In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr. It can be seen as constructed as a rectified tetrahexagonal tiling, r, as well as an expanded order-4 hexagonal tiling or expanded order-6 square tiling.
There are two uniform constructions of this tiling, one from [6,4] or (*642) symmetry, and secondly removing the mirror middle, [6,1<sup>+</sup>,4], gives a rectangular fundamental domain [∞,3,∞], (*3222).
| Name | Rhombitetrahexagonal tiling | ||
|---|---|---|---|
| Image | |||
| Symmetry | [6,4] (
| [6,1<sup>+</sup>,4] = [∞,3,∞] (
= | |
| Schläfli symbol | rr | t0,1,2,3 | |
| 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<sup>+</sup>] (4*3) symmetry. sees the yellow squares as rectangles, with two color edges, with [6<sup>+</sup>,4] (6*2) symmetry. A final quarter symmetry combines these colorings, with [6<sup>+</sup>,4<sup>+</sup>] (32×) symmetry, with 2 and 3 fold gyration points and glide reflections.
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 .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<sup>+</sup>,4], (*3222). Removing half of the blue mirrors doubles the domain again into *3322 symmetry.