And there's at least one extra level: the ability to think about things that can't be imagined. It's not that difficult to visualize a hyperbolic plane and lea

And there's at least one extra level: the ability to think about things that can't be imagined. It's not that difficult to visualize a hyperbolic plane and learn to visualize hyperbolic space. It's much, much harder to visualize the real projective plane, and essentially impossible to visualize a quaternionic projective space. Yet these can still be worked with using geometric and topological intuition in a way that feels (to me) fundamentally different from manipulating abstract formulae or matrices.