Order-4 octagonal tiling
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.
Uniform constructions
There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the kaleidoscope. Removing the mirror between the order 2 and 4 points, , gives 884 symmetry|, symmetry. Removing two mirrors as , leaves remaining mirrors *4444 symmetry.| Uniform Coloring | ||||
| Symmetry | = | = = = | = | |
| Symbol | r | r = r | r = r | |
| Coxeter diagram | = = | = = = |
Symmetry
This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting as edges of a regular hexagon. This symmetry by orbifold notation is called or with 8 order-2 mirror intersections. In Coxeter notation can be represented as , removing two of three mirrors in the symmetry. Adding a bisecting mirror through 2 vertices of an octagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 4 bisecting mirrors through the vertices defines *444 symmetry. Adding 4 bisecting mirrors through the edge defines *4222 symmetry. Adding all 8 bisectors leads to full *842 symmetry.*444 | *4222 | *832 |
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.
Related polyhedra and 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.
Octahedron| | Square tiling| | Order-4 pentagonal tiling| | ... |