You can perfectly recreate any triangle shape just by using the dots on a standard piece of graph paper.
April 2, 2026
Original Paper
From discrete to dense: explorations in the moduli space of triangles
arXiv · 2604.00373
The Takeaway
While you might think a fixed grid would limit the types of triangles you can draw, mathematicians proved that 'lattice triangles' (those with vertices on grid points) are dense enough to mimic any shape imaginable. However, they discovered that these triangles don't appear evenly; the grid actually 'favors' certain shapes over others as you look at larger areas.
From the 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