# Tetrahexagonal tiling

Tetrahexagonal tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic uniform tiling |
| Vertex configuration | (4.6)$^2$ |
| Schläfli symbol | r{6,4} or ${\begin{Bmatrix}6\\4\end{Bmatrix}}$ rr{6,6} r(4,4,3) t$_{0,1,2,3}$(∞,3,∞,3) |
| Wythoff symbol | 2 | 6 4 |
| Coxeter diagram | or or |
| Symmetry group | [6,4], (*642) [6,6], (*662) [(4,4,3)], (*443) [(∞,3,∞,3)], (*3232) |
| Dual | Order-6-4 quasiregular rhombic tiling |
| Properties | Vertex-transitive edge-transitive |
In geometry, the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol r{6,4}.

## Constructions

There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [6,4] kaleidoscope. Removing the last mirror, [6,4,1$^+$], gives [6,6], (*662). Removing the first mirror [1$^+$,6,4], gives [(4,4,3)], (*443). Removing both mirror as [1$^+$,6,4,1$^+$], leaving [(3,∞,3,∞)] (*3232).
Four uniform constructions of 4.6.4.6 :
| Uniform Coloring | | | | |
| Fundamental Domains | | | | |
| Schläfli | r{6,4} | r{4,6}1⁄2 | r{6,4}1⁄2 | r{6,4}1⁄4 |
| Symmetry | [6,4] (*642) | [6,6] = [6,4,1$^+$] (*662) | [(4,4,3)] = [1$^+$,6,4] (*443) | [(∞,3,∞,3)] = [1$^+$,6,4,1$^+$] (*3232) or |
| Symbol | r{6,4} | rr{6,6} | r(4,3,4) | t$_{0,1,2,3}$(∞,3,∞,3) |
| Coxeter diagram | | = | = | = or |

## Symmetry

The dual tiling, called a rhombic tetrahexagonal tiling, with face configuration V4.6.4.6, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*3232), shown here in two different centered views. Adding a 2-fold rotation point in the center of each rhombi represents a (2*32) orbifold.

## Related polyhedra and tiling

*n42 symmetry mutations of quasiregular tilings: (4.n)$^2$ vte :
| Symmetry *4n2 [n,4] | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracompact | Noncompact |
| Symmetry *4n2 [n,4] | *342 [3,4] | *442 [4,4] | *542 [5,4] | *642 [6,4] | *742 [7,4] | *842 [8,4]... | *∞42 [∞,4] | [ni,4] |
| Figures | | | | | | | | |
| Config. | (4.3)$^2$ | (4.4)$^2$ | (4.5)$^2$ | (4.6)$^2$ | (4.7)$^2$ | (4.8)$^2$ | (4.∞)$^2$ | (4.ni)$^2$ |
Symmetry mutation of quasiregular tilings: 6.n.6.n vte :
| Symmetry *6n2 [n,6] | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracompact | Noncompact |
| Symmetry *6n2 [n,6] | *632 [3,6] | *642 [4,6] | *652 [5,6] | *662 [6,6] | *762 [7,6] | *862 [8,6]... | *∞62 [∞,6] | [iπ/λ,6] |
| Quasiregular figures configuration | 6.3.6.3 | 6.4.6.4 | 6.5.6.5 | 6.6.6.6 | 6.7.6.7 | 6.8.6.8 | 6.∞.6.∞ | 6.∞.6.∞ |
| Dual figures |||||||||
| Rhombic figures configuration | V6.3.6.3 | V6.4.6.4 | V6.5.6.5 | V6.6.6.6 | V6.7.6.7 | V6.8.6.8 | V6.∞.6.∞ | |
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} |
Uniform hexahexagonal tilings vte :
| Symmetry: [6,6], (*662) |||||||
| {6,6} = h{4,6} | t{6,6} = h$_2${4,6} | r{6,6} {6,4} | t{6,6} = h$_2${4,6} | {6,6} = h{4,6} | rr{6,6} r{6,4} | tr{6,6} t{6,4} |
| Uniform duals |||||||
| V6$^6$ | V6.12.12 | V6.6.6.6 | V6.12.12 | V6$^6$ | V4.6.4.6 | V4.12.12 |
| Alternations |||||||
| [1$^+$,6,6] (*663) | [6$^+$,6] (6*3) | [6,1$^+$,6] (*3232) | [6,6$^+$] (6*3) | [6,6,1$^+$] (*663) | [(6,6,2$^+$)] (2*33) | [6,6]$^+$ (662) |
| h{6,6} | s{6,6} | hr{6,6} | s{6,6} | h{6,6} | hrr{6,6} | sr{6,6} |
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 |
Similar H2 tilings in *3232 symmetry vte :
| Coxeter diagrams | | | | | | | | |
| Vertex figure | 6$^6$ | 6$^6$ | (3.4.3.4)$^2$ | (3.4.3.4)$^2$ | 3.4.6.6.4 | 3.4.6.6.4 | 6.4.6.4 | 6.4.6.4 |
| Image | | | | | | | | |
| Dual | | | | | | | | |