TILINGS OF THE SPHERE BY CONGRUENT QUADRILATERALS II: EDGE COMBINATION a3b WITH RATIONAL ANGLES

Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This second one applies the powerful tool of trigonometric Diophantine equations to classify the case of a(3)b-quadrilaterals with all angles being rational degrees. There are 12 sporadic and 3 infinite sequences of quadrilaterals admitting the two-layer earth map tilings together with their modifications, and 3 sporadic quadrilaterals admitting 4 exceptional tilings. Among them only three quadrilaterals are convex. New interesting non-edge-to-edge triangular tilings are obtained as a byproduct.

Automated summary

This paper completely classifies edge-to-edge tilings of the sphere by congruent quadrilaterals in a series of three papers. The second paper uses trigonometric Diophantine equations to classify the case of a(3)b-quadrilaterals with all angles being rational degrees. There are 12 sporadic and 3 infinite sequences of quadrilaterals allowing two-layer earth map tilings and their modifications, as well as 3 sporadic quadrilaterals allowing 4 exceptional tilings. Among them, only three quadrilaterals are convex. New interesting non-edge-to-edge triangular tilings are also obtained as a byproduct.