In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of .
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.
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 | 4, 3 and 2 colors | ||||
|---|---|---|---|---|---|---|---|
| [∞,4], (*∞42) | [∞,∞], (*∞∞2) | [(∞,∞,∞)], (*∞∞∞) | (*∞∞∞∞) | ||||
| r = | t0,2(∞,∞,∞) = r | t0,1,2,3(∞,∞,∞,∞) = r = | |||||
(1111) | (1212) | (1213) | (1112) | (1234) | (1123) | (1122) | |
| = | = = | = = | |||||
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, and Coxeter diagram, with n progressing to infinity.