Tilings of the sphere by congruent quadrilaterals III: edge combination a3ba^3b 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 $a^3b$-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.Comment: 29 pages, 22 figures, 12 tabl

Tilings of the sphere by congruent quadrilaterals I: edge combination a2bca^2bc

The edge-to-edge tilings of the sphere by congruent quadrilaterals of Type $a^2bc$ are classified as $3$ classes: a sequence of two-parameter families of $2$-layer earth map tilings with $2n$ $(n\ge3)$ tiles, a one-parameter family of quadrilateral subdivisions of the octahedron with $24$ tiles together with a flip modification for a special parameter, and a sequence of $3$-layer earth map tilings with $8n$ $(n\ge2)$ tiles together with two flip modifications for odd $n$. We also describe the moduli and calculate the geometric data.Comment: Final version: 44 pages, 42 figures, 2 tables. Thank two junior students Fangbin Chen and Nan Zhang for showing us how to draw Fig. 2 using GeoGebra. Thank one referee for the long and detailed suggestions so much, which essentially improved the writing