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
4, 3 and 2 colors
[∞,4], (*∞42)
[∞,∞], (*∞∞2)
[(∞,∞,∞)], (*∞∞∞)
(*∞∞∞∞)
{∞,4}
r{∞,∞} = {∞,4}1⁄2
t0,2(∞,∞,∞) = r{∞,∞}1⁄2
t0,1,2,3(∞,∞,∞,∞) = 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.
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
"Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN0-486-40919-8. LCCN99035678.