String theory: Why assume additional dimensions are 'rolled up?

[Gibsense]

String theory has extra dimensions (9 or so) rolled up. I have never understood why they have to be hidden in this way. We are three-dimensional time (spacetime if you like maybe) with our Universe embedded in unlimited dimensions (?). If nothing exists there how can embedding space be limited?

Even if our Universe is multidimensional why would we expect to perceive it as such (except mathematically). We might be a 3 Torus or a 4 Torus or an (n + x) Torus but we are still only capable of perceiving our 3D status. The other dimensions could still be spatial (not tiny curled) Where am I going wrong?

 

[Classical Motion]

All one needs to do, is twist a line to get relative length, area and density.