Much more niche than the hairy ball theorem is the Cox-Zucker Machine, supposedly they decided during the first year of undergrad that they eventually had to work together.
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!)
Heard a mathematician friend call this the “hairy sphere theorem” once. At first I thought he was being a prude, but now I appreciate that the theorem is about spheres, as opposed to balls.
S^2 isn't a special case though: Brouwer's showed the theorem can be easily extended to high dimensions, hence today we usually consider the more general statement that there is a nonzero tangent vector field on the n-sphere S^n iff n is odd.
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.
It’s really easy to see that such a vector field exists on odd dimensional spheres, though, by extending the construction on S^1: f(x, y) = (-y, x). In higher dimensions, you do the same thing, swap elements pair wise and multiply one of the elements of the pair by -1. This works in odd dimensional sphere because you can pair up coordinates.
If you're talking about C(p, s): consider how lines of latitude create a sequence of circles on Earth: the curve C(p, s) is the "circle of latitude" given by fixing p on S^2 as your North Pole, and 's' as (up to rescaling) the "latitude" relative to the North Pole. More specifically, when 's' = 0, C(p, s) is the Equator relative to the North Pole, and when 's' approaches 1, imagine these circles of latitude getting closer and closer to the North Pole.
I'm finding it a little harder to visualize rotation numbers, though. My best attempt at a description is to imagine continuously tracing the curve '\gamma(t)', going through every point that it passes through, while looking top-down on it. At every point on the curve, the vector field 'v' produces a vector 'v(\gamma(t))' that begins at '\gamma(t)', lies flat on the sphere (i.e. is tangent to the sphere), and is of nonzero length. (The last assumption is the assumption we are making for contradiction).
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.
I am confused how we can define a rotation number of the map from S^1 to R^3 defined at the end of the second paragraph. R^3 is nullhomotopic, after all...
I majored in mathematics and remember encountering this theorem in a topology course. I giggled then, and 20 years later I giggle again.
Much more niche than the hairy ball theorem is the Cox-Zucker Machine, supposedly they decided during the first year of undergrad that they eventually had to work together.
In German it’s called the Hedgehog Theorem.
Clearly, what they say about Germans is true.
Ooh go on, what do they say about Germans?
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!)
Also known as the Combed Hedgehog Theorem (which i like a bit better)
Have you considered a job in Defense? They love acronyms you can’t say out loud.
Heard a mathematician friend call this the “hairy sphere theorem” once. At first I thought he was being a prude, but now I appreciate that the theorem is about spheres, as opposed to balls.
S^2 isn't a special case though: Brouwer's showed the theorem can be easily extended to high dimensions, hence today we usually consider the more general statement that there is a nonzero tangent vector field on the n-sphere S^n iff n is odd.
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.
It’s really easy to see that such a vector field exists on odd dimensional spheres, though, by extending the construction on S^1: f(x, y) = (-y, x). In higher dimensions, you do the same thing, swap elements pair wise and multiply one of the elements of the pair by -1. This works in odd dimensional sphere because you can pair up coordinates.
It may be a short proof, but it somewhat implicitly asks that the reader has some background in geometry.
I didn't quite understand the curves that they are constructing on S^2. Some figures would be nice.
If you're talking about C(p, s): consider how lines of latitude create a sequence of circles on Earth: the curve C(p, s) is the "circle of latitude" given by fixing p on S^2 as your North Pole, and 's' as (up to rescaling) the "latitude" relative to the North Pole. More specifically, when 's' = 0, C(p, s) is the Equator relative to the North Pole, and when 's' approaches 1, imagine these circles of latitude getting closer and closer to the North Pole.
I'm finding it a little harder to visualize rotation numbers, though. My best attempt at a description is to imagine continuously tracing the curve '\gamma(t)', going through every point that it passes through, while looking top-down on it. At every point on the curve, the vector field 'v' produces a vector 'v(\gamma(t))' that begins at '\gamma(t)', lies flat on the sphere (i.e. is tangent to the sphere), and is of nonzero length. (The last assumption is the assumption we are making for contradiction).
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.
I am confused how we can define a rotation number of the map from S^1 to R^3 defined at the end of the second paragraph. R^3 is nullhomotopic, after all...