# Order-8 hexagonal tiling

Order-8 hexagonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 68
Schläfli symbol {6,8}
Wythoff symbol 8 | 6 2
Coxeter diagram
Symmetry group [8,6], (*862)
Dual Order-6 octagonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-8 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,8}.

## Uniform constructions

There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,6] kaleidoscope. Removing the mirror between the order 2 and 6 points, [6,8,1+], gives [(6,6,4)], (*664). Removing the mirror between the order 8 and 6 points, [6,1+,8], gives (*4232). Removing two mirrors as [6,8*], leaves remaining mirrors (*33333333).

UniformColoring Symmetry Symbol Coxeterdiagram [6,8](*862) [6,8,1+] = [(6,6,4)](*664)      = [6,1+,8](*4232)      = [6,8*](*33333333) {6,8} {6,8}​1⁄2 r(8,6,8) {6,8}​1⁄8 = =

## Symmetry

This tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting as edges of a square, with eight squares around every vertex. This symmetry by orbifold notation is called (*444444) with 6 order-4 mirror intersections. In Coxeter notation can be represented as [8,6*], removing two of three mirrors (passing through the square center) in the [8,6] symmetry.