Consider the following two foliations of torus:


1)The Kronecker foliation with slope $\sqrt{2}$


2)The Kronecker foliation with slope $\pi$


As I learn from the literature, these two foliations are not topological equivalent. The proof is that the K theory of their corresponding $C^{\*}$ algebras are not isomorphic.In fact two Kronecker foliations with slopes $\alpha$ and $\beta$ are not equivalent if $\alpha$ and $\beta$ are not on the same orbit of action of $Sl\_{2}(\mathbb{Z})$. See Non Commutative Geometry by Alain Connes.


But **intuitively** it is difficult to imagine that these two foliations are different. because in both foliations all leaves are dense! one can not distinguish these two foliations via visible topological behavior. 



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> Is there an intuitive and geometric proof for this fact(without using K theory, $C^{\*}$ algebras of foliation,etc.)?
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I think, if there is no an intuitive proof, this shows the deep power of the role of $C^{\*}$ algebra of foliations, at least in this example.



"The" kronecker foliation or "a" kronecker foliation?
Just what the doctor ordered, a proof using diffeology instead of $\mathrm C^\*$-algebras:


MR0799609 Donato, Paul(F-CNRS-T); Iglésias, Patrick(F-CNRS-T)
 **Exemples de groupes difféologiques: flots irrationnels sur le tore.** (French. English summary) [Examples of diffeological groups: irrational flows on the torus] 
[C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 4, 127–130](http://gallica.bnf.fr/ark:/12148/bpt6k5494341n/f43). 



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> Author summary (translated from the French): "We illustrate J.-M. Souriau's technique of `diffeological spaces and groups' in the case of the quotient $T\_α$ of the standard torus by the irrational flow with slope $α$. Computing the universal covering $\mathbf R$
>  and the fundamental group $\mathbf Z^2$ of $T\_α$ allows us to classify these tori diffeologically: $T\_α$ and $T\_β$ are diffeomorphic if and only if $α∼β$ modulo $\mathrm{GL}(2,\mathbf Z)$; moreover, the computation of $\mathrm{Diff}(T\_α)$ reveals a difference between quadratic irrationals and other irrationals.''
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