From discrete to dense: explorations in the moduli space of triangles

TL;DR

This paper investigates the space of triangle shapes, proving that lattice triangles are dense in this space but do not distribute uniformly within bounded regions, revealing connections to various mathematical fields.

Contribution

It establishes the density of lattice triangles in the moduli space and analyzes their distribution properties within bounded regions, linking geometry, number theory, analysis, and probability.

Findings

Lattice triangles are dense in the moduli space of all triangles.

Lattice triangles in bounded regions do not become uniformly distributed as the region size increases.

Connections with geometry, number theory, analysis, and probability are explored.

Abstract

The moduli space of triangles is a two-dimensional space that records triangle shapes in the plane, considered up to similarity. We study the subset corresponding to \textit{lattice triangles}, which are triangles whose vertices have integer coordinates. We prove that this subset is \textit{dense}, that is, every triangle shape can be approximated arbitrarily well by lattice triangles. However, when one restricts to lattice triangles in the square $[-N,N]^2$, their shapes do \textit{not} become uniformly distributed in the moduli space as $N$ grows. Along the way, we encounter connections with geometry, number theory, analysis, and probability.