Snub heptaheptagonal tiling
In geometry, the snub heptaheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{7,7}, constructed from two regular heptagons and three equilateral triangles around every vertex.
| Snub heptaheptagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 3.3.7.3.7 |
| Schläfli symbol | sr{7,7} or |
| Wythoff symbol | | 7 7 2 |
| Coxeter diagram |     |
| Symmetry group | [7,7]+, (772) [7+,4], (7*2) |
| Dual | Order-7-7 floret pentagonal tiling |
| Properties | Vertex-transitive |
Images
Drawn in chiral pairs, with edges missing between black triangles:
Symmetry
A double symmetry coloring can be constructed from [7,4] symmetry with only one color heptagon.
Related tilings
| Uniform heptaheptagonal tilings | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry: [7,7], (*772) | [7,7]+, (772) | ||||||||||
   =  |
= |
 =  |
 = |
= |
 = |
= |
 = | ||||
 |
 |
 |
 |
 |
 |
 |
| ||||
| {7,7} | t{7,7} |
r{7,7} | 2t{7,7}=t{7,7} | 2r{7,7}={7,7} | rr{7,7} | tr{7,7} | sr{7,7} | ||||
| Uniform duals | |||||||||||
 |
|
|
|
|
|
|
 | ||||
|
 |
 |
|
|
 |
 |
|||||
| V77 | V7.14.14 | V7.7.7.7 | V7.14.14 | V77 | V4.7.4.7 | V4.14.14 | V3.3.7.3.7 | ||||
| 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 | ||
| 4n2 symmetry mutations of snub tilings: 3.3.n.3.n | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry 4n2 |
Spherical | Euclidean | Compact hyperbolic | Paracompact | |||||||
| 222 | 322 | 442 | 552 | 662 | 772 | 882 | ∞∞2 | ||||
| Snub figures |
 |
 |
 |
 |
 |
|
 |
 | |||
| Config. | 3.3.2.3.2 | 3.3.3.3.3 | 3.3.4.3.4 | 3.3.5.3.5 | 3.3.6.3.6 | 3.3.7.3.7 | 3.3.8.3.8 | 3.3.∞.3.∞ | |||
| Gyro figures |
 |
 |
 |
 | |||||||
| Config. | V3.3.2.3.2 | V3.3.3.3.3 | V3.3.4.3.4 | V3.3.5.3.5 | V3.3.6.3.6 | V3.3.7.3.7 | V3.3.8.3.8 | V3.3.∞.3.∞ | |||
See also
 |
Wikimedia Commons has media related to Uniform tiling 3-3-7-3-7. |
- Square tiling
- Uniform tilings in hyperbolic plane
- List of regular polytopes
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, 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
- 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 article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.