| Truncated order-8 hexagonal tiling | |
|---|---|
 Poincaré disk model of thehyperbolic 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 |
Ingeometry, thetruncated order-8 hexagonal tiling is a semiregular tiling of the hyperbolic plane. It hasSchläfli symbol of t{6,8}.
This tiling can also be constructed from *664 symmetry, as t{(6,6,4)}.
From aWythoff construction there are fourteen hyperbolicuniform 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 | ||||||
|---|---|---|---|---|---|---|
| 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 | ||||||
 | | | | | | |
 |  |  |  |  |  |  |
| V86 | V6.16.16 | V(6.8)2 | V8.12.12 | V68 | 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 | V3.45 | V3.3.6.3.8 |
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 to862 symmetry by adding a bisecting mirror across the fundamental domains. Thesubgroup index-8 group, [(1+,6,1+,6,1+,4)] (332332) is thecommutator subgroup of [(6,6,4)].
A large subgroup is constructed [(6,6,4*)], index 8, as (4*33) with gyration points removed, becomes (*38), and another large subgroup is constructed [(6,6*,4)], index 12, as (6*32) with gyration points removed, becomes (*(32)6).
| Fundamental domains |  |  |   |   | | |
|---|---|---|---|---|---|---|
| Subgroup index | 1 | 2 | 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 | *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 | 4*33 | 3*3232 | |||
| Direct subgroups | ||||||
| Subgroup index | 2 | 4 | 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)] = | |
| Orbifold | 664 | 6362 | 4343 | 332332 | ||
