What was your biggest recent surprise?

DataPacRat

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?

It can't be 2n-2 because it's 3 when n=3. I get 2n-3 because the first vector is chosen with n-1 degrees of freedom, then the second with n-2, then subtract one because of the equivalence class of rotations, then add one for choosing how far to rotate.

EDIT: More generally, I think that the dimension of k-dimensional subspaces of an n-dimensional spaces is k(n-k), so where k=2 you get 2n-4, then add one for choosing how far to rotate. I'd feel better if I knew what I meant by "dimension" here though; it's not a vector space.

EDIT: More generally, I think that the dimension of k-dimensional subspaces of an n-dimensional spaces is k(n-k)

This is correct.

4gjm14y

Meh, you're right: the dimension of the space of 2-dimensional subspaces of n-space is 2n-4, not 2n-3. The reason why my handwavy dimension-counting above was wrong is ("of course") that I failed to "subtract one because of the equivalence class of rotations". And yes, you're right that in general it's k(n-k). "Dimension" here means: locally the set looks like a that-many-dimensional vector space. That is, e.g., any element of SO(n) has a neighbourhood that's topologically the same as a neighbourhood in R^(n(n-1)/2).

2A1987dM14y

The number of parameters you need to label each element (provided the labelling is a continuous function, otherwise you can label points of R^2 with a single parameter e.g. (3.1415..., 2.7182...) -> 32.174118...)