In geometry, the truncated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{ 6,4}. A secondary construction tr{ 6,6} is called a truncated hexahexagonal tiling with two colors of dodecagons.
Constructions
There are two uniform constructions of this tiling, first from [6,4] kaleidoscope, and a lower symmetry by removing the last mirror, [6,4,1+], gives [6,6], (*662).
Dual tiling
Related polyhedra and tiling
Symmetry
The dual of the tiling represents the fundamental domains of (*662) orbifold symmetry. From [6,6] (*662) symmetry, there are 15 small index subgroup (12 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1+,6,1+,6,1+] (3333) is the commutator subgroup of [6,6].
Larger subgroup constructed as [6,6*], removing the gyration points of (6*3), index 12 becomes (*333333).
The symmetry can be doubled to 642 symmetry by adding a mirror to bisect the fundamental domain.
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, 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