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.
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<sup>+</sup>,8,1<sup>+</sup>,6,1<sup>+</sup>] (4343) is the commutator subgroup of [8,6].
A radical subgroup is constructed as [8,6*], index 12, as [8,6<sup>+</sup>], (6*4) with gyration points removed, becomes (*444444), and another [8*,6], index 16 as [8<sup>+</sup>,6], (8*3) with gyration points removed as (*33333333).
| Index | 1 | 2 | 4 | |||
|---|---|---|---|---|---|---|
| Diagram | ||||||
| Coxeter | [8,6] = | [1<sup>+</sup>,8,6] = | [8,6,1<sup>+</sup>] = = | [8,1<sup>+</sup>,6] = | [1<sup>+</sup>,8,6,1<sup>+</sup>] = | [8<sup>+</sup>,6<sup>+</sup>] |
| Orbifold |
|
|
|
|
| 43× |
| Semidirect subgroups | ||||||
| Diagram | ||||||
| Coxeter | [8,6<sup>+</sup>] | [8<sup>+</sup>,6] | [(8,6,2<sup>+</sup>)] | [8,1<sup>+</sup>,6,1<sup>+</sup>] = = = = | [1<sup>+</sup>,8,1<sup>+</sup>,6] = = = = | |
| Orbifold | 6*4 | 8*3 | 2*43 | 3*44 | 4*33 | |
| Direct subgroups | ||||||
| Index | 2 | 4 | 8 | |||
| Diagram | ||||||
| Coxeter | [8,6]+ = | [8,6<sup>+</sup>]+ = | [8<sup>+</sup>,6]+ = | [8,1<sup>+</sup>,6]+ = | [8<sup>+</sup>,6<sup>+</sup>]+ = [1<sup>+</sup>,8,1<sup>+</sup>,6,1<sup>+</sup>] = = = | |
| Orbifold | 862 | 664 | 883 | 4232 | 4343 | |
| Radical subgroups | ||||||
| Index | 12 | 24 | 16 | 32 | ||
| Diagram | ||||||
| Coxeter | [8,6*] | [8*,6] | [8,6*]+ | [8*,6]+ | ||
| Orbifold |
|
| 444444 | 33333333 | ||
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.