 6 | ||
| bgcolor=#e7dcc3|Dual polyhedron | trigonal antiprism | ||
| bgcolor=#e7dcc3|Properties | convex, equilateral polygon, face-transitive, zonohedron, parallelohedron |
In geometry, a trigonal trapezohedron is a polyhedron with six congruent quadrilateral faces, which may be scalene or rhomboid.{{cite book
| last = Cromwell | first = Peter R.
| isbn = 978-0-521-55432-9
| location = Cambridge
| mr = 1458063
| pages = 302, 304, 367
| publisher = Cambridge University Press
| title = Polyhedra | title-link = Polyhedra (book)
| year = 1997}}{{cite book
| last1 = Cundy | first1 = H. Martyn | author1-link = Martyn Cundy
| last2 = Rollett | first2 = A.P.
| edition = 2nd
| location = Oxford
| mr = 0124167
| publisher = Clarendon Press
| title = Mathematical models
| title-link = Mathematical Models (Cundy and Rollett)
| year = 1961
| page = 117}} The variety with rhombus-shaped faces faces is a rhombohedron.{{Cite book
|title=Solid geometry: with chapters on space-lattices, sphere-packs and crystals
|last=Lines
|first=L
|publisher=Dover Publications
|year=1965}}{{cite journal
| last = Grünbaum | first = Branko | author-link = Branko Grünbaum
| 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
}}
An alternative name for the same shape is the trigonal deltohedron.{{cite journal
| last1 = Futamura | first1 = F. | author1-link = Fumiko Futamura
| last2 = Frantz | first2 = M.
| last3 = Crannell | first3 = A. | author3-link = Annalisa Crannell
| 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}}
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 figures, meaning that they have symmetries that take any face to any other face.
{{anchor|Golden_rhombohedron}} Special cases
A cube is a special case of a trigonal trapezohedron, since a square is a special case of a rhombus.{{cite journal
| last1 = Chilton | first1 = B. L.
| last2 = Coxeter | first2 = H. S. M. | author2-link = Harold Scott MacDonald Coxeter
| doi = 10.1080/00029890.1963.11992147
| journal = The American Mathematical Monthly
| jstor = 2313051
| mr = 157282
| pages = 946–951
| title = Polar zonohedra
| volume = 70
| year = 1963}}
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 faces.
Copies of these can be assembled to form other convex polyhedra with golden rhombus faces, including the Bilinski dodecahedron and rhombic triacontahedron.{{cite book
| last = Senechal | first = Marjorie | author-link = Marjorie Senechal
| 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}}
{{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. The same rhombohedra also tile space in the trigonal trapezohedral honeycomb.{{cite book
| last1 = Conway | first1 = John H. | author1-link = John Horton Conway
| 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}}
Related polyhedra
The trigonal trapezohedra are special cases of trapezohedra, polyhedra with an even number of congruent kite-shaped faces. When this number of faces is six, the kites degenerate to rhombi, and the result is a trigonal trapezohedron. As with the rhombohedra more generally, the trigonal trapezohedra are also special cases of parallelepipeds, and are the only parallelepipeds with six congruent faces. Parallelepipeds are zonohedra, and Evgraf Fedorov proved that the trigonal trapezohedra are the only infinite family of zonohedra whose faces are all congruent rhombi.
Dürer's solid is generally presumed to be a truncated triangular trapezohedron, a trigonal trapezohedron with two opposite vertices truncated, although its precise shape is still a matter for debate.
See also
References
{{reflist}}
External links
- {{mathworld | urlname = TrigonalTrapezohedron | title = Trigonal Trapezohedron}}
{{Trapezohedra}}
Category:Space-filling polyhedra
{{Polyhedron-stub}}