Truncated tetraheptagonal tiling
From Wikipedia the free encyclopedia
| Truncated tetraheptagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 4.8.14 |
| Schläfli symbol | tr{7,4} or |
| Wythoff symbol | 2 7 4 | |
| Coxeter diagram |    |
| Symmetry group | [7,4], (*742) |
| Dual | Order-4-7 kisrhombille tiling |
| Properties | Vertex-transitive |
In geometry, the truncated tetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of tr{4,7}.
Images[edit]
Poincaré disk projection, centered on 14-gon:
Symmetry[edit]
The dual to this tiling represents the fundamental domains of [7,4] (*742) symmetry. There are three small index subgroups constructed from [7,4] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
| Small index subgroups of [7,4] (*742) | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Index | 1 | 2 | 14 | ||||||||
| Diagram |  |  |  |  | |||||||
| Coxeter (orbifold) | [7,4] = (*742) | [7,4,1+] =  =   (*772) | [7+,4] =  (7*2) | [7*,4] =   (*2222222) | |||||||
| Index | 2 | 4 | 28 | ||||||||
| Diagram |  |  |  | ||||||||
| Coxeter (orbifold) | [7,4]+ = (742) | [7+,4]+ = =   (772) | [7*,4]+ = (2222222) | ||||||||
Related polyhedra and tiling[edit]
| Uniform heptagonal/square tilings | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry: [7,4], (*742) | [7,4]+, (742) | [7+,4], (7*2) | [7,4,1+], (*772) | ||||||||
 | | | | | | |  | | | ||
 |  |  |  |  |  |  |  |  |  | ||
| {7,4} | t{7,4} | r{7,4} | 2t{7,4}=t{4,7} | 2r{7,4}={4,7} | rr{7,4} | tr{7,4} | sr{7,4} | s{7,4} | h{4,7} | ||
| Uniform duals | |||||||||||
 | | | | | | |  | | | ||
| |  |  |  | |  |  |  | ||||
| V74 | V4.14.14 | V4.7.4.7 | V7.8.8 | V47 | V4.4.7.4 | V4.8.14 | V3.3.4.3.7 | V3.3.7.3.7 | V77 | ||
| *n42 symmetry mutation of omnitruncated tilings: 4.8.2n | ||||||||
|---|---|---|---|---|---|---|---|---|
| Symmetry *n42 [n,4] | Spherical | Euclidean | Compact hyperbolic | Paracomp. | ||||
| *242 [2,4] | *342 [3,4] | *442 [4,4] | *542 [5,4] | *642 [6,4] | *742 [7,4] | *842 [8,4]... | *∞42 [∞,4] | |
| Omnitruncated figure |  4.8.4 |  4.8.6 |  4.8.8 |  4.8.10 |  4.8.12 | 4.8.14 |  4.8.16 |  4.8.∞ |
| Omnitruncated duals |  V4.8.4 |  V4.8.6 |  V4.8.8 |  V4.8.10 |  V4.8.12 | V4.8.14 |  V4.8.16 |  V4.8.∞ |
| *nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry *nn2 [n,n] | Spherical | Euclidean | Compact hyperbolic | Paracomp. | ||||||||||
| *222 [2,2] | *332 [3,3] | *442 [4,4] | *552 [5,5] | *662 [6,6] | *772 [7,7] | *882 [8,8]... | *∞∞2 [∞,∞] | |||||||
| Figure |  |  | |  |  |  |  |  | ||||||
| Config. | 4.4.4 | 4.6.6 | 4.8.8 | 4.10.10 | 4.12.12 | 4.14.14 | 4.16.16 | 4.∞.∞ | ||||||
| Dual |  |  | |  |  |  |  | | ||||||
| Config. | V4.4.4 | V4.6.6 | V4.8.8 | V4.10.10 | V4.12.12 | V4.14.14 | V4.16.16 | V4.∞.∞ | ||||||
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.
See also[edit]
Wikimedia Commons has media related to Uniform tiling 4-8-14.
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
 | This hyperbolic geometry-related article is a stub. You can help Wikipedia by expanding it. |