Tetraheptagonal tiling
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| Tetraheptagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | (4.7)2 |
| Schläfli symbol | r{7,4} or rr{7,7} |
| Wythoff symbol | 2 | 7 4 7 7 | 2 |
| Coxeter diagram |     |
| Symmetry group | [7,4], (*742) [7,7], (*772) |
| Dual | Order-7-4 rhombille tiling |
| Properties | Vertex-transitive edge-transitive |
In geometry, the tetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of r{4,7}.
Symmetry
[edit] A half symmetry [1+,4,7] = [7,7] construction exists, which can be seen as two colors of heptagons. This coloring can be called a rhombiheptaheptagonal tiling. |  The dual tiling is made of rhombic faces and has a face configuration V4.7.4.7. |
Related polyhedra and tiling
[edit]| *n42 symmetry mutations of quasiregular tilings: (4.n)2 | ||||||||
|---|---|---|---|---|---|---|---|---|
| Symmetry *4n2 [n,4] | Spherical | Euclidean | Compact hyperbolic | Paracompact | Noncompact | |||
| *342 [3,4] | *442 [4,4] | *542 [5,4] | *642 [6,4] | *742 [7,4] | *842 [8,4]... | *∞42 [∞,4] | [ni,4] | |
| Figures |  |  |  |  | |  |  | |
| Config. | (4.3)2 | (4.4)2 | (4.5)2 | (4.6)2 | (4.7)2 | (4.8)2 | (4.∞)2 | (4.ni)2 |
| 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 | ||
| 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 | ||||
| Dimensional family of quasiregular polyhedra and tilings: 7.n.7.n | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry *7n2 [n,7] | Hyperbolic... | Paracompact | Noncompact | ||||||||
| *732 [3,7] | *742 [4,7] | *752 [5,7] | *762 [6,7] | *772 [7,7] | *872 [8,7]... | *∞72 [∞,7] | [iπ/λ,7] | ||||
| Coxeter |  | |  |  | |  |  |  | |||
| Quasiregular figures configuration |  3.7.3.7 | 4.7.4.7 |  7.5.7.5 |  7.6.7.6 |  7.7.7.7 |  7.8.7.8 |  7.∞.7.∞ | 7.∞.7.∞ | |||
See also
[edit]
Wikimedia Commons has media related to Uniform tiling 4-7-4-7.
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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