In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of .
This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain. This symmetry by orbifold notation is called *222222 with 6 order-2 mirror intersections. In Coxeter notation can be represented as [6<sup>*</sup>,4], removing two of three mirrors (passing through the hexagon center). Adding a bisecting mirror through 2 vertices of a hexagonal fundamental domain defines a trapezohedral
There are 7 distinct uniform colorings for the order-4 hexagonal tiling. They are similar to 7 of the uniform colorings of the square tiling, but exclude 2 cases with order-2 gyrational symmetry. Four of them have reflective constructions and Coxeter diagrams while three of them are undercolorings.
| 1 color | 2 colors | 3 and 2 colors | 4, 3 and 2 colors | |||||
|---|---|---|---|---|---|---|---|---|
| Uniform Coloring | (1111) | (1212) | (1213) | (1113) | (1234) | (1123) | (1122) | |
| Symmetry | [6,4] (
| [6,6] (
= | [(6,6,3)] = [6,6,1<sup>+</sup>] (
= | [1<sup>+</sup>,6,6,1<sup>+</sup>] (
= = | ||||
| Symbol | r = 1/2 | r(6,3,6) = r1/2 | r1/4 | |||||
| Coxeter diagram | = | = | = = | |||||
The regular map 3 or (4,0) can be seen as a 4-coloring on the tiling. It also has a representation as a petrial octahedron, , an abstract polyhedron with vertices and edges of an octahedron, but instead connected by 4 Petrie polygon faces.
This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol, and Coxeter diagram, progressing to infinity.
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol, and Coxeter diagram, with n progressing to infinity.