Cyclotomic points on varieties and all rational $a^3b$-monotiles

TL;DR

This paper introduces a novel method for identifying rational $a^3b$-monotiles on spheres by computing cyclotomic points on algebraic varieties, leading to a complete classification of certain tilings.

Contribution

It presents an independent, efficient approach to classify edge-to-edge monohedral quadrilateral tilings using cyclotomic points, improving upon previous methods that relied on multiple older works.

Findings

Complete classification of rational $a^3b$-monotiles for the sphere.

New method based on cyclotomic points on algebraic varieties.

Resolved gaps and corrected typos in previous classifications.

Abstract

By computing all cyclotomic points on some algebraic varieties, we get an independent and efficient way to find all rational $a^3b$-monotiles for the sphere, thereby completing the classification of edge-to-edge monohedral quadrilateral tilings. Both of the previous classifications \cite{lw2} and \cite{cl} depended on many old works of different authors while quite a few typos and gaps were found.