# Order-4 hexagonal tiling

Order-4 hexagonal tiling Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex figure64
Schläfli symbol{6,4}
Wythoff symbol4 | 6 2
Coxeter diagram     Symmetry group[6,4], (*642)
DualOrder-6 square tiling
PropertiesVertex-transitive, edge-transitive, face-transitive

In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}.

## Symmetry

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*,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 *3322 symmetry. Adding 3 bisecting mirrors through the vertices defines *443 symmetry. Adding 3 bisecting mirrors through the edge defines *3222 symmetry. Adding all 6 bisectors leads to full *642 symmetry. *222222 *443 *3222 *642

## Uniform colorings

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.

Uniform constructions of 6.6.6.6
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]
(*642)     [6,6]
(*662)   =     [(6,6,3)] = [6,6,1+]
(*663)   =     [1+,6,6,1+]
(*3333)   =     =     Symbol {6,4} r{6,6} = {6,4}1/2 r(6,3,6) = r{6,6}1/2 r{6,6}1/4
Coxeter
diagram        =        =        =     =     This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, 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 {n,4}, and Coxeter diagram     , with n progressing to infinity. Wikimedia Commons has media related to Order-4 hexagonal tiling.