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arXiv · 2205.14936

Tilings of the sphere by congruent quadrilaterals II: edge combination $a^3 b$ with rational angles

Yixi Liao·Erxiao Wang
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Abstract

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 $a^3b$-quadrilaterals with all angles being rational degrees. There are $12$ sporadic and $3$ infinite sequences of quadrilaterals admitting the $2$-layer earth map tilings together with their modifications, and $3$ sporadic quadrilaterals admitting $4$ exceptional tilings. Among them only $3$ quadrilaterals are convex. New interesting non-edge-to-edge triangular tilings are obtained as a byproduct.

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Yixi Liao, Erxiao Wang. 2022-05-30. Tilings of the sphere by congruent quadrilaterals II: edge combination $a^3 b$ with rational angles. https://arxiv.org/abs/2205.14936

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