The three gap theorem and the space of lattices

The three gap theorem (or Steinhaus conjecture) asserts that there are at most three distinct gap lengths in the fractional parts of the sequence $\alpha ,2\alpha ,\dots ,N\alpha$, for any integer $N$ and real number $\alpha$. This statement was proved in the 1950s independently by various authors. Here we present a different approach using the space of two-dimensional Euclidean lattices.