## February 21, 2006

### Hey Michi

My friend Michael Johansson has a nice blog if you like math and recipes in Swedish. I think it is characteristic that he uses the Borsuk-Ulam theorem to prove that there are two antipodal points on Earth with the same temperature. It reminds me that the Brouwer fixed point theorem shows that there is one point on Earth where the air moves exactly vertically.

Posted by Anders3 at February 21, 2006 06:23 PM

Hrm. You've spent tooo much time in UK by now. I write with a 'k', not 'ch'. ;)
How do you set it up for Brouwer to give that result?

Posted by: Mikael at February 21, 2006 11:32 PM

The theorem says that any map from the ball to itself has a fixed point. So take the air velocity vector at a point and use it as a Gauss map to map that point to the surface point with the corresponding normal. The fixed point is the place where the air velocity is perfectly vertical.

Posted by: Anders at February 22, 2006 03:36 AM

Not two points?

Posted by: Anton Sherwood at February 22, 2006 04:48 AM

The result seems to follow more easily from the intermediate value theorem.

Cut out a circle and let T(a) be the difference in temperature between diametrically opposite points on an angle a measured against some fixed angle. Since T(0) = -T(2 pi) for each a, it follows thatthere is at least one angle a for which T(a) = 0.

In general, there will be a (n-1) dimensional manifold of such points on a n-dimensional sphere, if I am thinking correctly.

Posted by: Herr Vöslauer at February 23, 2006 10:31 PM

It seems to me that the mapping from ball to ball must be continuous for this theorem to hold?

Of course, because the atmosphere has thickness, it doesn't hold at all. There doesn't have to be a point where air velocity is vertical.

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