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Topic: Hairy ball theorem

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	Hairy ball theorem - Wikipedia, the free encyclopedia
		The hairy ball theorem of algebraic topology states that, in layman's terms, "one cannot comb the hair on a ball in a smooth manner".
		This is a consequence of the Poincaré–Hopf theorem.
		A consequence of the hairy ball theorem is that any continuous function that maps a sphere into itself has either a fixed point or a point that maps onto its own antipodal point.
	en.wikipedia.org /wiki/Hairy_ball_theorem (894 words)

	Universal Book of Mathematics: list of entries
		Brouwer's theorem says that there must be at least one point on the top sheet that is in exactly the same position relative the bottom sheet as it was originally.
		Brower's theorem insists that there must be some point in the coffee that is in exactly the same spot as it was before you started stirring (though it might have moved around in between).
		This theorem generalizes to higher-dimensional ham sandwiches, when it essentially becomes the Borsuk-Ulam theorem: in n-dimensional space in which there are n globs of positive volume, there is always a hyperplane that cuts all the globs exactly in half.
	www.daviddarling.info /works/Mathematics/mathematics_samples.html (5710 words)

	PlanetMath: hairy ball theorem
		The second proof gives the hairy ball theorem as a corollary of the Poincaré-Hopf index theorem.
		Near a zero of a vector field, we can consider a small sphere around the zero, and restrict the vector field to that.
		This is version 9 of hairy ball theorem, born on 2002-12-04, modified 2006-09-15.
	planetmath.org /encyclopedia/HairyBallTheorem.html (267 words)

	BBC - h2g2 - The Hairy Ball Theorem
		This states that if you have a ball that is completely covered in hair, you can't comb it so that all the hair lies flat and neat on the ball.
		The rugby ball and the American football both are elongated, but this does not affect the theorem.
		So, if we have a hairy doughnut, it will be extremely unpleasant to eat, but we can think about whether it is possible to comb the hair on it so that it lies flat everywhere.
	www.bbc.co.uk /dna/collective/A9632964 (827 words)

	blob
		Not unexpectedly, in the case of n=2 the theorem is called the pancake theorem, since it is obviously dealing with having to cut two infinitesimally thin pancakes on a plate each in half with a single cut.
		Of course, we should not confuse this with the ham-less sandwich theorem in graph theory and squeezing theorem (also called sandwich theorem) in calculus.
		And here I thought it was bad when I heard about the hairy ball theorem (alias the uncombed ball theorem, as it was presented to me) which states that “one cannot comb the hair on a ball in a smooth manner”.
	kanarcek.wordpress.com (487 words)

	hairy ball theorem
		If a sphere is covered with hair or fur, like a tennis ball, the hair cannot be brushed so that it lies flat at every point.
		In mathematical terms: any continuous tangent vector field on the sphere must have a point where the vector is zero.
		This theorem also means that somewhere on the Earth's surface there has to be a point where the horizontal wind speed is zero, even if it's windy everywhere else.
	www.daviddarling.info /encyclopedia/H/hairy_ball_theorem.html (250 words)

	Topology (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-12)
		Similarly, the hairy ball theorem of algebraic Topology says that "one cannot comb the hair on a ball smooth".
		The Tietze extension theorem: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.
		The Baire category theorem: If X is a complete metric space or a locally compact Hausdorff space, Then the interior of every union of countably many nowhere dense sets is empty.
	topology.iqnaut.net.cob-web.org:8888 (1336 words)

	Amazon.com: "Hairy Ball Theorem": Key Phrase page (Site not responding. Last check: 2007-10-12)
		hairy ball theorem If a sphere is covered with hair or fur, like a tennis ball, the hair cannot be brushed so that...
		This is sometimes called the Hairy Ball Theorem: a hairy ball cannot be combed flat.
		The theorem stating this fact is called the "Hairy Ball Theorem," the name arising because one cannot "comb" smoothly the tangent vectors on spheres of even dimension.
	www.amazon.com /phrase/Hairy-Ball-Theorem (531 words)

	hairy - The Jargon File v4.4.7
		There is a theorem in simplicial homology theory which states that any continuous tangent field on a 2-sphere is null at least in a point.
		Mathematically literate hackers tend to associate the term ‘hairy’ with the informal version of this theorem; “You can't comb a hairy ball smooth.
		In British mainstream use, “hairy” means “dangerous”, and consequently, in British programming terms, “hairy” may be used to denote complicated and/or incomprehensible code, but only if that complexity or incomprehesiveness is also considered dangerous.
	www.retrologic.com /jargon/H/hairy.html (238 words)

