Truncated heptagonal tiling
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| Truncated heptagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 3.14.14 |
| Schläfli symbol | t{7,3} |
| Wythoff symbol | 2 3 | 7 |
| Coxeter diagram |     |
| Symmetry group | [7,3], (*732) |
| Dual | Order-7 triakis triangular tiling |
| Properties | Vertex-transitive |
In geometry, the truncated heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are one triangle and two tetradecagons on each vertex. It has Schläfli symbol of t{7,3}. The tiling has a vertex configuration of 3.14.14.
Dual tiling
[edit]The dual tiling is called an order-7 triakis triangular tiling, seen as an order-7 triangular tiling with each triangle divided into three by a center point.
Related polyhedra and tilings
[edit]This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.
| *n32 symmetry mutation of truncated tilings: t{n,3} | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry *n32 [n,3] | Spherical | Euclid. | Compact hyperb. | Paraco. | Noncompact hyperbolic | ||||||
| *232 [2,3] | *332 [3,3] | *432 [4,3] | *532 [5,3] | *632 [6,3] | *732 [7,3] | *832 [8,3]... | *∞32 [∞,3] | [12i,3] | [9i,3] | [6i,3] | |
| Truncated figures |  |  |  |  |  | |  |  |  |  |  |
| Symbol | t{2,3} | t{3,3} | t{4,3} | t{5,3} | t{6,3} | t{7,3} | t{8,3} | t{∞,3} | t{12i,3} | t{9i,3} | t{6i,3} |
| Triakis figures |  |  |  |  |  |  |  |  | |||
| Config. | V3.4.4 | V3.6.6 | V3.8.8 | V3.10.10 | V3.12.12 | V3.14.14 | V3.16.16 | V3.∞.∞ | |||
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 eight forms.
| Uniform heptagonal/triangular tilings | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry: [7,3], (*732) | [7,3]+, (732) | ||||||||||
| | | | | | | |  | ||||
 | |  |  |  |  |  |  | ||||
| {7,3} | t{7,3} | r{7,3} | t{3,7} | {3,7} | rr{7,3} | tr{7,3} | sr{7,3} | ||||
| Uniform duals | |||||||||||
 | | | | | | |  | ||||
| | |  |  | |  |  |  | ||||
| V73 | V3.14.14 | V3.7.3.7 | V6.6.7 | V37 | V3.4.7.4 | V4.6.14 | V3.3.3.3.7 | ||||
See also
[edit]
References
[edit]- 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)
- "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
[edit]- Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
- Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
- 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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