Classification of rational angles in plane lattices

TL;DR

This paper investigates the classification of angles that are rational multiples of pi in plane lattices, revealing complex structures and parametrizations involving algebraic curves of positive genus.

Contribution

It provides a detailed classification of rational angles in plane lattices, including parametrizations involving algebraic curves of positive genus.

Findings

Limited number of rational angles in a given lattice

Parametrizations involve rational points on algebraic curves of positive genus

Classification is more complex than initially expected

Abstract

This paper is concerned with configurations of points in a plane lattice which determine angles that are rational multiples of $\pi$. We shall study how many such angles may appear in a given lattice and in which positions, allowing the lattice to vary arbitrarily. This classification turns out to be much less simple than could be expected, leading even to parametrizations involving rational points on certain algebraic curves of positive genus.Bulletin of the American Mathematical Society