# Truncated order-6 square tiling

Truncated order-6 square tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 8.8.6 |
| Schläfli symbol | t{4,6} |
| Wythoff symbol | 2 6 | 4 |
| Coxeter diagram | |
| Symmetry group | [6,4], (*642) [(3,3,4)], (*334) |
| Dual | Order-4 hexakis hexagonal tiling |
| Properties | Vertex-transitive |
In geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,6}.

## Uniform colorings

* The half symmetry [1$^+$,6,4] = [(4,4,3)] can be shown with alternating two colors of octagons, with as Coxeter diagram .

## Symmetry

The dual tiling represents the fundamental domains of the *443 orbifold symmetry. There are two reflective subgroup kaleidoscopic constructed from [(4,4,3)] by removing one or two of three mirrors. In these images fundamental domains are alternately colored black and cyan, and mirrors exist on the boundaries between colors.
A larger subgroup is constructed [(4,4,3*)], index 6, as (3*22) with gyration points removed, becomes (*222222).
The symmetry can be doubled as 642 symmetry by adding a mirror bisecting the fundamental domain.
Small index subgroups of [(4,4,3)] (*443) :
| Index | 1 | 2 | 2 | 6 |
| Diagram | | | | |
| Coxeter (orbifold) | [(4,4,3)] = (*443) | [(4,1$^+$,4,3)] = = (*3232) | [(4,4,3$^+$)] = (3*22) | [(4,4,3*)] = (*222222) |
| Direct subgroups |||||
| Index | 2 | 4 | 4 | 12 |
| Diagram | | | | |
| Coxeter (orbifold) | [(4,4,3)]$^+$ = (443) | [(4,4,3$^+$)]$^+$ = = (3232) | [(4,4,3$^+$)]$^+$ = = (3232) | [(4,4,3*)]$^+$ = (222222) |

## Related polyhedra and tilings

From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular order-4 hexagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.
Uniform tetrahexagonal tilings vte :
| Symmetry: [6,4], (*642) (with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries) (And [(∞,3,∞,3)] (*3232) index 4 subsymmetry) |||||||
| {6,4} | t{6,4} | r{6,4} | t{4,6} | {4,6} | rr{6,4} | tr{6,4} |
| Uniform duals |||||||
| V6$^4$ | V4.12.12 | V(4.6)$^2$ | V6.8.8 | V4$^6$ | V4.4.4.6 | V4.8.12 |
| Alternations |||||||
| [1$^+$,6,4] (*443) | [6$^+$,4] (6*2) | [6,1$^+$,4] (*3222) | [6,4$^+$] (4*3) | [6,4,1$^+$] (*662) | [(6,4,2$^+$)] (2*32) | [6,4]$^+$ (642) |
| h{6,4} | s{6,4} | hr{6,4} | s{4,6} | h{4,6} | hrr{6,4} | sr{6,4} |
It can also be generated from the (4 4 3) hyperbolic tilings:
Uniform (4,4,3) tilings vte :
| Symmetry: [(4,4,3)] (*443) | Symmetry: [(4,4,3)] (*443) | Symmetry: [(4,4,3)] (*443) | Symmetry: [(4,4,3)] (*443) | Symmetry: [(4,4,3)] (*443) | Symmetry: [(4,4,3)] (*443) | Symmetry: [(4,4,3)] (*443) | [(4,4,3)]$^+$ (443) | [(4,4,3$^+$)] (3*22) | [(4,1$^+$,4,3)] (*3232) | [(4,1$^+$,4,3)] (*3232) |
| h{6,4} t$_0$(4,4,3) | h$_2${6,4} t$_{0,1}$(4,4,3) | {4,6}$^1$/$_2$ t$_1$(4,4,3) | h$_2${6,4} t$_{1,2}$(4,4,3) | h{6,4} t$_2$(4,4,3) | r{6,4}$^1$/$_2$ t$_{0,2}$(4,4,3) | t{4,6}$^1$/$_2$ t$_{0,1,2}$(4,4,3) | s{4,6}$^1$/$_2$ s(4,4,3) | hr{4,6}$^1$/$_2$ hr(4,3,4) | h{4,6}$^1$/$_2$ h(4,3,4) | q{4,6} h$_1$(4,3,4) |
| Uniform duals |||||||||||
| V(3.4)$^4$ | V3.8.4.8 | V(4.4)$^3$ | V3.8.4.8 | V(3.4)$^4$ | V4.6.4.6 | V6.8.8 | V3.3.3.4.3.4 | V(4.4.3)$^2$ | V6$^6$ | V4.3.4.6.6 |
*n42 symmetry mutation of truncated tilings: n.8.8 vte :
| Symmetry *n42 [n,4] | Spherical | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracompact |
| 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] |
| Truncated figures | | | | | | | | |
| Config. | 2.8.8 | 3.8.8 | 4.8.8 | 5.8.8 | 6.8.8 | 7.8.8 | 8.8.8 | ∞.8.8 |
| n-kis figures | | | | | | | | |
| Config. | V2.8.8 | V3.8.8 | V4.8.8 | V5.8.8 | V6.8.8 | V7.8.8 | V8.8.8 | V∞.8.8 |
*n32 symmetry mutation of omnitruncated tilings: 6.8.2n vte :
| Sym. *n43 [(n,4,3)] | Spherical | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paraco. |
| Sym. *n43 [(n,4,3)] | *243 [4,3] | *343 [(3,4,3)] | *443 [(4,4,3)] | *543 [(5,4,3)] | *643 [(6,4,3)] | *743 [(7,4,3)] | *843 [(8,4,3)] | *∞43 [(∞,4,3)] |
| Figures | | | | | | | | |
| Config. | 4.8.6 | 6.8.6 | 8.8.6 | 10.8.6 | 12.8.6 | 14.8.6 | 16.8.6 | ∞.8.6 |
| Duals | | | | | | | | |
| Config. | V4.8.6 | V6.8.6 | V8.8.6 | V10.8.6 | V12.8.6 | V14.8.6 | V16.8.6 | V6.8.∞ |