# Truncated tetraoctagonal tiling

Truncated tetraoctagonal tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 4.8.16 |
| Schläfli symbol | tr{8,4} or $t{\begin{Bmatrix}8\\4\end{Bmatrix}}$ |
| Wythoff symbol | 2 8 4 | |
| Coxeter diagram | or |
| Symmetry group | [8,4], (*842) |
| Dual | Order-4-8 kisrhombille tiling |
| Properties | Vertex-transitive |
In geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,4}.

## Dual tiling

* The dual tiling is called an order-4-8 kisrhombille tiling, made as a complete bisection of the order-4 octagonal tiling, here with triangles are shown with alternating colors. This tiling represents the fundamental triangular domains of [8,4] (*842) symmetry. |

### Symmetry

There are 15 subgroups constructed from [8,4] by mirror removal and alternation. 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$^+$,4,1$^+$] (4242) is the commutator subgroup of [8,4].
A larger subgroup is constructed as [8,4*], index 8, as [8,4$^+$], (4*4) with gyration points removed, becomes (*4444) or (*4$^4$), and another [8*,4], index 16 as [8$^+$,4], (8*2) with gyration points removed as (*22222222) or (*2$^8$). And their direct subgroups [8,4*]$^+$, [8*,4]$^+$, subgroup indices 16 and 32 respectively, can be given in orbifold notation as (4444) and (22222222).
Small index subgroups of [8,4] (*842) :
| Index | 1 | 2 | 2 | 2 | 4 | 4 |
| Diagram | | | | | | |
| Coxeter | [8,4] = | [1$^+$,8,4] = | [8,4,1$^+$] = = | [8,1$^+$,4] = | [1$^+$,8,4,1$^+$] = | [8$^+$,4$^+$] |
| Orbifold | *842 | *444 | *882 | *4222 | *4242 | 42× |
| Semidirect subgroups |||||||
| Diagram | | | | | | |
| Coxeter | | [8,4$^+$] | [8$^+$,4] | [(8,4,2$^+$)] | [8,1$^+$,4,1$^+$] = = = = | [1$^+$,8,1$^+$,4] = = = = |
| Orbifold | | 4*4 | 8*2 | 2*42 | 2*44 | 4*22 |
| Direct subgroups |||||||
| Index | 2 | 4 | 4 | 4 | 8 | 8 |
| Diagram | | | | | | |
| Coxeter | [8,4]$^+$ = | [8,4$^+$]$^+$ = | [8$^+$,4]$^+$ = | [8,1$^+$,4]$^+$ = | [8$^+$,4$^+$]$^+$ = [1$^+$,8,1$^+$,4,1$^+$] = = = | [8$^+$,4$^+$]$^+$ = [1$^+$,8,1$^+$,4,1$^+$] = = = |
| Orbifold | 842 | 444 | 882 | 4222 | 4242 | 4242 |
| Radical subgroups ||||||
| Index | | 8 | 16 | 16 | 32 |
| Diagram | | | | | |
| Coxeter | | [8,4*] = | [8*,4] | [8,4*]$^+$ = | [8*,4]$^+$ |
| Orbifold | | *4444 | *22222222 | 4444 | 22222222 |

## Related polyhedra and tilings

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-4 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 [8,4] symmetry, and 7 with subsymmetry.
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 |
*n42 symmetry mutation of omnitruncated tilings: 4.8.2n vte :
| Symmetry *n42 [n,4] | Spherical | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracomp. |
| Symmetry *n42 [n,4] | *242 [2,4] | *342 [3,4] | *442 [4,4] | *542 [5,4] | *642 [6,4] | *742 [7,4] | *842 [8,4]... | *∞42 [∞,4] |
| Omnitruncated figure | 4.8.4 | 4.8.6 | 4.8.8 | 4.8.10 | 4.8.12 | 4.8.14 | 4.8.16 | 4.8.∞ |
| Omnitruncated duals | V4.8.4 | V4.8.6 | V4.8.8 | V4.8.10 | V4.8.12 | V4.8.14 | V4.8.16 | V4.8.∞ |
*nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n vte :
| Symmetry *nn2 [n,n] | Spherical | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracomp. |
| Symmetry *nn2 [n,n] | *222 [2,2] | *332 [3,3] | *442 [4,4] | *552 [5,5] | *662 [6,6] | *772 [7,7] | *882 [8,8]... | *∞∞2 [∞,∞] |
| Figure | | | | | | | | |
| Config. | 4.4.4 | 4.6.6 | 4.8.8 | 4.10.10 | 4.12.12 | 4.14.14 | 4.16.16 | 4.∞.∞ |
| Dual | | | | | | | | |
| Config. | V4.4.4 | V4.6.6 | V4.8.8 | V4.10.10 | V4.12.12 | V4.14.14 | V4.16.16 | V4.∞.∞ |