In geometry, the order-6 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of { 6,6} and is self-dual.
Symmetry
This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain. This symmetry by orbifold notation is called *333333 with 6 order-3 mirror intersections. In Coxeter notation can be represented as [6*,6], removing two of three mirrors (passing through the hexagon center) in the [6,6] symmetry.
The even/odd fundamental domains of this kaleidoscope can be seen in the alternating colorings of the tiling:
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with order-6 vertices with Schläfli symbol { n,6}, and Coxeter diagram , progressing to infinity.
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.
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings Hyperbolic Planar Tessellations, Don Hatch