# Truncated hexaoctagonal tiling

Truncated hexaoctagonal tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 4.12.16 |
| Schläfli symbol | tr{8,6} or $t{\begin{Bmatrix}8\\6\end{Bmatrix}}$ |
| Wythoff symbol | 2 8 6 | |
| Coxeter diagram | or |
| Symmetry group | [8,6], (*862) |
| Dual | Order-6-8 kisrhombille tiling |
| Properties | Vertex-transitive |
In geometry, the truncated hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one dodecagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,6}.

## Dual tiling

* The dual tiling is called an order-6-8 kisrhombille tiling, made as a complete bisection of the order-6 octagonal tiling, here with triangles are shown with alternating colors. This tiling represents the fundamental triangular domains of [8,6] (*862) symmetry. |

## Symmetry

There are six reflective subgroup kaleidoscopic constructed from [8,6] by removing one or two of three mirrors. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1$^+$,8,1$^+$,6,1$^+$] (4343) is the commutator subgroup of [8,6].
A radical subgroup is constructed as [8,6*], index 12, as [8,6$^+$], (6*4) with gyration points removed, becomes (*444444), and another [8*,6], index 16 as [8$^+$,6], (8*3) with gyration points removed as (*33333333).
Small index subgroups of [8,6] (*862) :
| Index | 1 | 2 | 2 | 2 | 4 | 4 |
| Diagram | | | | | | |
| Coxeter | [8,6] = | [1$^+$,8,6] = | [8,6,1$^+$] = = | [8,1$^+$,6] = | [1$^+$,8,6,1$^+$] = | [8$^+$,6$^+$] |
| Orbifold | *862 | *664 | *883 | *4232 | *4343 | 43× |
| Semidirect subgroups |||||||
| Diagram | | | | | | |
| Coxeter | | [8,6$^+$] | [8$^+$,6] | [(8,6,2$^+$)] | [8,1$^+$,6,1$^+$] = = = = | [1$^+$,8,1$^+$,6] = = = = |
| Orbifold | | 6*4 | 8*3 | 2*43 | 3*44 | 4*33 |
| Direct subgroups |||||||
| Index | 2 | 4 | 4 | 4 | 8 | 8 |
| Diagram | | | | | | |
| Coxeter | [8,6]$^+$ = | [8,6$^+$]$^+$ = | [8$^+$,6]$^+$ = | [8,1$^+$,6]$^+$ = | [8$^+$,6$^+$]$^+$ = [1$^+$,8,1$^+$,6,1$^+$] = = = | [8$^+$,6$^+$]$^+$ = [1$^+$,8,1$^+$,6,1$^+$] = = = |
| Orbifold | 862 | 664 | 883 | 4232 | 4343 | 4343 |
| Radical subgroups ||||||
| Index | | 12 | 24 | 16 | 32 |
| Diagram | | | | | |
| Coxeter | | [8,6*] | [8*,6] | [8,6*]$^+$ | [8*,6]$^+$ |
| Orbifold | | *444444 | *33333333 | 444444 | 33333333 |

## Related polyhedra and tilings

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 octagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [8,6] symmetry, and 7 with subsymmetry.
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 |