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 rectified Order-3 heptagonal tiling. There are two triangles and two heptagons alternating on each vertex. It has Schläfli symbol of r{7,3}.

Images

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|240px
Klein disk model of this tiling preserves straight lines, but distorts angles

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The dual tiling is called an Order-7-3 rhombille tiling, made from rhombic faces, alternating 3 and 7 per vertex.

{{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]⁺, (732)

| Face_Type=V3.7.3.7

| Dual=Triheptagonal tiling

}}

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

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7-3 rhombile tiling in band model

The triheptagonal tiling can be seen in a sequence of quasiregular polyhedrons and tilings:

From a Wythoff construction there are eight hyperbolic 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.

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}}

{{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]