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 six 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 two vertices of a hexagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding three bisecting mirrors through the vertices defines *443 symmetry. Adding three bisecting mirrors through the edge defines *3222 symmetry. Adding all six bisectors leads to full *642 symmetry. *222222*443*3222*642
Uniform colorings There are seven distinct uniform colorings for the order-4 hexagonal tiling. They are similar to seven of the uniform colorings of the square tiling, but exclude two cases with order-2 gyrational symmetry. Four of them have reflective constructions and Coxeter diagrams while three of them are undercolorings.
+ Uniform constructions of 6.6.6.61 color2 colors3 and 2 colors4, 3 and 2 colorsUniformColoring(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}1/2r(6,3,6) = r{6,6}1/2r{6,6}1/4Coxeterdiagram = = = =
Regular maps The regular map {6,4}3 or {6,4}(4,0) can be seen as a 4-coloring on the {6,4} tiling. It also has a representation as a petrial octahedron, {3,4}, an abstract polyhedron with vertices and edges of an octahedron, but instead connected by 4 Petrie polygon faces.
Related polyhedra and tiling
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.
See also
Square tiling Tilings of regular polygons List of uniform planar tilings List of regular polytopes
References John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings Hyperbolic Planar Tessellations, Don Hatch