mmmmmmmike's comments

Yeah, and even given that, there’s the question of how exactly it deforms from its flattened shape to make a spiral (and if this changes the area). I wouldn’t agree with the “correct” answer if the tape was very thick, but given that the answer is .005 cm, it’s probably thin enough that such an approximation is okay.

I think that’s more about what he chose to publish. He was known for discovering things and keeping them to himself.

As I understand it, he actually first came to conjecture quadratic reciprocity after doing incredible amounts of calculation by hand and noticing the pattern.

One thing I distinctly remember they _weren't_ doing better was letting you just watch your video in piece. The controls were obtrusive and always visible, the background was bright white, and the suggested videos took up a huge chunk of the screen. I don't know how much that may have played to their advantage (by encouraging you to surf new videos) but I remember avoiding YouTube links in favor of basically anything else for a good long while.

Sort of. A Euclidean structure has “addition” but not “multiplication”. Having an additional operation that’s required to be associative, have inverses, distribute over addition, and play nicely with the norm turns out to be such a constraint that, as he mentions in the previous paragraph, the real numbers, complex numbers, and quaternions are the only such structures.

Literally speaking then, “... when R^d is an associative normed division algebra” just means “when d = 1, 2, or 4”, except of course that the idea is to use the multiplicative structure in the proof.

Many theorems let you know that something exists but don't tell you how to compute it efficiently (an orthogonal basis, eigenvalues, inverses, etc.)

Also, sometimes algorithms can be invented and empirically shown to have good complexity properties before there are proofs.