Cyclotomic points on varieties and all rational $a^{3}b$-monotiles

By computing all cyclotomic points on some algebraic varieties, we get an independent and efficient way to find all rational $a^{3}b$-monotiles for the sphere, thereby completing the classification of edge-to-edge monohedral quadrilateral tilings. Both of the previous classifications \cite{lw2} and \cite{cl} depended on many old works of different authors while quite a few typos and gaps were found.