Not only does it generalize to higher n, it also shows a bit more: not only that the lack of such vector field for an even n, but the also the existence of such for odds.
Even better, the (4n - 1)-sphere (so think of S^3, S^7, S^11, ...) can be thought of as a certain subset of H^n (same thing as before but with quaternions instead of complex numbers), where multiplication by i, j and k are available! And now in this case you have not only one nowhere-vanishing vector field on the sphere, but three everywhere pairwise orthogonal vector fields. This in particular shows that S^3 is 'parallelisable' — a property it shares with S^1 and means that there exists a continuous global choice of basis for each tangent space.
I didn't quite understand the curves that they are constructing on S^2. Some figures would be nice.
The idea is that, as we trace the curve '\gamma(t)', we are constantly measuring the angle - with a positive-negative sign - between (a) the tangent vector 'v(\gamma(t))', and (b) the current velocity vector of '\gamma(t)'. As we trace the curve, if this angle rotates counterclockwise 0...90...180...270...0, we add "1" to our rotation number, and we subtract one for a clockwise rotation 0...-90...-180...-270...0.
SegfaultSeagull•5mo ago
hinkley•5mo ago
xanderlewis•5mo ago
Clearly, what they say about Germans is true.
fxwin•5mo ago
gerdesj•5mo ago
I lived in West Germany for some years back in the day and I don't recall the locals being too shy. Frankly the Germans and the Dutch seemed to have had a rather more ribald sense of humour than the "oo err Missus" efforts we Brits fielded back then.
To be fair we could robustly swear on telly after 2100, provided it didn't involve too many rude bits and you could not misspell one variant of King Canute's name or Matron would be jolly upset.
Anyway, I'm pretty sure someone called this the "dog's arse" (it has to go somewhere!)
IAmBroom•5mo ago
That they don't have a healthy sense of humor.
I remember a comedian who toured Europe, and said it wasn't true - Germans laughed as much as anyone else at his jokes. However, afterwards they took him aside and explained, "It was very funny, you see, the joke about combs being like salad forks, but we just want you that we don't discriminate against forks here. That was an unfortunate incident from during the War, but today we are much more enlightened and invite all kinds of cutlery!"
gerdesj•5mo ago
We have a comedian on the circuit in the UK called Henning Wehn. He is German and suitably daft and hilarious.
xanderlewis•5mo ago
cool_dude85•5mo ago