In geometry, the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol r.
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<sup>+</sup>], gives [6,6], (*662). Removing the first mirror [1<sup>+</sup>,6,4], gives [(4,4,3)], (*443). Removing both mirror as [1<sup>+</sup>,6,4,1<sup>+</sup>], leaving [(3,∞,3,∞)] (*3232).
| Uniform Coloring | |||||
|---|---|---|---|---|---|
| Fundamental Domains | |||||
| Schläfli | r | r | r | r | |
| Symmetry | [6,4] (*642) | [6,6] = [6,4,1<sup>+</sup>] (*662) | [(4,4,3)] = [1<sup>+</sup>,6,4] (*443) | [(∞,3,∞,3)] = [1<sup>+</sup>,6,4,1<sup>+</sup>] (*3232) or | |
| Symbol | r | rr | r(4,3,4) | t0,1,2,3(∞,3,∞,3) | |
| Coxeter diagram | = | = | = or |
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.