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.

Is there an intuitive and geometric proof for this fact(without using K theory, $C^{*}$ algebras of foliation,etc.)?

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.