Truncated tetraoctagonal tiling
In geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,4}.
Dual tiling
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The dual tiling is called an order-4-8 kisrhombille tiling, made as a complete bisection of the order-4 octagonal tiling, here with triangles are shown with alternating colors. This tiling represents the fundamental triangular domains of [8,4] (*842) symmetry. |
Symmetry
There are 15 subgroups constructed from [8,4] by mirror removal and alternation. 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 subgroup index-8 group, [1+,8,1+,4,1+] (4242) is the commutator subgroup of [8,4].
A larger subgroup is constructed as [8,4*], index 8, as [8,4+], (4*4) with gyration points removed, becomes (*4444) or (*44), and another [8*,4], index 16 as [8+,4], (8*2) with gyration points removed as (*22222222) or (*28). And their direct subgroups [8,4*]+, [8*,4]+, subgroup indices 16 and 32 respectively, can be given in orbifold notation as (4444) and (22222222).
Small index subgroups of [8,4] (*842) | |||||||||||
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Index | 1 | 2 | 4 | ||||||||
Diagram | ![]() |
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Coxeter | [8,4]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[1+,8,4]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[8,4,1+]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[8,1+,4]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[1+,8,4,1+]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[8+,4+]![]() ![]() ![]() ![]() ![]() | |||||
Orbifold | *842 | *444 | *882 | *4222 | *4242 | 42× | |||||
Semidirect subgroups | |||||||||||
Diagram | ![]() |
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Coxeter | [8,4+]![]() ![]() ![]() ![]() ![]() |
[8+,4]![]() ![]() ![]() ![]() ![]() |
[(8,4,2+)]![]() ![]() ![]() |
[8,1+,4,1+]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[1+,8,1+,4]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||||
Orbifold | 4*4 | 8*2 | 2*42 | 2*44 | 4*22 | ||||||
Direct subgroups | |||||||||||
Index | 2 | 4 | 8 | ||||||||
Diagram | ![]() |
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Coxeter | [8,4]+![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[8,4+]+![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[8+,4]+![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[8,1+,4]+![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[8+,4+]+ = [1+,8,1+,4,1+]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||||
Orbifold | 842 | 444 | 882 | 4222 | 4242 | ||||||
Radical subgroups | |||||||||||
Index | 8 | 16 | 32 | ||||||||
Diagram | ![]() |
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Coxeter | [8,4*]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[8*,4]![]() ![]() ![]() ![]() ![]() ![]() |
[8,4*]+![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[8*,4]+![]() ![]() ![]() ![]() ![]() ![]() | |||||||
Orbifold | *4444 | *22222222 | 4444 | 22222222 |
Related polyhedra and tilings
From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-4 octagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [8,4] symmetry, and 7 with subsymmetry.
See also
- Tilings of regular polygons
- List of uniform planar tilings
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.
External links
- Weisstein, Eric W.. "Hyperbolic tiling". http://mathworld.wolfram.com/HyperbolicTiling.html.
- Weisstein, Eric W.. "Poincaré hyperbolic disk". http://mathworld.wolfram.com/PoincareHyperbolicDisk.html.
- Hyperbolic and Spherical Tiling Gallery
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch
![]() | Original source: https://en.wikipedia.org/wiki/Truncated tetraoctagonal tiling.
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