From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: A combed hairy ball has AT LEAST TWO bald spots!!
Date: 3 Mar 1995 21:47:37 GMT
In article <3j7ts4$tsj@bubba.ucc.okstate.edu>,
John Chandler wrote:
>There have been several posts in sci.math recently
>saying, in effect,
> A combed hairy ball has at least one bald spot.
>A much more difficult and beautiful result was proved by Brouwer:
> A combed hairy ball has at least two bald spots!!
Well, since I gave an exercise to show this is NOT true I guess it's up
to me to make the correction. This is true _if_ the "bald spots" are
counted according to multiplicity, but not otherwise. Here's a solution
with one bald spot:
Since the sphere with one point removed is homeomorphic to the plane,
use the fact that a constant (and thus non-zero) vector field on the plane
exists, so one gets a vector field on the once-punctured sphere.
In pictures, do this. Have the vector field point along a great
circle from the North pole, through Greenwich, the Southpole, the dateline,
and back. The rest of the vector field also points through circles beginning
at the north pole. Each circle departs a little more from the first one, with
the worst deviation being at its southernmost point. Eventually the circles
look like small circle centered at points a little away from the north pole
lying on the 90-degree and 270-degree lines of longitude.
You can recognize the bald spot at the north pole as a double point
by thinking of the vector field I just described as the limit of the
following ones: Begin with a vector field which points along circles which
are centered on the sphere's, uh, east-west axis. Then, gradually drag both
the bald spots (at the East and West poles, I guess) until they both
reach the north pole; at each stage, the vector field should consist of a
bunch of parallel lines lined up between the bald spots, then swung around
the earth (spreading out a little as they head south). These lines point in
the direction of the magnetic field caused by a current running in a wire
passing through the bald points and the earth's center (I think).
Well, I guess this proves one thing: a picture's worth at least
300 words. :-)
dave
==============================================================================
Date: Fri, 3 Mar 1995 16:42:26 -0800
From: [Permission pending]
To: rusin@washington.math.niu.edu
Subject: Re: A combed hairy ball has AT LEAST TWO bald spots!!
In article <3j82pp$f69@watson.math.niu.edu> you write:
>In article <3j7ts4$tsj@bubba.ucc.okstate.edu>,
>John Chandler wrote:
>>There have been several posts in sci.math recently
>>saying, in effect,
>> A combed hairy ball has at least one bald spot.
>>A much more difficult and beautiful result was proved by Brouwer:
>> A combed hairy ball has at least two bald spots!!
>
>Well, since I gave an exercise to show this is NOT true I guess it's up
>to me to make the correction. This is true _if_ the "bald spots" are
>counted according to multiplicity, but not otherwise.
I would have omitted this last sentence, lest it confuse the bewildered.
Here's a solution
>
>dave
-------------------------------------------------------------
I would have just said: Consider a line L tangent to the unit sphere S in R^3
at the point N.
For every plane P contining L, consider the intersection of P with S. These
create a family of circles C on S all disjoint from each other, except that
they all contain the point N.
Now define a vector field on S by at each point x, use the tangent direction of
the unique circle C through x, with magnitude equal to the distance from the
point N. Except if x = N, in which case let it be the zero vector.
[Permission pending]
==============================================================================
Date: Mon, 6 Mar 1995 09:00:45 -0800
From: [Permission pending]
To: rusin@math.niu.edu (Dave Rusin)
Subject: Re: A combed hairy ball has AT LEAST TWO bald spots!!
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. Probably
(distance to N)^2 would lead to a C^oo field, or maybe even C^w.
[Permission pending]
==============================================================================
From: jpc@a.cs.okstate.edu (John Chandler)
Newsgroups: sci.math
Subject: > A combed hairy ball has AT LEAST TWO bald spots!!
Date: 6 Mar 1995 20:05:42 GMT
Last week I paraphrased from memory L. E. J. Brouwer's Hairy Ball Theorem.
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.
-- L. E. J. Brouwer, II KNAW Proc. 12 (1910) 716-734
This can also be found on page 301 of
L. E. J. Brouwer -- Collected Works
Volume 2: Geometry, Analysis, Topology, and Mechanics
Edited by Hans Freudenthal
North-Holland Publishing Company, 1976
--
John Chandler
jpc@a.cs.okstate.edu