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

Aahana Aggarwal, Subhojoy Gupta, Ajay K. Nair

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.

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

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 , their shapes do \textit{not} become uniformly distributed in the moduli space as grows. Along the way, we encounter connections with geometry, number theory, analysis, and probability.

Paper Structure

This paper contains 15 sections, 12 theorems, 27 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattice triangles are not equilateral
  4. Moduli space of triangles
  5. Simultaneous Dirichlet approximation
  6. Proof of the Density result
  7. Approximating equilateral triangles
  8. Approximating general triangles
  9. Distribution in moduli space
  10. The question of equidistribution
  11. Proof of Theorem \ref{['thm:nequi']}
  12. Epilogue
  13. Forgetting weights
  14. Rate of convergence
  15. Higher dimensions

Key Result

Theorem 1.1

The set of all lattice triangles $\mathcal{L}$ determines a dense set in $\mathcal{M}_\Delta$. $\blacktriangleleft$$\blacktriangleleft$

Figures (6)

  • Figure 1: Vector $\overrightarrow{AC}$ is obtained by rotating $\overrightarrow{AB}$ counterclockwise by $60^\circ$; if $A,B$ are lattice points, then $C$ is not.
  • Figure 2: The Teichmüller space of triangles $\mathcal{T}_\Delta$ as parameterized in Lemma \ref{['param']}, together with its projection to the $ab$-plane. The subset corresponding to $\mathcal{M}_\Delta$ is also shown, shaded darker.
  • Figure 3: The space $\mathcal{T}_\Delta$ is parametrized by two normalized side-lengths $a,b$, since the third side-length $c=2-a-b$. The loci of isosceles triangles (where two side-lengths are equal) are the three line segments (shown in bold); they meet at the equilateral triangle (shown in red). The fundamental domain of the action of the permutation group $S_3$ (shown shaded in blue), is the moduli space $\mathcal{M}_\Delta$.
  • Figure 4: The subset of $\mathcal{T}_\Delta$ corresponding to obtuse triangles defines the three regions $O_a,O_b$ and $O_c$ (shaded grey). The quotient of $O_a \cup O_a \cup O_b$ by the action of $S_3$ determines the subset $\mathcal{O}$ of similarity classes of obtuse triangles in $\mathcal{M}_\Delta$ (shaded purple).
  • Figure 5: Points in $\mathcal{T}_\Delta$ corresponding to a random sample of $10^3$ points (left), $5\times 10^3$ points (middle), and $1.5 \times 10^4$ points (right) in $\mathcal{S}_{100}$, i.e., lattice triangles in $[-100, 100] \times [-100,100]$. In the middle figure notice that more points seem to concentrate close to the sides; this is expected as by Langford's result, $\approx 72\%$ of obtuse triangles needs to fit into $\approx 68\%$ of the area of $\mathcal{T}_\Delta$. The distribution away from the sides is visually close to being uniform, which is confirmed by density plots.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • Definition 2.5: Lattice triangle $\to$ Points in $\mathcal{M}_\Delta$
  • ...and 13 more