Hairy donuts and spherical dogs?

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Offline damocles

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Hairy donuts and spherical dogs?
« on: 24/07/2011 10:15:07 »
An allegedly important result in topology -- you cannot comb a (long-haired) spherical dog without producing at least two crowns or cowlicks. It is a simple matter, on the other hand, to comb a very hairy donut without producing a crown or cowlick.

Heard about this result many years ago, and intuitively it seems right, but I have not managed to find a reference on the web.
1 4 6 4 1
4 4 9 4 4     
a perfect perfect square square
6 9 6 9 6
4 4 9 4 4
1 4 6 4 1

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Offline CliffordK

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Hairy donuts and spherical dogs?
« Reply #1 on: 24/07/2011 11:02:37 »
I usually try to avoid hair on my donuts.

Does your dog have a nose and eyes?

I can imagine combing a donut shaped object, either around the circle the long direction, or combing around it and through the hole.  Either way, one ends up at the starting place at the end of the stroke, and one can comb the whole thing with an essentially parallel or spiral pattern without crossing a line, and ending up back at the starting point.  Thus the hair is always falling in a parallel path.

If you think of a sphere.  Like the globe.  Two methods for "combing" would to follow the longitude lines, and thus end up crossing and a mess at the poles.  If you comb with a half-circle stroke then pick up the comb, I think it is a cowlick and a crown.  Combing around it in a spiral-circle is good until you get back to the beginning when you have a down-stroke and an up-stroke adjoining each other.

Or follow the latitudes, and end up with a swirl at the poles.

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Offline imatfaal

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Hairy donuts and spherical dogs?
« Reply #2 on: 25/07/2011 10:52:55 »
Damocles

You need to search on Hairy Ball theorem - it was one of Poincare's originals ovre two hundred yeas ago (although he probably didnt call it that).  It is highly important when considering continuous tangential vector fields a 2-sphere always has a zero point.  odd-dimensional spheres can be combed - whereas even dimension spheres cannot!

http://en.wikipedia.org/wiki/Hairy_ball_theorem
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