View source for Heptagonal tiling

{{Short description|Tiling of the hyperbolic plane}}
{{Uniform hyperbolic tiles db|Reg hyperbolic tiling stat table|U73_0}}
In [[geometry]], the '''heptagonal tiling''' is a [[regular hyperbolic tiling|regular tiling]] of the [[Hyperbolic geometry|hyperbolic plane]]. It is represented by [[Schläfli symbol]] of {7,3}, having three regular [[heptagon]]s around each vertex.
== Images ==
{| class=wikitable width=480
|- align=center
|[[File:PavageDemiPlanPoincare.svg|160px]]<BR>[[Poincaré half-plane model]]
|[[File:PavageHypPoincare2.svg|160px]]<BR>[[Poincaré disk model]]
|[[File:PavageKleinBeltrami.svg|160px]]<BR>[[Beltrami-Klein model]]
|}

== Related polyhedra and tilings ==
This tiling is topologically related as a part of sequence of regular polyhedra with [[Schläfli symbol]] {n,3}.
{{Order-3 tiling 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}}

== Hurwitz surfaces ==
[[File:3-7 kisrhombille.svg|thumb|The symmetry group of the heptagonal tiling has fundamental domain the (2,3,7) [[Schwarz triangle]], which yields this tiling.]]
{{see|Hurwitz surface}}
The symmetry group of the tiling is the [[(2,3,7) triangle group]], and a [[fundamental domain]] for this action is the (2,3,7) [[Schwarz triangle]]. This is the smallest hyperbolic Schwarz triangle, and thus, by the proof of [[Hurwitz's automorphisms theorem]], the tiling is the universal tiling that covers all [[Hurwitz surface]]s (the [[Riemann surface]]s with maximal symmetry group), giving them a tiling by heptagons whose symmetry group equals their automorphism group as Riemann surfaces. The smallest Hurwitz surface is the [[Klein quartic]] (genus 3, automorphism group of order 168), and the induced tiling has 24 heptagons, meeting at 56 vertices.

The dual [[order-7 triangular tiling]] has the same symmetry group, and thus yields [[Polygon triangulation|triangulation]]s of Hurwitz surfaces.

==See also==
{{Commonscat|Order-3 heptagonal tiling}}
*[[Hexagonal tiling]]
*[[Tilings of regular polygons]]
*[[List of uniform planar tilings]]
*[[List of regular polytopes]]

==References==
{{reflist}}
{{refbegin}}
* [[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}}
{{refend}}

== 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] {{Webarchive|url=https://web.archive.org/web/20130324095520/http://bork.hampshire.edu/~bernie/hyper/ |date=2013-03-24 }}
* [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:Heptagonal tilings| ]]
[[Category:Hyperbolic tilings]]
[[Category:Isogonal tilings]]
[[Category:Isohedral tilings]]
[[Category:Regular tilings]]