| Order-4 octagonal tiling | |
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 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 8 |
| Schläfli symbol | {8,4} r{8,8} |
| Wythoff symbol | 4 | 8 2 |
| Coxeter diagram |     or  
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| Symmetry group | , (*842) , (*882) |
| Dual | Order-8 square tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}. Its checkerboard coloring can be called a octaoctagonal tiling, and Schläfli symbol of r{8,8}.
Uniform constructions
There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the kaleidoscope. Removing the mirror between the order 2 and 4 points, , gives , (*884) symmetry. Removing two mirrors as , leaves remaining mirrors *4444 symmetry.
| Uniform Coloring |
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| Symmetry | (*842)   
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(*882)  =
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(*4444)   =
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| Symbol | {8,4} | r{8,8} | r(8,4,8) = r{8,8}1⁄2 | r{8,4}1⁄8 = r{8,8}1⁄4 |
| Coxeter diagram |
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Symmetry
This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting as edges of a regular hexagon. This symmetry by orbifold notation is called (*22222222) or (*2) with 8 order-2 mirror intersections. In Coxeter notation can be represented as , removing two of three mirrors (passing through the octagon center) in the symmetry. Adding a bisecting mirror through 2 vertices of an octagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 4 bisecting mirrors through the vertices defines *444 symmetry. Adding 4 bisecting mirrors through the edge defines *4222 symmetry. Adding all 8 bisectors leads to full *842 symmetry.
 *444 |
 *4222 |
 *832 |
The kaleidoscopic domains can be seen as bicolored octagonal tiling, representing mirror images of the fundamental domain. This coloring represents the uniform tiling r{8,8}, a quasiregular tiling and it can be called a octaoctagonal tiling.
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Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram 
, progressing to infinity.
| *n42 symmetry mutation of regular tilings: {n,4} | |||||||
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| Spherical | Euclidean | Hyperbolic tilings | |||||
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| 2 | 3 | 4 | 5 | 6 | 7 | 8 | ...∞ |
| Regular tilings: {n,8} | |||||||||||
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| Spherical | Hyperbolic tilings | ||||||||||
 {2,8} 
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 {3,8} 
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 {4,8}
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 {5,8} 
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 {6,8} 
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 {7,8} 
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... |  {∞,8} 
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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.
 {3,4}
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{4,4}
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{5,4}
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 {6,4}
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 {7,4}
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 {8,4}
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... | {∞,4}
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| Uniform octagonal/square tilings | |||||||||||
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| , (*842) (with (*882), (*444) , (*4222) index 2 subsymmetries) (And (*4242) index 4 subsymmetry) | |||||||||||
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| {8,4} | t{8,4} |
r{8,4} | 2t{8,4}=t{4,8} | 2r{8,4}={4,8} | rr{8,4} | tr{8,4} | |||||
| Uniform duals | |||||||||||
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| V8 | V4.16.16 | V(4.8) | V8.8.8 | V4 | V4.4.4.8 | V4.8.16 | |||||
| Alternations | |||||||||||
(*444) |
(8*2) |
(*4222) |
(4*4) |
(*882) |
(2*42) |
(842) | |||||
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| h{8,4} | s{8,4} | hr{8,4} | s{4,8} | h{4,8} | hrr{8,4} | sr{8,4} | |||||
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| V(4.4) | V3.(3.8) | V(4.4.4) | V(3.4) | V8 | V4.4 | V3.3.4.3.8 | |||||
| Uniform octaoctagonal tilings | |||||||||||
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| Symmetry: , (*882) | |||||||||||
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| {8,8} | t{8,8} |
r{8,8} | 2t{8,8}=t{8,8} | 2r{8,8}={8,8} | rr{8,8} | tr{8,8} | |||||
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| V8 | V8.16.16 | V8.8.8.8 | V8.16.16 | V8 | V4.8.4.8 | V4.16.16 | |||||
| Alternations | |||||||||||
(*884) |
(8*4) |
(*4242) |
(8*4) |
(*884) |
(2*44) |
(882) | |||||
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| h{8,8} | s{8,8} | hr{8,8} | s{8,8} | h{8,8} | hrr{8,8} | sr{8,8} | |||||
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| V(4.8) | V3.4.3.8.3.8 | V(4.4) | V3.4.3.8.3.8 | V(4.8) | V4 | V3.3.8.3.8 | |||||
See also
References
- 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
- 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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