Truncated triapeirogonal tiling
| 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 |
| Symmetry group | [∞,3], (*∞32) |
| Dual | |
| 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 [∞,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*∞). 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 | ||
|---|---|---|---|---|---|---|---|---|---|---|
| Diagrams | ||||||||||
| Coxeter (orbifold) | [∞,3] = (*∞32) | [1+,∞,3] = (*∞33) | [∞,3+] (3*∞) | [∞,∞] (*∞∞2) | [(∞,∞,3)] () | [∞,3*] = (*∞3) | [∞,1+,∞] | [(∞,1+,∞,3)] | [1+,∞,∞,1+] (*∞4) | [(∞,∞,3*)] |
| Direct subgroups | ||||||||||
| Index | 2 | 4 | 6 | 8 | 12 | 16 | 24 | 48 | ||
| Diagrams | ||||||||||
| 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
| Paracompact uniform tilings in [∞,3] family | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry: [∞,3], (*∞32) | [∞,3]+ (∞32) | [1+,∞,3] (*∞33) | [∞,3+] (3*∞) | |||||||
= | = | = | = or | = or | = | |||||
| {∞,3} | t{∞,3} | r{∞,3} | t{3,∞} | {3,∞} | rr{∞,3} | tr{∞,3} | sr{∞,3} | h{∞,3} | h2{∞,3} | s{3,∞} |
| Uniform duals | ||||||||||
| V∞3 | V3.∞.∞ | V(3.∞)2 | V6.6.∞ | V3∞ | V4.3.4.∞ | V4.6.∞ | V3.3.3.3.∞ | V(3.∞)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 | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Sym. *n32 [n,3] | Spherical | Euclid. | Compact hyperb. | Paraco. | Noncompact hyperbolic | |||||||
| *232 [2,3] | *332 [3,3] | *432 [4,3] | *532 [5,3] | *632 [6,3] | *732 [7,3] | *832 [8,3] | *∞32 [∞,3] | [12i,3] | [9i,3] | [6i,3] | [3i,3] | |
| Figures | ||||||||||||
| 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 |
| Duals | ||||||||||||
| 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
- List of uniform planar tilings
- Tilings of regular polygons
- Uniform tilings in hyperbolic plane
External links
- Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
- Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.