Order-8 square tiling
| Order-8 square tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 48 |
| Schläfli symbol | {4,8} |
| Wythoff symbol | 8 | 4 2 |
| Coxeter diagram |    
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| Symmetry group | [8,4], (*842) |
| Dual | Order-4 octagonal tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,8}.
Symmetry
[edit]This tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting as edges of a square, with eight squares around every vertex. This symmetry by orbifold notation is called (*4444) with 4 order-4 mirror intersections. In Coxeter notation can be represented as [1+,8,8,1+], (*4444 orbifold) removing two of three mirrors (passing through the square center) in the [8,8] symmetry. The *4444 symmetry can be doubled by bisecting the fundamental domain (square) by a mirror, creating *884 symmetry.
This bicolored square tiling shows the even/odd reflective fundamental square domains of this symmetry. This bicolored tiling has a wythoff construction (4,4,4), or {4[3]}, 
:
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Related polyhedra and tiling
[edit]This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).
| *n42 symmetry mutation of regular tilings: {4,n} |
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| Uniform octagonal/square tilings |
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| Uniform (4,4,4) tilings |
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See also
[edit]
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.
