truncated tetraoctagonal tiling
{{short description|Semiregular tiling in geometry}}
{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|U84_012}}
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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colspan=2|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=
File:Truncated_tetraoctagonal_tiling_with_mirrors.png
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).
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!colspan=12| Small index subgroups of [8,4] (*842) |
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!1 !colspan=3|2 !colspan=2|4 |
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!Diagram |
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|[8,4] |[1+,8,4] |[8,4,1+] |[8,1+,4] |[1+,8,4,1+] |[8+,4+] |
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|*842 |*444 |*882 |42× |
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!colspan=7|Semidirect subgroups |
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!Diagram | |
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!Coxeter | |[8,4+] |[8+,4] |[(8,4,2+)] |[8,1+,4,1+] |[1+,8,1+,4] |
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!Orbifold | |4*4 |8*2 |2*42 |2*44 |4*22 |
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!colspan=7|Direct subgroups |
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!Index !2 !colspan=3|4 !colspan=2|8 |
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!Diagram |colspan=2|120px |
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!Coxeter |[8,4]+ |[8,4+]+ |[8+,4]+ |[8,1+,4]+ |colspan=2|[8+,4+]+ = [1+,8,1+,4,1+] |
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!Orbifold |842 |444 |882 |4222 |colspan=2|4242 |
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!colspan=6|Radical subgroups |
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!Index ! !8 !colspan=2|16 !32 |
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!Diagram | |
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!Coxeter | |[8,4*] |[8*,4] |[8,4*]+ |[8*,4]+ |
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!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.
{{Order 8-4 tiling table}}
{{Omnitruncated4 table}}
{{Omnitruncated_symmetric_table}}
See also
{{Commons category|Uniform tiling 4-8-16}}
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)
- {{Cite book|title=The Beauty of Geometry: Twelve Essays|year=1999|publisher=Dover Publications|lccn=99035678|isbn=0-486-40919-8|chapter=Chapter 10: Regular honeycombs in hyperbolic space}}
External links
- {{MathWorld | urlname= HyperbolicTiling | title = Hyperbolic tiling}}
- {{MathWorld | urlname=PoincareHyperbolicDisk | title = Poincaré hyperbolic disk }}
- [http://bork.hampshire.edu/~bernie/hyper/ Hyperbolic and Spherical Tiling Gallery]
- [http://geometrygames.org/KaleidoTile/index.html KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings]
- [http://www.hadron.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch]
{{Tessellation}}