For a given real number $$extract_itex$$\alpha$$/extract_itex$$, let us place the fractional parts of the points $$extract_itex$$0, \alpha, 2 \alpha, \cdots, N \alpha$$/extract_itex$$ on the unit circle.
These points partition the unit circle into $$extract_itex$$N+1$$/extract_itex$$ intervals having at most three lengths, one being the sum of the other two. This is the three distance theorem. We consider a two-dimensional version of the three distance theorem obtained by placing the points $$extract_itex$$n\alpha+ m\beta$$/extract_itex$$, for $$extract_itex$$0 \leq n,m \leq N$$/extract_itex$$, on the unit circle. We provide examples of pairs of real numbers $$extract_itex$$(\alpha,\beta)$$/extract_itex$$ for which there are finitely many lengths between successive points (with $$extract_itex$$1,\alpha, \beta$$/extract_itex$$ rationally independent and not badly approximable), as well as examples for which there are infinitely many of them.