I recently flipped through the "Cartoon Guide to Physics", expecting an easy-to-understand rehash of ideas I was long familiar with; and that's what I got - right up to the last few pages, where I was presented with a fairly fundamental concept that's been absent from the popular science media I've enjoyed over the years. (Specifically, that the uncertainty principle, when expressed as linking energy and time, explains what electromagnetic fields actually /are/, as the propensity for virtual photons of various strengths to happen.) I find myself happy to try to integrate this new understanding - and at least mildly disturbed that I'd been missing it for so long, and with an increased curiosity about how I might find any other such gaps in my understanding of how the universe works.
So: what's the biggest, or most surprising, or most interesting concept /you/ have learned of, after you'd already gotten a handle on the basics?
Wow, I didn't know that. It makes sense now I think about it though; SO(n) must be something like an n(n-1)/2 dimensional space, but the space of rotations about an (n-2)-subspace must be ... err ... something smaller - maybe 2n-3 dimensional? I may be abusing the idea of dimension here...
First of all, terminology. SO(n) is orientation-preserving orthogonal transformations on n-space, or equivalently the orientation-preserving symmetries of an (n-1)-sphere in n-space. So Joshua's statement is about SO(n) for n>3.
OK. So the obvious way to interpret "rotation about an axis" in many dimensions is: you choose a 2-dimensional subspace V, then represent an arbitrary vector as v+w with v in V and w in its orthogonal complement, and then you rotate v. The dimension of the set of these things is (n-1)+(n-2) from choosing V -- you can pi... (read more)