Tilings of the sphere by congruent quadrilaterals. III: Edge combination \(a^3b\) with general angles (Q6612583)

This is part III terminating the series of papers classifying all edge-to-edge tilings of the sphere by congruent quadrilaterals, started by \textit{Y. Liao} et al. [Chin. Ann. Math., Ser. B 45, No. 5, 733--766 (2024; Zbl 07938240)] and \textit{Y. Liao} and \textit{E. Wang} [Nagoya Math. J. 253, 128--163 (2024; Zbl 1545.52013)].\N\NAs final edge configuration the case \(a^3b\) is considered, i.e. tiles with three edges of equal length, with at least one angle an irrational multiple of \(\pi\). The main result is that on the one hand there are 5 sporadic cases among which 3 with 16 tiles and one of 12 and 24 tiles each; on the other hand, for each even number of tiles \(\geq 6\) a continuous 1-parameter family of 2-layer earth map a3b-tilings exist, together with a discrete number of flip modifications.\N\NThe paper ends with an overview of the complete classification of spherical edge-to-edge tilings by congruent quadrilaterals.