Truncated triapeirogonal tiling
In geometry, the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr{∞,3}.
Symmetry
The dual of this tiling represents the fundamental domains of [∞,3], *∞32 symmetry. There are 3 small index subgroup constructed from [∞,3] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
A special index 4 reflective subgroup, is [(∞,∞,3)], (*∞∞3), and its direct subgroup [(∞,∞,3)]+, (∞∞3), and semidirect subgroup [(∞,∞,3+)], (3*∞).[1] Given [∞,3] with generating mirrors {0,1,2}, then its index 4 subgroup has generators {0,121,212}.
An index 6 subgroup constructed as [∞,3*], becomes [(∞,∞,∞)], (*∞∞∞).
| Index | 1 | 2 | 3 | 4 | 6 | 8 | 12 | 24 | ||
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| Diagrams | 
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| Coxeter (orbifold) |
[∞,3]    =    (*∞32) |
[1+,∞,3] =    (*∞33) |
[∞,3+] (3*∞) |
[∞,∞] (*∞∞2) |
[(∞,∞,3)] (*∞∞3) |
[∞,3*]  =   (*∞3) |
[∞,1+,∞] (*(∞2)2) |
[(∞,1+,∞,3)] (*(∞3)2) |
[1+,∞,∞,1+] (*∞4) |
[(∞,∞,3*)] (*∞6) |
| Direct subgroups | ||||||||||
| Index | 2 | 4 | 6 | 8 | 12 | 16 | 24 | 48 | ||
| Diagrams | 
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| Coxeter (orbifold) |
[∞,3]+ =  (∞32) |
[∞,3+]+ = (∞33) |
[∞,∞]+ (∞∞2) |
[(∞,∞,3)]+ (∞∞3) |
[∞,3*]+ = (∞3) |
[∞,1+,∞]+ (∞2)2 |
[(∞,1+,∞,3)]+ (∞3)2 |
[1+,∞,∞,1+]+ (∞4) |
[(∞,∞,3*)]+ (∞6) | |
Related polyhedra and tiling
This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram 
. For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.
See also
- List of uniform planar tilings
- Tilings of regular polygons
- Uniform tilings in hyperbolic plane
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.
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