Truncated order-8 octagonal tiling
| Truncated order-8 octagonal tiling | |
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 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 8.16.16 |
| Schläfli symbol | t{8,8} t{(8,8,4)} |
| Wythoff symbol | 2 8 | 4 |
| Coxeter diagram |      |
| Symmetry group | [8,8], (*882) [(8,8,4)], (*884) |
| Dual | Order-8 octakis octagonal tiling |
| Properties | Vertex-transitive |
In geometry, the truncated order-8 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,8}.
Uniform colorings
This tiling can also be constructed in *884 symmetry with 3 colors of faces:
Related polyhedra and tiling
| Symmetry: [8,8], (*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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| V88 | V8.16.16 | V8.8.8.8 | V8.16.16 | V88 | V4.8.4.8 | V4.16.16 | |||||
| Alternations | |||||||||||
| [1+,8,8] (*884) |
[8+,8] (8*4) |
[8,1+,8] (*4242) |
[8,8+] (8*4) |
[8,8,1+] (*884) |
[(8,8,2+)] (2*44) |
[8,8]+ (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)8 | V3.4.3.8.3.8 | V(4.4)4 | V3.4.3.8.3.8 | V(4.8)8 | V46 | V3.3.8.3.8 | |||||
Symmetry
The dual of the tiling represents the fundamental domains of (*884) orbifold symmetry. From [(8,8,4)] (*884) symmetry, there are 15 small index subgroup (11 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled to 882 symmetry by adding a bisecting mirror across the fundamental domains. The subgroup index-8 group, [(1+,8,1+,8,1+,4)] (442442) is the commutator subgroup of [(8,8,4)].
| Fundamental domains |
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| Subgroup index | 1 | 2 | 4 | |||||
| Coxeter | [(8,8,4)] |
[(1+,8,8,4)]  |
[(8,8,1+,4)] |
[(8,1+,8,4)]  |
[(1+,8,8,1+,4)] |
[(8+,8+,4)] | ||
| orbifold | *884 | *8482 | *4444 | 2*4444 | 442× | |||
| Coxeter | [(8,8+,4)]  |
[(8+,8,4)] |
[(8,8,4+)] |
[(8,1+,8,1+,4)] |
[(1+,8,1+,8,4)] | |||
| Orbifold | 8*42 | 4*44 | 4*4242 | |||||
| Direct subgroups | ||||||||
| Subgroup index | 2 | 4 | 8 | |||||
| Coxeter | [(8,8,4)]+ |
[(1+,8,8+,4)] |
[(8+,8,1+,4)] |
[(8,1+,8,4+)] |
[(1+,8,1+,8,1+,4)] = [(8+,8+,4+)] | |||
| Orbifold | 844 | 8482 | 4444 | 442442 | ||||
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
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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