Diffeology and Arithmetic of Irrational Tori

The irrational torus, ${\mathrm{T}}_{\alpha }$, originally introduced as a geometric model for quasicrystals, is a foundational object in the theory of diffeology. This paper, after recalling its main algebraic properties, provides a comprehensive analysis of a new geometric invariant for this singular space: the group of flows, $\mathbf{F}\mathbf{l}({\mathrm{T}}_{\alpha },\mathbf{R})$. This invariant, which is trivial for all manifolds, arises as the core of the obstruction to the de Rham theorem in the diffeological setting. We provide a complete computation and geometric interpretation of this group, proving the isomorphism $\mathbf{F}\mathbf{l}({\mathrm{T}}_{\alpha },\mathbf{R})\simeq \mathbf{R}\times \mathrm{c}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}({\mathrm{\Delta }}_{\alpha })$.