The tilings of the sphere by polygons have been studied by mathematicians for more than a century. One major achievement was the complete classification of edge-to-edge monohedral tilings of the sphere by triangles. The tiles in other edge-to-edge monohedral tilings of the sphere must be quadrilaterals or pentagons. Little is known about quadrilateral and pentagon tilings. The minimal case of the pentagon tiling has been completely classified, by studying the combinatorial, edge length and angle aspects separately, and then combining the corresponding classifications together. This thesis attempts to study the numerical part of the angle combinations beyond the minimal case. We completely classify all possible angle combinations at degree 3 vertices. We further completely classify the a...[ Read more ]

The tilings of the sphere by polygons have been studied by mathematicians for more than a century. One major achievement was the complete classification of edge-to-edge monohedral tilings of the sphere by triangles. The tiles in other edge-to-edge monohedral tilings of the sphere must be quadrilaterals or pentagons. Little is known about quadrilateral and pentagon tilings. The minimal case of the pentagon tiling has been completely classified, by studying the combinatorial, edge length and angle aspects separately, and then combining the corresponding classifications together. This thesis attempts to study the numerical part of the angle combinations beyond the minimal case. We completely classify all possible angle combinations at degree 3 vertices. We further completely classify the angle combinations at all vertices for the cases of one or two distinct angles. For three distinct angles, we classify all but four cases. And for four distinct angles, we get all 94 angle combinations for one special case, and we also know many other cases that do not allow suitable angle combinations. In the process, we developed some new techniques to tackle the new phenomena. Such techniques could be useful for further study.