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

TL;DR

This paper classifies all edge-to-edge tilings of the sphere by congruent quadrilaterals of type $a^2bc$, revealing three main classes including earth map tilings and subdivisions of the octahedron, with detailed geometric data.

Contribution

It provides a complete classification of sphere tilings by congruent quadrilaterals of type $a^2bc$, including explicit families and geometric descriptions.

Findings

Three classes of tilings identified: two-parameter earth map tilings, octahedron subdivisions, and three-layer earth map tilings.

Explicit formulas for the number of tiles in each tiling class.

Descriptions of moduli and geometric parameters for each tiling type.

Abstract

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.