View source for Triheptagonal tiling

{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|U73_1}}
In [[geometry]], the '''triheptagonal tiling''' is a semiregular tiling of the hyperbolic plane, representing a [[Rectification (geometry)|rectified]] [[Order-3 heptagonal tiling]]. There are two [[triangle]]s and two [[heptagon]]s alternating on each [[vertex (geometry)|vertex]]. It has [[Schläfli symbol]] of r{7,3}.

Compare to [[trihexagonal tiling]] with [[vertex configuration]] ''3.6.3.6''.

== Images ==
{| class=wikitable width=480
|- valign=top
|[[File:Uniform tiling 73-t1 klein.png|240px]]<BR>[[Klein disk model]] of this tiling preserves straight lines, but distorts angles
|[[File:7-3 rhombille tiling.svg|240px]]<BR>The dual tiling is called an ''Order-7-3 rhombille tiling'', made from rhombic faces, alternating 3 and 7 per vertex.
|}
== 7-3 Rhombille ==
{{Infobox face-uniform tiling
|  name=7-3 rhombille tiling
|  Image_File=File:7-3 rhombille tiling.svg
|  Type=
|  Cox={{CDD|node|3|node_f1|7|node}}
|  Face_List=Rhombi
|  Symmetry_Group=[7,3], *732
|  Rotation_Group= [7,3]<sup>+</sup>, (732)
|  Face_Type=V3.7.3.7
|  Dual=Triheptagonal tiling
|  Property_List=[[edge-transitive]] [[face-transitive]]
}}

In [[geometry]], the '''7-3 rhombille tiling''' is a [[tessellation]] of identical [[rhombus|rhombi]] on the [[hyperbolic plane]]. Sets of three and seven rhombi meet two classes of vertices.

[[File:Order 7-3 rhombic tiling in the Band Model.png|640px]]<BR>7-3 rhombile tiling in [[band model]]
{{-}}

== Related polyhedra and tilings ==
The triheptagonal tiling  can be seen in a sequence of [[quasiregular polyhedron]]s and tilings:
{{Quasiregular3 table}}

From a [[Wythoff construction]] there are eight hyperbolic [[Uniform tilings in hyperbolic plane|uniform tilings]] that can be based from the regular heptagonal tiling. 

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.
{{Heptagonal tiling table}}

{{Quasiregular7 table}}

== See also ==
{{Commonscat|Uniform tiling 3-7-3-7}}
* [[Trihexagonal tiling]] - 3.6.3.6 tiling
** [[Rhombille tiling]] - dual V3.6.3.6 tiling
* [[Tilings of regular polygons]]
* [[List of uniform tilings]]

==References==
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{isbn|978-1-56881-220-5}} (Chapter 19, The Hyperbolic Archimedean Tessellations)
* {{Cite book|title=The Beauty of Geometry: Twelve Essays|year=1999|publisher=Dover Publications|lccn=99035678|isbn=0-486-40919-8|chapter=Chapter 10: Regular honeycombs in hyperbolic space}}

== External links ==
*{{MathWorld | urlname= HyperbolicTiling | title = Hyperbolic tiling}}
*{{MathWorld | urlname=PoincareHyperbolicDisk | title = Poincaré hyperbolic disk }}
* [http://bork.hampshire.edu/~bernie/hyper/ Hyperbolic and Spherical Tiling Gallery]
* [http://geometrygames.org/KaleidoTile/index.html KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings]
* [http://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch]

{{Tessellation}}

[[Category:Hyperbolic tilings]]
[[Category:Isogonal tilings]]
[[Category:Isotoxal tilings]]
[[Category:Quasiregular polyhedra]]
[[Category:Semiregular tilings]]


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