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This would make a beautiful visualization of the hairy ball problem: "The hairy ball theorem of algebraic topology states that there is no nonvanishing continu
by dfc 14y ago
This would make a beautiful visualization of the hairy ball problem:
"The hairy ball theorem of algebraic topology states that there is no nonvanishing continuous tangent vector field on even dimensional n-spheres. For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in R3 to every point p on a sphere such that f(p) is always tangent to the sphere at p, then there is at least one p such that f(p) = 0. In other words, whenever one attempts to comb a hairy ball flat, there will always be at least one tuft of hair at one point on the ball. The theorem was first stated by Henri Poincaré in the late 19th century."
https://en.wikipedia.org/wiki/Hairy_ball_theorem https://en.wikipedia.org/wiki/Hairy_ball_theorem
- the_cat_kittles 14y agoProof be damned, I can't help but try and think of ways to comb it flat... :)
- nealabq 14y agoHairy bagels, on the other hand, can be combed without cowlicks.
- Devilboy 14y agoWhat about a bagel with two holes?
- sbi 14y agoThe Euler characteristic of a compact orientable genus g surface X is nonzero if g is not 1 (the general formula is 2-2g), so the Euler class e(TX) of the tangent bundle TX of X is nonzero. This implies that TX cannot have a continuous nowhere-vanishing section. If it did, e(TX) would be 0. The single-holed torus is the only orientable compact surface that possesses a continuous nonvanishing vector field.
- brownbat 14y agoAh, the dilemma of Wikipedia math articles, which all kick off with mouthfuls like "nonvanishing continuous tangent vector field." If you don't have a PhD in mathematics, they're unreadable. And if you do, they're unnecessary.
- gsteinb88 14y ago[Tangent about math articles on Wikipedia] I'd like to respectfully disagree. 'Nonvanishing continuous tangent vector field' is a phrase that can be understood after freshman level multivariable calculus. More generally, Wikipedia articles on math do tend to rely on the accepted terminology of the field for the opening sentence. It's generally better, however, to skim that bit, pick out the words you know to make sure you're in generally the right place, and then skip around until you find something you can latch onto. If you're really lost, you might need to click around a bit. It's actually pretty amazing what you can pick up just from looking at the pictures and trying to get the general gist of what's going on rather than getting caught up in the details. But the details have to be there -- without the precision and proper terminology, these articles would not only be wrong; they would also be useless to anyone trying to actually apply them.