Order-7 triangular tiling
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| Order-7 triangular tiling | |
|---|---|
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| Type | Hyperbolic regular tiling |
| Vertex configuration | |
| Schläfli symbol | |
| Wallpaper group | , |
| Dual | heptagonal tiling |
| Properties | vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,7}.
Hurwitz surfaces
Script error: No such module "labelled list hatnote". 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 surfaces (the Riemann surfaces with maximal symmetry group), giving them a triangulation whose symmetry group equals their automorphism group as Riemann surfaces.
The smallest of these is the Klein quartic, the most symmetric genus 3 surface, together with a tiling by 56 triangles, meeting at 24 vertices, with symmetry group the simple group of order 168, known as PSL(2,7). The resulting surface can in turn be polyhedrally immersed into Euclidean 3-space, yielding the small cubicuboctahedron.[1]
The dual order-3 heptagonal tiling has the same symmetry group, and thus yields heptagonal tilings of Hurwitz surfaces.
| File:3-7 kisrhombille.svg The symmetry group of the order-7 triangular tiling has fundamental domain the (2,3,7) Schwarz triangle, which yields this tiling. |
File:Small cubicuboctahedron.png The small cubicuboctahedron is a polyhedral immersion of the Klein quartic,[1] which, like all Hurwitz surfaces, is a quotient of this tiling. |
Related polyhedra and tiling
It is related to two star-tilings by the same vertex arrangement: the order-7 heptagrammic tiling, {7/2,7}, and heptagrammic-order heptagonal tiling, {7,7/2}.
This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {3,p}.
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|---|---|---|---|---|---|---|---|---|---|---|---|
| Spherical | Euclid. | Compact hyper. | Paraco. | Noncompact hyperbolic | |||||||
| File:Trigonal dihedron.svg | File:Uniform tiling 332-t2.svg | File:Uniform tiling 432-t2.svg | File:Uniform tiling 532-t2.svg | File:Uniform polyhedron-63-t2.svg | File:Order-7 triangular tiling.svg | File:H2-8-3-primal.svg | File:H2 tiling 23i-4.png | File:H2 tiling 23j12-4.png | File:H2 tiling 23j9-4.png | File:H2 tiling 23j6-4.png | File:H2 tiling 23j3-4.png |
| 3.3 | 33 | 34 | 35 | 36 | 37 | 38 | 3∞ | 312i | 39i | 36i | 33i |
This tiling is a part of regular series {n,7}:
| Tiles of the form {n,7} | ||||||||
|---|---|---|---|---|---|---|---|---|
| Spherical | Hyperbolic tilings Lua error in package.lua at line 80: module 'Module:Navbar/configuration' not found. | |||||||
| File:Spherical heptagonal hosohedron.svg {2,7} Template:CDD |
File:Order-7 triangular tiling.svg {3,7} Template:CDD |
File:H2 tiling 247-4.png {4,7} Template:CDD |
File:H2 tiling 257-4.png {5,7} Template:CDD |
File:H2 tiling 267-4.png {6,7} Template:CDD |
File:H2 tiling 277-1.png {7,7} Template:CDD |
File:H2 tiling 278-1.png {8,7} Template:CDD |
... | File:H2 tiling 27i-1.png {∞,7} Template:CDD |
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.
See also
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- Order-7 tetrahedral honeycomb
- List of regular polytopes
- List of uniform planar tilings
- Tilings of regular polygons
- Triangular tiling
- Uniform tilings in hyperbolic plane
References
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- ^ a b Lua error in package.lua at line 80: module 'Module:Footnotes/anchor_id_list' not found. Note each face in the polyhedron consist of multiple faces in the tiling – two triangular faces constitute a square face and so forth, as per this explanatory image Script error: No such module "webarchive"..
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- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Template:Isbn (Chapter 19, The Hyperbolic Archimedean Tessellations)
- Page Module:Citation/CS1/styles.css has no content."Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
- Page Module:Citation/CS1/styles.css has no content.Richter, David A., How to Make the Mathieu Group M24, archived from the original on 2010-01-16, retrieved 2010-04-15
External links
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- Hyperbolic and Spherical Tiling Gallery
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch
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