From discrete to dense: explorations in the moduli space of triangles
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866908928318636032 |
|---|---|
| author | Aggarwal, Aahana Gupta, Subhojoy Nair, Ajay K. |
| author_facet | Aggarwal, Aahana Gupta, Subhojoy Nair, Ajay K. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_00373 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | From discrete to dense: explorations in the moduli space of triangles Aggarwal, Aahana Gupta, Subhojoy Nair, Ajay K. Metric Geometry Probability 51M04, 60D05, 28A33 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. |
| title | From discrete to dense: explorations in the moduli space of triangles |
| topic | Metric Geometry Probability 51M04, 60D05, 28A33 |
| url | https://arxiv.org/abs/2604.00373 |