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 $a^{3}b$-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.