Order-4 heptagonal tiling
| Order-4 heptagonal tiling | |
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 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex figure | 74 |
| Schläfli symbol | {7,4} r{7,7} |
| Wythoff symbol | 4 | 7 2 2 | 7 7 |
| Coxeter diagram |     |
| Symmetry group | [7,4], (*742) [7,7], (*772) |
| Dual | Order-7 square tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-4 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,4}.
Symmetry
This tiling represents a hyperbolic kaleidoscope of 7 mirrors meeting as edges of a regular heptagon. This symmetry by orbifold notation is called *2222222 with 7 order-2 mirror intersections. In Coxeter notation can be represented as [1+,7,1+,4], removing two of three mirrors (passing through the heptagon center) in the [7,4] symmetry.
The kaleidoscopic domains can be seen as bicolored heptagons, representing mirror images of the fundamental domain. This coloring represents the uniform tiling t1{7,7} and as a quasiregular tiling is called a heptaheptagonal tiling.
Related polyhedra and tiling
| Symmetry: [7,4], (*742) | [7,4]+, (742) | [7+,4], (7*2) | [7,4,1+], (*772) | |||||||
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| {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} | |
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| 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 | |
| Symmetry: [7,7], (*772) | [7,7]+, (772) | ||||||
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| {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} |
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| V77 | V7.14.14 | V7.7.7.7 | V7.14.14 | V77 | V4.7.4.7 | V4.14.14 | V3.3.7.3.7 |
This tiling is topologically related as a part of sequence of regular tilings with heptagonal faces, starting with the heptagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram 
, progressing to infinity.
 {7,3}  |
{7,4} |
 {7,5}  |
 {7,6}  |
{7,7} |
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
| *n42 symmetry mutation of regular tilings: {n,4} | |||||||
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| Spherical | Euclidean | Hyperbolic tilings | |||||
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| 24 | 34 | 44 | 54 | 64 | 74 | 84 | ...∞4 |
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.
See also
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Wikimedia Commons has media related to Order-4 heptagonal tiling. |
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
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