A two dimensional analog of the three gap theorem

For a two dimensional vector $\bar{v}=(\alpha ,\beta )$, where $\alpha >0,\beta >0$ are irrational numbers independent over $\mathbb{Q}$, we consider the set $D_n=\{(i\alpha \mathrm{m}\mathrm{o}\mathrm{d}1,i\beta \mathrm{m}\mathrm{o}\mathrm{d}1),i=1,\dots ,n\}$ in a two dimensional torus and the partition of this torus into Voronoi cells. Areas and forms of these cells are the subject of this experimental work.