Tetraoctagonal tiling
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| Tetraoctagonal tiling | |
|---|---|
| Tetraoctagonal tiling Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | (4.8)2 |
| Schläfli symbol | r{8,4} or rr{8,8} rr(4,4,4) t0,1,2,3(∞,4,∞,4) |
| Wythoff symbol | 2 | 8 4 |
| Coxeter diagram | File:CDel node.pngFile:CDel 8.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel node 1.pngFile:CDel split1-84.pngFile:CDel nodes.png File:CDel node 1.pngFile:CDel 8.pngFile:CDel node.pngFile:CDel 8.pngFile:CDel node 1.png or File:CDel node.pngFile:CDel split1-88.pngFile:CDel nodes 11.png File:CDel label4.pngFile:CDel branch 11.pngFile:CDel split2-44.pngFile:CDel node.png File:CDel labelinfin.pngFile:CDel branch 11.pngFile:CDel 4a4b.pngFile:CDel branch 11.pngFile:CDel labelinfin.png |
| Symmetry group | [8,4], (*842) [8,8], (*882) [(4,4,4)], (*444) [(∞,4,∞,4)], (*4242) |
| Dual | Order-8-4 quasiregular rhombic tiling |
| Properties | Vertex-transitive edge-transitive |
In geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.
Constructions
There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,4] or (*842) orbifold symmetry. Removing the mirror between the order 2 and 4 points, [8,4,1+], gives [8,8], (*882). Removing the mirror between the order 2 and 8 points, [1+,8,4], gives [(4,4,4)], (*444). Removing both mirrors, [1+,8,4,1+], leaves a rectangular fundamental domain, [(∞,4,∞,4)], (*4242).
Symmetry
The dual tiling has face configuration V4.8.4.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4242), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*42) orbifold.
| File:Ord84 qreg rhombic til.png | File:H2chess 248e.png |
Related polyhedra and tiling
| *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 | File:Uniform tiling 432-t1.png | File:Uniform tiling 44-t1.png | File:H2-5-4-rectified.svg | File:H2 tiling 246-2.png | File:H2 tiling 247-2.png | File:H2 tiling 248-2.png | File:H2 tiling 24i-2.png | |
| Config. | (4.3)2 | (4.4)2 | (4.5)2 | (4.6)2 | (4.7)2 | (4.8)2 | (4.∞)2 | (4.ni)2 |
See also
Wikimedia Commons has media related to Uniform tiling 4-8-4-8.
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.