# Truncated order-8 hexagonal tiling

Truncated order-8 hexagonal tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 8.12.12 |
| Schläfli symbol | t{6,8} |
| Wythoff symbol | 2 8 | 6 |
| Coxeter diagram | |
| Symmetry group | [8,6], (*862) |
| Dual | Order-6 octakis octagonal tiling |
| Properties | Vertex-transitive |
In geometry, the truncated order-8 hexagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of t{6,8}.

## Uniform colorings

This tiling can also be constructed from *664 symmetry, as t{(6,6,4)}.

## 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 |

### Symmetry

The dual of the tiling represents the fundamental domains of (*664) orbifold symmetry. From [(6,6,4)] (*664) symmetry, there are 15 small index subgroup (11 unique) by mirror removal and alternation operators. 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 symmetry can be doubled to 862 symmetry by adding a bisecting mirror across the fundamental domains. The subgroup index-8 group, [(1$^+$,6,1$^+$,6,1$^+$,4)] (332332) is the commutator subgroup of [(6,6,4)].
A large subgroup is constructed [(6,6,4$^*$)], index 8, as (4*33) with gyration points removed, becomes (*3$^8$), and another large subgroup is constructed [(6,6$^*$,4)], index 12, as (6*32) with gyration points removed, becomes (*(32)$^6$).
Small index subgroups of [(6,6,4)] (*664) :
| Fundamental domains | | | | | | |
| Subgroup index | 1 | 2 | 2 | 2 | 4 | 4 |
| Coxeter | [(6,6,4)] | [(1$^+$,6,6,4)] | [(6,6,1$^+$,4)] | [(6,1$^+$,6,4)] | [(1$^+$,6,6,1$^+$,4)] | [(6$^+$,6$^+$,4)] |
| Orbifold | *664 | *6362 | *6362 | *4343 | 2*3333 | 332× |
| Coxeter | | [(6,6$^+$,4)] | [(6$^+$,6,4)] | [(6,6,4$^+$)] | [(6,1$^+$,6,1$^+$,4)] | [(1$^+$,6,1$^+$,6,4)] |
| Orbifold | | 6*32 | 6*32 | 4*33 | 3*3232 | 3*3232 |
| Direct subgroups |||||||
| Subgroup index | 2 | 4 | 4 | 4 | 8 | 8 |
| Coxeter | [(6,6,4)]$^+$ | [(1$^+$,6,6$^+$,4)] | [(6$^+$,6,1$^+$,4)] | [(6,1$^+$,6,4$^+$)] | [(6$^+$,6$^+$,4$^+$)] = [(1$^+$,6,1$^+$,6,1$^+$,4)] = | [(6$^+$,6$^+$,4$^+$)] = [(1$^+$,6,1$^+$,6,1$^+$,4)] = |
| Orbifold | 664 | 6362 | 6362 | 4343 | 332332 | 332332 |