Order-4 apeirogonal tiling

This tiling shows the mirror lines of the symmetry group written in orbifold notation as *2∞. Its dual tiling corresponds to the fundamental domains of the symmetry group written as *∞∞∞∞. In that case, the fundamental domain is a square with four ideal vertices. : == Uniform colorings ==

As in the Euclidean square tiling, this tiling has nine uniform colorings. Three of these arise from triangular reflective symmetry domains. A fourth comes from square symmetry written as *∞∞∞∞, with four different colors meeting at each vertex. The checkerboard coloring, denoted r{∞,∞}, defines the fundamental domains of the symmetry group written as [(∞,4,4)] or *∞44, and is usually shown as alternating black and white regions corresponding to reflections. == Related polyhedra and tiling ==

This tiling is also related to a sequence of regular polyhedra and tilings where four faces meet at each vertex. The sequence starts with the octahedron, which has the Schläfli symbol {n,4}, and a corresponding Coxeter diagram , and continues with larger tilings as n increases toward infinity. ==See also==