Hairy ball theorem. 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,
Hairy Ball Theorem. There does not exist an everywhere nonzero tangent vector field on the 2-sphere . This implies that somewhere on the surface of the Earth, there is a point with zero horizontal wind velocity. The theorem can be generalized to the statement that the …
Nov 27, 2011 · Ever tried to comb a hairy ball? Math says you failed! Trying out a new feature: English Transcript! Let me know how it works Tweet it – http://bit.ly/sKAjpS
Hairy Ball Theorem. Another fun theorem from topology is the Hairy Ball Theorem. It states that given a ball with hairs all over it, it is impossible to comb the hairs continuously and have all the hairs lay flat. Some hair must be sticking straight up! A more formal version says …
The answer is negative, and is called the hairy ball theorem (since it “explains” why one cannot continuously comb the hair on a ball without a bald spot): Theorem 1.1. A smooth vector ﬁeld on Sn must vanish somewhere if n is even.
The Hairy Ball Theorem. The Hairy Ball theorem is also a consequence of a more general theorem of Poincaré on vector fields on surfaces. Let X be a continuous vector field on a surface with an isolated zero at p, that is, such that p is the only point in its neighbourhood where X vanishes.
One consequence of the hairy ball theorem: There is always a place on the surface of Earth where the wind is still (such as in the eye of a cyclone ). This applies on a scale of roughly 10 km, the height of the weather; to that approximation, the velocity of the wind is a …
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One of these is the so-called hairy ball theorem, which says that any tangent vector eld on a 2n-sphere must vanish somewhere. Another is the ham sandwich theorem, which says that given any collection of nobjects of nite measure in n-space, there is an (n 1)-hyperplane which slices each object in half.
May 17, 2013 · Of Hairy Balls, Fixed Points and Coffee The title of this post must be inducing a giggle or two from the readers, and yes, you read that right; I will be discussing the ‘Hairy Ball Theorem…
The hairy ball theorem is an abstract century-old result in algebraic topology popularly known for its amusing name and wonderfully intuitive visualization: rather than thinking of continuous tangent vector fields on the 2-sphere, imagine hairs on a ball, and try to comb them all smooth and flat with no whorls or cowlicks. Hairy ball theorem
THE HAIRY BALL THEOREM. We now use Lemma 2 to prove the Hairy Ball Theorem. We begin by assuming that a continuous non-zero tangent vector ﬁeld on S2 does exist, and use this supposed vector ﬁeld to produce a labeling of a triangulated polygon.