In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of . Its checkerboard coloring can be called a octaoctagonal tiling, and Schläfli symbol of r.
There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,8] kaleidoscope. Removing the mirror between the order 2 and 4 points, [8,8,1<sup>+</sup>], gives [(8,8,4)], (*884) symmetry. Removing two mirrors as [8,4<sup>*</sup>], leaves remaining mirrors
| Uniform Coloring | |||||
|---|---|---|---|---|---|
| Symmetry | [8,4] (*842) | [8,8] (*882) = | [(8,4,8)] = [8,8,1<sup>+</sup>] (*884) = = | [1<sup>+</sup>,8,8,1<sup>+</sup>] (*4444) = | |
| Symbol | r | r(8,4,8) = r | r = r | ||
| Coxeter diagram | = = | = = = |
This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting as edges of a regular hexagon. This symmetry by orbifold notation is called (*22222222) or (*28) with 8 order-2 mirror intersections. In Coxeter notation can be represented as [8<sup>*</sup>,4], removing two of three mirrors (passing through the octagon center) in the [8,4] symmetry. Adding a bisecting mirror through 2 vertices of an octagonal fundamental domain defines a trapezohedral
The kaleidoscopic domains can be seen as bicolored octagonal tiling, representing mirror images of the fundamental domain. This coloring represents the uniform tiling r, a quasiregular tiling and it can be called a octaoctagonal tiling.
This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol, and Coxeter diagram, progressing to infinity.
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.