# Hexaoctagonal tiling

hexaoctagonal tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic uniform tiling |
| Vertex configuration | (6.8)$^2$ |
| Schläfli symbol | r{8,6} or ${\begin{Bmatrix}8\\6\end{Bmatrix}}$ |
| Wythoff symbol | 2 | 8 6 |
| Coxeter diagram | |
| Symmetry group | [8,6], (*862) |
| Dual | Order-8-6 quasiregular rhombic tiling |
| Properties | Vertex-transitive edge-transitive |
In geometry, the hexaoctagonal tiling is a uniform tiling of the hyperbolic plane.

## 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 4 points, [8,6,1$^+$], gives [(8,8,3)], (*883). Removing the mirror between the order 2 and 8 points, [1$^+$,8,6], gives [(4,6,6)], (*664). Removing two mirrors as [8,1$^+$,6,1$^+$], leaves remaining mirrors (*4343).
Four uniform constructions of 6.8.6.8 :
| Uniform Coloring | | | | |
| Symmetry | [8,6] (*862) | [(8,3,8)] = [8,6,1$^+$] (*883) | [(6,4,6)] = [1$^+$,8,6] (*664) | [1$^+$,8,6,1$^+$] (*4343) |
| Symbol | r{8,6} | r{(8,3,8)} | r{(6,4,6)} | |
| Coxeter diagram | | = | = | = |

## Symmetry

The dual tiling has face configuration V6.8.6.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4343), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*43) orbifold. These are subsymmetries of.
| [1$^+$,8,4,1$^+$], (*4343) | [(8,4,2$^+$)], (2*43) |

## Related polyhedra and tiling

Uniform octagonal/hexagonal tilings vte :
| Symmetry: [8,6], (*862) |||||||
| {8,6} | t{8,6} | r{8,6} | 2t{8,6}=t{6,8} | 2r{8,6}={6,8} | rr{8,6} | tr{8,6} |
| Uniform duals |||||||
| V8$^6$ | V6.16.16 | V(6.8)$^2$ | V8.12.12 | V6$^8$ | V4.6.4.8 | V4.12.16 |
| Alternations |||||||
| [1$^+$,8,6] (*466) | [8$^+$,6] (8*3) | [8,1$^+$,6] (*4232) | [8,6$^+$] (6*4) | [8,6,1$^+$] (*883) | [(8,6,2$^+$)] (2*43) | [8,6]$^+$ (862) |
| h{8,6} | s{8,6} | hr{8,6} | s{6,8} | h{6,8} | hrr{8,6} | sr{8,6} |
| Alternation duals |||||||
| V(4.6)$^6$ | V3.3.8.3.8.3 | V(3.4.4.4)$^2$ | V3.4.3.4.3.6 | V(3.8)$^8$ | V3.4$^5$ | V3.3.6.3.8 |
Symmetry mutation of quasiregular tilings: 6.n.6.n vte :
| Symmetry *6n2 [n,6] | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracompact | Noncompact |
| Symmetry *6n2 [n,6] | *632 [3,6] | *642 [4,6] | *652 [5,6] | *662 [6,6] | *762 [7,6] | *862 [8,6]... | *∞62 [∞,6] | [iπ/λ,6] |
| Quasiregular figures configuration | 6.3.6.3 | 6.4.6.4 | 6.5.6.5 | 6.6.6.6 | 6.7.6.7 | 6.8.6.8 | 6.∞.6.∞ | 6.∞.6.∞ |
| Dual figures |||||||||
| Rhombic figures configuration | V6.3.6.3 | V6.4.6.4 | V6.5.6.5 | V6.6.6.6 | V6.7.6.7 | V6.8.6.8 | V6.∞.6.∞ | |
Dimensional family of quasiregular polyhedra and tilings: (8.n)$^2$ vte :
| Symmetry *8n2 [n,8] | Hyperbolic... | Hyperbolic... | Hyperbolic... | Hyperbolic... | Hyperbolic... | Hyperbolic... | Paracompact | Noncompact |
| Symmetry *8n2 [n,8] | *832 [3,8] | *842 [4,8] | *852 [5,8] | *862 [6,8] | *872 [7,8] | *882 [8,8]... | *∞82 [∞,8] | [iπ/λ,8] |
| Coxeter | | | | | | | | |
| Quasiregular figures configuration | 3.8.3.8 | 4.8.4.8 | 8.5.8.5 | 8.6.8.6 | 8.7.8.7 | 8.8.8.8 | 8.∞.8.∞ | 8.∞.8.∞ |