Tilings of the sphere by congruent quadrilaterals III: Edge combination a 3 b with general angles

Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This last one classifies the case of a 3 ⁢ b {a^{3}b} -quadrilaterals with some irrational angle: there are a sequence of 1-parameter families of quadrilaterals admitting 2-layer earth map tilings together with their basic flip modifications under extra condition, and 5 sporadic quadrilaterals each admitting a special tiling. A summary of the full classification is presented in the end.