# Truncated order-4 octagonal tiling

Truncated order-4 octagonal tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 4.16.16 |
| Schläfli symbol | t{8,4} tr{8,8} or $t{\begin{Bmatrix}8\\8\end{Bmatrix}}$ |
| Wythoff symbol | 2 8 | 8 2 8 8 | |
| Coxeter diagram | or |
| Symmetry group | [8,4], (*842) [8,8], (*882) |
| Dual | Order-8 tetrakis square tiling |
| Properties | Vertex-transitive |
In geometry, the truncated order-4 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t$_{0,1}${8,4}. A secondary construction t$_{0,1,2}${8,8} is called a truncated octaoctagonal tiling with two colors of hexakaidecagons.

## Constructions

There are two uniform constructions of this tiling, first by the [8,4] kaleidoscope, and second by removing the last mirror, [8,4,1$^+$], gives [8,8], (*882).
Two uniform constructions of 4.8.4.8 :
| Name | Tetraoctagonal | Truncated octaoctagonal |
| Image | | |
| Symmetry | [8,4] (*842) | [8,8] = [8,4,1$^+$] (*882) = |
| Symbol | t{8,4} | tr{8,8} |
| Coxeter diagram | | |

## Dual tiling

* The dual tiling, Order-8 tetrakis square tiling has face configuration V4.16.16, and represents the fundamental domains of the [8,8] symmetry group. |

## Symmetry

The dual of the tiling represents the fundamental domains of (*882) orbifold symmetry. From [8,8] symmetry, there are 15 small index subgroup 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 unique mirrors are colored red, green, and blue, and alternatively colored triangles show the location of gyration points. The [8$^+$,8$^+$], (44×) subgroup has narrow lines representing glide reflections. The subgroup index-8 group, [1$^+$,8,1$^+$,8,1$^+$] (4444) is the commutator subgroup of [8,8].
One larger subgroup is constructed as [8,8*], removing the gyration points of (8*4), index 16 becomes (*44444444), and its direct subgroup [8,8*]$^+$, index 32, (44444444).
The [8,8] symmetry can be doubled by a mirror bisecting the fundamental domain, and creating *884 symmetry.
Small index subgroups of [8,8] (*882) :
| Index | 1 | 2 | 2 | 2 | 4 | 4 |
| Diagram | | | | | | |
| Coxeter | [8,8] | [1$^+$,8,8] = | [8,8,1$^+$] = | [8,1$^+$,8] = | [1$^+$,8,8,1$^+$] = | [8$^+$,8$^+$] |
| Orbifold | *882 | *884 | *884 | *4242 | *4444 | 44× |
| Semidirect subgroups |||||||
| Diagram | | | | | | |
| Coxeter | | [8,8$^+$] | [8$^+$,8] | [(8,8,2$^+$)] | [8,1$^+$,8,1$^+$] = = = = | [1$^+$,8,1$^+$,8] = = = = |
| Orbifold | | 8*4 | 8*4 | 2*44 | 4*44 | 4*44 |
| Direct subgroups |||||||
| Index | 2 | 4 | 4 | 4 | 8 | 8 |
| Diagram | | | | | | |
| Coxeter | [8,8]$^+$ | [8,8$^+$]$^+$ = | [8$^+$,8]$^+$ = | [8,1$^+$,8]$^+$ = | [8$^+$,8$^+$]$^+$ = [1$^+$,8,1$^+$,8,1$^+$] = = = | [8$^+$,8$^+$]$^+$ = [1$^+$,8,1$^+$,8,1$^+$] = = = |
| Orbifold | 882 | 884 | 884 | 4242 | 4444 | 4444 |
| Radical subgroups |||||||
| Index | | 16 | 16 | 32 | 32 |
| Diagram | | | | | |
| Coxeter | | [8,8*] | [8*,8] | [8,8*]$^+$ | [8*,8]$^+$ |
| Orbifold | | *44444444 | *44444444 | 44444444 | 44444444 |

## Related polyhedra and tiling

*n42 symmetry mutation of truncated tilings: 4.2n.2n vte :
| Symmetry *n42 | Spherical | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracomp. |
| Symmetry *n42 | *242 [2,4] | *342 [3,4] | *442 [4,4] | *542 [5,4] | *642 [6,4] | *742 [7,4] | *842 [8,4]... | *∞42 [∞,4] |
| Truncated figures | | | | | | | | |
| Config. | 4.4.4 | 4.6.6 | 4.8.8 | 4.10.10 | 4.12.12 | 4.14.14 | 4.16.16 | 4.∞.∞ |
| n-kis figures | | | | | | | | |
| Config. | V4.4.4 | V4.6.6 | V4.8.8 | V4.10.10 | V4.12.12 | V4.14.14 | V4.16.16 | V4.∞.∞ |
Uniform octagonal/square tilings vte :
| [8,4], (*842) (with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries) (And [(∞,4,∞,4)] (*4242) index 4 subsymmetry) |||||||
| {8,4} | t{8,4} | r{8,4} | 2t{8,4}=t{4,8} | 2r{8,4}={4,8} | rr{8,4} | tr{8,4} |
| Uniform duals |||||||
| V8$^4$ | V4.16.16 | V(4.8)$^2$ | V8.8.8 | V4$^8$ | V4.4.4.8 | V4.8.16 |
| Alternations |||||||
| [1$^+$,8,4] (*444) | [8$^+$,4] (8*2) | [8,1$^+$,4] (*4222) | [8,4$^+$] (4*4) | [8,4,1$^+$] (*882) | [(8,4,2$^+$)] (2*42) | [8,4]$^+$ (842) |
| h{8,4} | s{8,4} | hr{8,4} | s{4,8} | h{4,8} | hrr{8,4} | sr{8,4} |
| Alternation duals |||||||
| V(4.4)$^4$ | V3.(3.8)$^2$ | V(4.4.4)$^2$ | V(3.4)$^3$ | V8$^8$ | V4.4$^4$ | V3.3.4.3.8 |
Uniform octaoctagonal tilings vte :
| Symmetry: [8,8], (*882) |||||||
| {8,8} | t{8,8} | r{8,8} | 2t{8,8}=t{8,8} | 2r{8,8}={8,8} | rr{8,8} | tr{8,8} |
| Uniform duals |||||||
| V8$^8$ | V8.16.16 | V8.8.8.8 | V8.16.16 | V8$^8$ | V4.8.4.8 | V4.16.16 |
| Alternations |||||||
| [1$^+$,8,8] (*884) | [8$^+$,8] (8*4) | [8,1$^+$,8] (*4242) | [8,8$^+$] (8*4) | [8,8,1$^+$] (*884) | [(8,8,2$^+$)] (2*44) | [8,8]$^+$ (882) |
| h{8,8} | s{8,8} | hr{8,8} | s{8,8} | h{8,8} | hrr{8,8} | sr{8,8} |
| Alternation duals |||||||
| V(4.8)$^8$ | V3.4.3.8.3.8 | V(4.4)$^4$ | V3.4.3.8.3.8 | V(4.8)$^8$ | V4$^6$ | V3.3.8.3.8 |