Tilings of the sphere by congruent quadrilaterals I: edge combination $a^{2}bc$

Abstract

The edge-to-edge tilings of the sphere by congruent quadrilaterals of Type $a^{2}bc$ are classified as $3$ classes: a sequence of two-parameter families of $2$-layer earth map tilings with $2n$ $(n\ge 3)$ 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 $8n$ $(n\ge 2)$ tiles together with two flip modifications for odd $n$. We also describe the moduli and calculate the geometric data.

arXiv commentFinal 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 writings