Density of shapes of periodic tori in the cubic case

Consider the compact orbits of the ${\mathbb{R}}^2$ action of the diagonal group on $\mathrm{SL}(3,\mathbb{R})\mathrm{/}\mathrm{SL}(3,\mathbb{Z})$, the so-called periodic tori. For any periodic torus, the set of periods of the orbit forms a lattice in ${\mathbb{R}}^2$. Such a lattice, re-scaled to covolume one, gives a shape point in $\mathrm{SL}(2,\mathbb{R})\mathrm{/}\mathrm{SL}(2,\mathbb{Z})$. We prove that the shapes of all periodic tori are dense in $\mathrm{SL}(2,\mathbb{R})\mathrm{/}\mathrm{SL}(2,\mathbb{Z})$. This implies the density of shapes of the unit groups of totally real cubic orders.