	hairy billiard balls and eigenvectors
		it follows from the theorem that one cannot comb the hair on a billiard ball.
		Brouwer's fixed point theorem is referred to as the "hairy ball" theorem in L.C. Thomas' Games, Theory and Applications.
		Like I saie, I'm not totally sure that Brouwer's hairy ball theorem fits in, but that's what jumped at me when I first saw your thread.
	www.physicsforums.com /showthread.php?t=66647 (538 words)

	jargon, node: hairy (Site not responding. Last check: 2007-10-12)
		A well-known result in topology called the Brouwer Fixed-Point Theorem states that any continuous transformation of a surface into itself has at least one fixed point.
		The adjective `long-haired' is well-attested to have been in slang use among scientists and engineers during the early 1950s; it was equivalent to modern `hairy' senses 1 and 2, and was very likely ancestral to the hackish use.
		Both senses probably passed out of use when long hair was adopted as a signature trait by the 1960s counterculture, leaving hackish `hairy' as a sort of stunted mutant relic.
	www.jargon.net /jargonfile/h/hairy.html (163 words)

	Good Math, Bad Math : Connected Topologies and Fixed Points
		No matter how you crumple the paper into a ball, no matter where you put it down on the uncrumpled sheet, there will be at least one point on the crumpled ball of paper which is directly above the point with the same coordinate on the flat sheet.
		This topic also gives rise to the Hairy Ball theorem, and the result that there are always two antipodal points on earth with the exact same temperature and pressure.
		The hairy ball theorem is also different from these two, and is proven using the concepts of degrees of maps on spheres, and fixed points on spheres.
	scienceblogs.com /goodmath/2006/10/connected_topologies_and_fixed.php (1679 words)

	Hairy ball - 8.4.1.1 Solution concepts (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-12)
		That's a claw and ball foot, not a hairy paw foot.
		The hairy ball theorem is usually attributed to Brouwer or Poincaré.
		For example, due to the hairy ball theorem [836], it is known that no such vector field exists for $ {\mathbb{S}}^n$ for any even integer $ n$.
	anime-drawing-guide.surferfind.com.cob-web.org:8888 /?q=anime-drawing-guide-hairy-ball (252 words)

	Facts or Did You Know?
		So they'll talk about the Ham Sandwich Theorem rather than planes in 3-dimensional space, or the Hairy ball Theorem instead of tangent vectors on spheres which is much more boring.
		Imagine that you are combing a tennis ball which is hairy rather than fluffy.
		Since the earth is a ball, and the wind at any point has a direction, as if the air were being combed over the Earth's surface, it follows that there is always a cyclone somewhere.
	dspace.dial.pipex.com /town/way/po28/maths/facts.htm (1083 words)

	Hairy ball theorem (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-12)
		Hairy ball theorem - Google News (via CobWeb/3.1 planetlab2.cs.unc.edu)
		The hairy ball theorem of algebraic topology states that, in layman's terms, "one cannot comb the hair on a ball smooth".
		In the case of the 2-torus, the Euler characteristic is 0; and it is possible to 'comb a hairy torus flat'.
	publicliterature.org.cob-web.org:8888 /en/wikipedia/h/ha/hairy_ball_theorem.html (307 words)

	Math Night Module: Topology Tricks (Site not responding. Last check: 2007-10-12)
		It says that if you have a hairy ball, you can't comb the hair on it so that it all lies smoothly.
		Use the clay to make a shape which is topologically the same as the ball or the doughnut in the pictures.
		I've nearly finished the ball and doughnut I am making--I just need to turn them right-side-out so that none of the seams show.
	orion.math.iastate.edu /mathnight/activities/modules/topology (408 words)

	[No title]
		Actually my example works fine for a C^0 example, but it's not clear in what differentiability class it lies, due to the fact I used "distance to N" as the magnitude of the field.
		Unfortunately I may have simplified and/or generalized it to a form that has the disadvantage of not being true.
		Pardon me. The correct statement of the theorem is: Theorem 8: A reduced distribution on a sphere possesses at least two radiating points.
	www.math.niu.edu /~rusin/known-math/95/hairy (694 words)

