# Order-4 octagonal tiling

Order-4 octagonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 84
Schläfli symbol {8,4}
r{8,8}
Wythoff symbol 4 | 8 2
Coxeter diagram
or
Symmetry group [8,4], (*842)
[8,8], (*882)
Dual Order-8 square tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}, its checkerboard coloring can be called a octaoctagonal tiling, and Schläfli symbol of r{8,8}.

## Uniform constructions

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+], gives [(8,8,4)], (*884) symmetry. Removing two mirrors as [8,4*], leaves remaining mirrors *4444 symmetry.

UniformColoring Symmetry Symbol Coxeterdiagram [8,4](*842) [8,8](*882) = [(8,4,8)] = [8,8,1+](*884) = = [1+,8,8,1+](*4444) = {8,4} r{8,8} r(8,4,8) = r{8,8}​1⁄2 r{8,4}​1⁄8 = r{8,8}​1⁄4 = = = = =

## Symmetry

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*,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 *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{8,8}, 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 {8,n}, 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 {n,4}, and Coxeter diagram , with n progressing to infinity.

 {3,4} {4,4} {5,4} {6,4} {7,4} {8,4} ... {∞,4}