# Order-4 apeirogonal tiling

Order-4 apeirogonal tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic regular tiling |
| Vertex configuration | ∞$^4$ |
| Schläfli symbol | {∞,4} r{∞,∞} t(∞,∞,∞) t$_{0,1,2,3}$(∞,∞,∞,∞) |
| Wythoff symbol | 4 | ∞ 2 2 | ∞ ∞ ∞ ∞ | ∞ |
| Coxeter diagram | |
| Symmetry group | [∞,4], (*∞42) [∞,∞], (*∞∞2) [(∞,∞,∞)], (*∞∞∞) (*∞∞∞∞) |
| Dual | Infinite-order square tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive edge-transitive |
In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,4}.

## Symmetry

This tiling represents the mirror lines of *2$^∞$ symmetry. Its dual tiling represents the fundamental domains of orbifold notation *∞∞∞∞ symmetry, a square domain with four ideal vertices.

## Uniform colorings

Like the Euclidean square tiling there are 9 uniform colorings for this tiling, with 3 uniform colorings generated by triangle reflective domains. A fourth can be constructed from an infinite square symmetry (*∞∞∞∞) with 4 colors around a vertex. The checker board, r{∞,∞}, coloring defines the fundamental domains of [(∞,4,4)], (*∞44) symmetry, usually shown as black and white domains of reflective orientations.
| 1 color | 2 color | 3 and 2 colors | 3 and 2 colors | 4, 3 and 2 colors | 4, 3 and 2 colors | 4, 3 and 2 colors |
| [∞,4], (*∞42) | [∞,∞], (*∞∞2) | [(∞,∞,∞)], (*∞∞∞) | [(∞,∞,∞)], (*∞∞∞) | (*∞∞∞∞) | (*∞∞∞∞) | (*∞∞∞∞) |
| {∞,4} | r{∞,∞} = {∞,4}1⁄2 | t$_{0,2}$(∞,∞,∞) = r{∞,∞}1⁄2 | t$_{0,2}$(∞,∞,∞) = r{∞,∞}1⁄2 | t$_{0,1,2,3}$(∞,∞,∞,∞) = r{∞,∞}1⁄4 = {∞,4}1⁄8 | t$_{0,1,2,3}$(∞,∞,∞,∞) = r{∞,∞}1⁄4 = {∞,4}1⁄8 | t$_{0,1,2,3}$(∞,∞,∞,∞) = r{∞,∞}1⁄4 = {∞,4}1⁄8 |
| (1111) | (1212) | (1213) | (1112) | (1234) | (1123) | (1122) |

## Related polyhedra and tiling

This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
*n42 symmetry mutation of regular tilings: {n,4} vte :
| Spherical | Spherical | Euclidean | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings |
| 2$^4$ | 3$^4$ | 4$^4$ | 5$^4$ | 6$^4$ | 7$^4$ | 8$^4$ | ...∞$^4$ |
Paracompact uniform tilings in [∞,4] family vte :
| {∞,4} | t{∞,4} | r{∞,4} | 2t{∞,4}=t{4,∞} | 2r{∞,4}={4,∞} | rr{∞,4} | tr{∞,4} |
| Dual figures |||||||
| V∞$^4$ | V4.∞.∞ | V(4.∞)$^2$ | V8.8.∞ | V4$^∞$ | V4$^3$.∞ | V4.8.∞ |
| Alternations |||||||
| [1$^+$,∞,4] (*44∞) | [∞$^+$,4] (∞*2) | [∞,1$^+$,4] (*2∞2∞) | [∞,4$^+$] (4*∞) | [∞,4,1$^+$] (*∞∞2) | [(∞,4,2$^+$)] (2*2∞) | [∞,4]$^+$ (∞42) |
| h{∞,4} | s{∞,4} | hr{∞,4} | s{4,∞} | h{4,∞} | hrr{∞,4} | s{∞,4} |
| Alternation duals |||||||
| V(∞.4)$^4$ | V3.(3.∞)$^2$ | V(4.∞.4)$^2$ | V3.∞.(3.4)$^2$ | V∞$^∞$ | V∞.4$^4$ | V3.3.4.3.∞ |
Paracompact uniform tilings in [∞,∞] family vte :
| {∞,∞} | 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{∞,∞} | h$_2${∞,∞} | 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 vte :
| (∞,∞,∞) h{∞,∞} | r(∞,∞,∞) h$_2${∞,∞} | (∞,∞,∞) h{∞,∞} | r(∞,∞,∞) h$_2${∞,∞} | (∞,∞,∞) 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.∞ |