	AMERICAN MATHEMATICAL MONTHLY August-September 2004
		The “Spectral Mapping Theorem” is a simple and beautiful result from linear algebra that has many applications in algebra, geometry, and number theory.
		The spectral mapping theorem is, however, most natural and useful in the context of commutative rings with identity.
		Both proofs are based upon useful classical methods, the discriminant trick and a trick for splitting generic polynomials, that are of considerable interest in themselves.
	www.maa.org /pubs/monthly_augsept04_toc.html (439 words)

	TB Stumper Answers: 28 March 2003 - Howling Wind
		An application of the Brouwer Fixed Point Theorem known as the hairy ball theorem (really!) proves that it's not even possible to be windy everywhere!
		Since the Earth is a ball, and the wind always has a direction, like a hair, there must always be (at least) one high-pressure and another low-pressure cyclone somewhere.
		The combed hairy ball must have at least two bald spots.
	www.rain.org /~mkummel/stumpers/28mar03a.html (1120 words)

	[No title]
		hairy-ball +------------------------------------------------------------ hairy-ball A good topological theorem to mention any time is the theorem which, in essence, states that however you try to comb the hair on a hairy ball, you can never do it smoothly - the so-called 'hairy-ball' theorem.
		The mathematician had stacked the unopened cans into a surprising solution to the kissing problem; his dessicated corpse was propped calmly against a wall, and this was inscribed on the floor in blood: Theorem: If I can't open these cans, I'll die.
		Construct the middle orthogonal on AB in D and cut it with the line dividing the angle at C. Call the intersection E. Form the normal from E to AC in F and from E to BC in G. Draw the lines AE und BE.
	www.math.harvard.edu /~knill/sofia/data/citations.txt (7853 words)

	NSDL Metadata Record -- hairy ball theorem (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-12)
		The second proof gives the as a corollary of the Poincar\'e-Hopf index theorem.
		We define the index of the vector field at a zero to be the degree of that map.
		[Poincar\'e-Hopf index theorem] If X is a vector field on a compact manifold M with isolated zeroes, then...
	nsdl.org.cob-web.org:8888 /mr/1031927 (253 words)

	Course materials
		This occurred at the end of the first semester of the course (due in part to the presence of a significant population of mathematical economics minors who were unlikely to continue on to Analysis II).
		Quite a bit of adaptation was required to yield manageable questions for students with just one semester of analysis under their belts.
		Brouwer's fixed point theorem, based on the hairy ball theorem (in lecture only, based on Milnor's paper, requires stereographic projection and some other interesting but nontrivial steps)
	www.haverford.edu /math/rmanning/teaching.html (756 words)

	MAT 566 Differential Topology
		In the case of a Platonic solid this is just Euler's theorem.
		In differential topology we count the number of zeros of a smooth vector field, weighted by their indices, and once again get two.
		There can be no non-vanishing smooth vector field, a fact known as the hairy ball theorem.
	www.math.sunysb.edu /~sawon/MAT566/index.shtml (832 words)

	UW - Computer Sciences Events
		This separation between the representations of overall shape and fine detail spurs the need for good surface parameterizations - mappings to the plane that have few discontinuities, low distortion, and roughly uniform sampling.
		The fundamental challenge is that continuous, global parameterizations do not exist in general for arbitrary surfaces, not even for very simple objects: the celebrated "hairy ball theorem" states that one cannot comb a hairy ball without having a part or a cowlick.
		Fortunately, by relaxing the continuity constraint in favor of lower distortion and automatic generation, one can supplant global parameterizations with local parameterizations on a set of (possibly overlapping) patches.
	www.cs.wisc.edu /abstracts/4-1-2_0.html (258 words)

	Springer Online Reference Works
		» Encyclopaedia of Mathematics » P » Poincaré theorem
		This theorem has since been established for manifolds of all dimensions, [a1].
		of even dimension there is no continuous vector field without a zero (singularity), the Poincaré–Brouwer theorem, also called the hairy ball theorem.
	eom.springer.de /p/p073130.htm (138 words)

	Hairy Ball Theorem -- from Wolfram MathWorld (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-12)
		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
		Weisstein, Eric W. "Hairy Ball Theorem." From MathWorld--A Wolfram Web Resource.
	mathworld.wolfram.com.cob-web.org:8888 /HairyBallTheorem.html (101 words)

	loobloo » 2006 » October (Site not responding. Last check: 2007-10-12)
		I mean, they must have been intelligent to have made this into a theorem, but the idea is still pretty hilarious to me.
		I only met you this year in school because you were in two of my classes.
		I suggest that anyone that reads this that is in the Boston area go to the BAGLY Winter Ball.
	lostsockland.com /julia/blog2/?m=200610 (501 words)

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