# Rhombitetrahexagonal tiling

Rhombitetrahexagonal tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 4.4.6.4 |
| Schläfli symbol | rr{6,4} or $r{\begin{Bmatrix}6\\4\end{Bmatrix}}$ |
| Wythoff symbol | 4 | 6 2 |
| Coxeter diagram | |
| Symmetry group | [6,4], (*642) |
| Dual | Deltoidal tetrahexagonal tiling |
| Properties | Vertex-transitive |
In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{6,4}. It can be seen as constructed as a rectified tetrahexagonal tiling, r{6,4}, as well as an expanded order-4 hexagonal tiling or expanded order-6 square tiling.

## Constructions

There are two uniform constructions of this tiling, one from [6,4] or (*642) symmetry, and secondly removing the mirror middle, [6,1$^+$,4], gives a rectangular fundamental domain [∞,3,∞], (*3222).
Two uniform constructions of 4.4.4.6 :
| Name | Rhombitetrahexagonal tiling | |
| Image | | |
| Symmetry | [6,4] (*642) | [6,1$^+$,4] = [∞,3,∞] (*3222) = |
| Schläfli symbol | rr{6,4} | t$_{0,1,2,3}${∞,3,∞} |
| Coxeter diagram | | = |
There are 3 lower symmetry forms seen by including edge-colorings: sees the hexagons as truncated triangles, with two color edges, with [6,4$^+$] (4*3) symmetry. sees the yellow squares as rectangles, with two color edges, with [6$^+$,4] (6*2) symmetry. A final quarter symmetry combines these colorings, with [6$^+$,4$^+$] (32×) symmetry, with 2 and 3 fold gyration points and glide reflections.
Lower symmetry constructions :
| [6,4], (*632) | [6,4$^+$], (4*3) |
| [6$^+$,4], (6*2) | [6$^+$,4$^+$], (32×) |
This four color tiling is related to a semiregular infinite skew polyhedron with the same vertex figure in Euclidean 3-space with a prismatic honeycomb construction of .

## Symmetry

The dual tiling, called a deltoidal tetrahexagonal tiling, represents the fundamental domains of the *3222 orbifold, shown here from three different centers. Its fundamental domain is a Lambert quadrilateral, with 3 right angles. This symmetry can be seen from a [6,4], (*642) triangular symmetry with one mirror removed, constructed as [6,1$^+$,4], (*3222). Removing half of the blue mirrors doubles the domain again into *3322 symmetry.

## Related polyhedra and tiling

*n42 symmetry mutation of expanded tilings: n.4.4.4 vte :
| Symmetry [n,4], (*n42) | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracomp. |
| Symmetry [n,4], (*n42) | *342 [3,4] | *442 [4,4] | *542 [5,4] | *642 [6,4] | *742 [7,4] | *842 [8,4] | *∞42 [∞,4] |
| Expanded figures | | | | | | | |
| Config. | 3.4.4.4 | 4.4.4.4 | 5.4.4.4 | 6.4.4.4 | 7.4.4.4 | 8.4.4.4 | ∞.4.4.4 |
| Rhombic figures config. | V3.4.4.4 | V4.4.4.4 | V5.4.4.4 | V6.4.4.4 | V7.4.4.4 | V8.4.4.4 | V∞.4.4.4 |
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 tilings in symmetry *3222 vte :
| 6$^4$ | 6.6.4.4 | (3.4.4)$^2$ | 4.3.4.3.3.3 |
| 6.6.4.4 | 6.4.4.4 | 3.4.4.4.4 |
| (3.4.4)$^2$ | 3.4.4.4.4 | 4$^6$ |