New classes of monohedral spherical tilings by non-convex spherical hexagons and non-convex spherical Pentagons with GeoGebra

Abstract

In previous works we have ilustrate a procedure to obtain spherical tiling with GeoGebra. We have found new classes of monohedral spherical tiling by four spherical pentagons, and new class of dihedral spherical tiling by twelve spherical pentagons. One again, we would make use of GeoGebra to show how we can do generate new classes of monohedral non-convex hexagonal spherical tilings, H(C ,τ), changing the side gluing rules of the regular spherical octahedral tiling, by local action of particular subgroups of spherical isometries. In relation to one of the new classes, by hexagonal tiles, we describe some of its properties. We also show the existence of a a new family of monohedral pentagonal tiling which arises as a degenarated case associated to the family H(C ,0). All these classes of spherical tilings have emerged as a result of an interactive construction process, only possible by the use of newly produced GeoGebra tools and the dynamic interaction capabilities of this software.