In geometry, the hexaoctagonal tiling is a uniform tiling of the hyperbolic plane.
Constructions There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,6] kaleidoscope. Removing the mirror between the order 2 and 4 points, [8,6,1+], gives [(8,8,3)], (*883). Removing the mirror between the order 2 and 8 points, [1+,8,6], gives [(4,6,6)], (*664). Removing two mirrors as [8,1+,6,1+], leaves remaining mirrors (*4343). + Four uniform constructions of 6.8.6.8UniformColoringSymmetry[8,6](*862)[(8,3,8)] = [8,6,1+](*883)[(6,4,6)] = [1+,8,6](*664)[1+,8,6,1+](*4343)Symbolr{8,6}r{(8,3,8)}r{(6,4,6)}Coxeterdiagram = = =
Symmetry The dual tiling has face configuration V6.8.6.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4343), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*43) orbifold. These are subsymmetries of [8,6]. [1+,8,4,1+], (*4343)[(8,4,2+)], (2*43)
Related polyhedra and tiling
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