I took a differential geometry topics course from Gene Calabi about 12 years ago in grad school. I was surprised to see his name on the upcoming course calendar — I'd heard of him ("Calabi-Yau manifold") but hadn't expected he might still be teaching classes. (And ordinarily he wasn't — he was already long-retired at that point.)<p>Turned out the reason he was teaching a course that semester was that he was really excited to share a new idea he'd had about how to visualize the complex projective plane. There were two (or three?) students in attendance.<p>Okay, here's Calabi's visualization of the complex projective plane. The complex projective plane (as everyone knows) is the set of equivalence classes of complex 3-vectors, where two vectors are equivalent if one is a complex scalar multiple of the other. Calabi's visualization: for a given representative, take the real part and imaginary part of that 3-vector separately, now that's a pair of vectors in <i>real</i> 3-space, and those are much easier to visualize. Choose a particular representative for the equivalence class: use up the dilation part of your complex scalar to normalize your 3-vectors, then use up the rotation part to make them orthogonal. Now notice that multiplication by a unit-length complex scalar will rotate the real and imaginary 3-vectors through the plane they span, so we can visualize an element of the complex projective plane as an <i>oriented ellipse in real 3-space</i>.<p>Actually we have some singular ones — if the real and imaginary parts are the same length (and orthogonal) then instead of ellipses we have circles, and if the real or imaginary part is zero then we have line segments. But we have a 1-parameter family of oriented ellipses (the parameter is the eccentricity) where at each parameter we have a quotient of SO(3) (it's almost the special orthogonal group but not quite) and at one end of the family it reduces to a 2-sphere and at the other end to a real projective plane.<p>But the main point is instead of having to visualize 3 or 2 complex dimensions (6 or 4 real dimensions) now we get to visualize ellipses (1 real dimension!) inside real 3-space. Much more accessible to geometric intuition.
This is far outside of my area of competence, but I have always had a hard time understanding/imagining what people mean when they say string theory uses extra dimensions but that those are rolled up or tiny. Like, I know about projective and hyperbolic spaces and understand that a metric can be zero or negative. What I do not understand is how a dimension can be small, like what makes these different from our macro dimensions.<p>I know that string theory is a research program and that they first started out studying vibrating lines and circles because this avoids the singularity of the point which was a useful idea in quantum gravity but then this didn't work either because reasons so they keeps adding more "micro" dimensions.<p>Maybe someone can someone give me a drawing of one micro dimensions and one micro dimensions, sort of like the euclidian plane is two macro dimensions.