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

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 $\mathit{\text{lattice}}\mathit{\text{triangles}}$, which are triangles whose vertices have integer coordinates. We prove that this subset is $\mathit{\text{dense}}$, 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 $\mathit{\text{not}}$ become uniformly distributed in the moduli space as $N$ grows. Along the way, we encounter connections with geometry, number theory, analysis, and probability.