| Infinite-order apeirogonal tiling | |
|---|---|
 Poincaré disk model of thehyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | ∞∞ |
| Schläfli symbol | {∞,∞} |
| Wythoff symbol | ∞ | ∞ 2 ∞ ∞ | ∞ |
| Coxeter diagram |       |
| Symmetry group | [∞,∞], (*∞∞2) [(∞,∞,∞)], (*∞∞∞) |
| Dual | self-dual |
| Properties | Vertex-transitive,edge-transitive,face-transitive |
Theinfinite-order apeirogonal tiling is aregular tiling of thehyperbolic plane. It hasSchläfli symbol of {∞,∞}, which means it hascountably infinitely manyapeirogons around all its ideal vertices.
This tiling represents the fundamental domains of *∞∞ symmetry.
This tiling can also be alternately colored in the [(∞,∞,∞)] symmetry from 3 generator positions.
| Domains | 0 | 1 | 2 |
|---|---|---|---|
 symmetry: [(∞,∞,∞)] |  t0{(∞,∞,∞)}  |  t1{(∞,∞,∞)} |  t2{(∞,∞,∞)}  |
The union of this tiling and its dual can be seen as orthogonal red and blue lines here, and combined define the lines of a *2∞2∞ fundamental domain.
= ∪| Paracompact uniform tilings in [∞,∞] family | ||||||
|---|---|---|---|---|---|---|
=   =  | = =  | =  = | = = | = = | = | = |
|  |  |  |  |  |  |
| {∞,∞} | t{∞,∞} | r{∞,∞} | 2t{∞,∞}=t{∞,∞} | 2r{∞,∞}={∞,∞} | rr{∞,∞} | tr{∞,∞} |
| Dual tilings | ||||||
 | | | | | | |
 |  |  |  |  |  |  |
| V∞∞ | V∞.∞.∞ | V(∞.∞)2 | V∞.∞.∞ | V∞∞ | V4.∞.4.∞ | V4.4.∞ |
| Alternations | ||||||
| [1+,∞,∞] (*∞∞2) | [∞+,∞] (∞*∞) | [∞,1+,∞] (*∞∞∞∞) | [∞,∞+] (∞*∞) | [∞,∞,1+] (*∞∞2) | [(∞,∞,2+)] (2*∞∞) | [∞,∞]+ (2∞∞) |
 | | | | | | |
|  |  |  | |  | |
| h{∞,∞} | s{∞,∞} | hr{∞,∞} | s{∞,∞} | h2{∞,∞} | hrr{∞,∞} | sr{∞,∞} |
| Alternation duals | ||||||
 | | | | | | |
|  | |  | |||
| V(∞.∞)∞ | V(3.∞)3 | V(∞.4)4 | V(3.∞)3 | V∞∞ | V(4.∞.4)2 | V3.3.∞.3.∞ |
| Paracompact uniform tilings in [(∞,∞,∞)] family | ||||||
|---|---|---|---|---|---|---|
| | | | | | |
| | | | | | |
|  | |  | |  |  |
| (∞,∞,∞) h{∞,∞} | r(∞,∞,∞) h2{∞,∞} | (∞,∞,∞) h{∞,∞} | r(∞,∞,∞) h2{∞,∞} | (∞,∞,∞) h{∞,∞} | r(∞,∞,∞) r{∞,∞} | t(∞,∞,∞) t{∞,∞} |
| Dual tilings | ||||||
 |  |  |  |  |  | |
| V∞∞ | V∞.∞.∞.∞ | V∞∞ | V∞.∞.∞.∞ | V∞∞ | V∞.∞.∞.∞ | V∞.∞.∞ |
| Alternations | ||||||
| [(1+,∞,∞,∞)] (*∞∞∞∞) | [∞+,∞,∞)] (∞*∞) | [∞,1+,∞,∞)] (*∞∞∞∞) | [∞,∞+,∞)] (∞*∞) | [(∞,∞,∞,1+)] (*∞∞∞∞) | [(∞,∞,∞+)] (∞*∞) | [∞,∞,∞)]+ (∞∞∞) |
 | | |  | |  | |
| | | | | |  |
| Alternation duals | ||||||
| | | | | | |
| V(∞.∞)∞ | V(∞.4)4 | V(∞.∞)∞ | V(∞.4)4 | V(∞.∞)∞ | V(∞.4)4 | V3.∞.3.∞.3.∞ |
