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arXiv · 2604.00373

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

Aahana Aggarwal·Subhojoy Gupta·Ajay K. Nair
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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.

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Aahana Aggarwal, Subhojoy Gupta, Ajay K. Nair. 2026-04-01. From discrete to dense: explorations in the moduli space of triangles. https://arxiv.org/abs/2604.00373

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