| Truncated triapeirogonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 4.6.∞ |
| Schläfli symbol | tr{∞,3} or |
| Wythoff symbol | 2 ∞ 3 | |
| Coxeter diagram |    or  
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| Symmetry group | , (*∞32) |
| Dual | Order 3-infinite kisrhombille |
| Properties | Vertex-transitive |
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 , *∞32 symmetry. There are 3 small index subgroup constructed from 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), and its direct subgroup , (∞∞3), and semidirect subgroup , (3*∞). Given with generating mirrors {0,1,2}, then its index 4 subgroup has generators {0,121,212}.
An index 6 subgroup constructed as , becomes , (*∞∞∞).
| Index | 1 | 2 | 3 | 4 | 6 | 8 | 12 | 24 | ||
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| Diagrams | 
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| Coxeter (orbifold) |
  =   (*∞32) |
 =    (*∞33) |
 (3*∞) |
(*∞∞2) |
(*∞∞3) |
  =   (*∞) |
(*(∞2)) |
(*(∞3)) |
(*∞) |
(*∞) |
| Direct subgroups | ||||||||||
| Index | 2 | 4 | 6 | 8 | 12 | 16 | 24 | 48 | ||
| Diagrams | 
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| Coxeter (orbifold) |
=  (∞32) |
= (∞33) |
(∞∞2) |
(∞∞3) |
= (∞) |
(∞2) |
(∞3) |
(∞) |
(∞) | |
Related polyhedra and tiling
| Paracompact uniform tilings in family | ||||||||||
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| Symmetry: , (*∞32) | (∞32) |
(*∞33) |
(3*∞) | |||||||
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| {∞,3} | t{∞,3} | r{∞,3} | t{3,∞} | {3,∞} | rr{∞,3} | tr{∞,3} | sr{∞,3} | h{∞,3} | h2{∞,3} | s{3,∞} |
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| V∞ | V3.∞.∞ | V(3.∞) | V6.6.∞ | V3 | V4.3.4.∞ | V4.6.∞ | V3.3.3.3.∞ | V(3.∞) | V3.3.3.3.3.∞ | |
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.
| *n32 symmetry mutation of omnitruncated tilings: 4.6.2n | ||||||||||||
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| Sym. *n32 |
Spherical | Euclid. | Compact hyperb. | Paraco. | Noncompact hyperbolic | |||||||
| *232 |
*332 |
*432 |
*532 |
*632 |
*732 |
*832 |
*∞32 |
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| Config. | 4.6.4 | 4.6.6 | 4.6.8 | 4.6.10 | 4.6.12 | 4.6.14 | 4.6.16 | 4.6.∞ | 4.6.24i | 4.6.18i | 4.6.12i | 4.6.6i |
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| Config. | V4.6.4 | V4.6.6 | V4.6.8 | V4.6.10 | V4.6.12 | V4.6.14 | V4.6.16 | V4.6.∞ | V4.6.24i | V4.6.18i | V4.6.12i | V4.6.6i |
See also
References
- Norman W. Johnson and Asia Ivic Weiss, Quadratic Integers and Coxeter Groups, Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336
- 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.
External links
- Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
- Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
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