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william wu

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 Temperature Antipodes   « on: Nov 13th, 2003, 6:18pm » Quote Modify

Consider the earth to be a perfect sphere. The temperature at any location on the earth is given by some continuous temperature distribution. Prove that you can always find a pair of antipodal points with the same temperature!

Source: Paul Jung
 « Last Edit: Nov 13th, 2003, 6:18pm by william wu » IP Logged

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Icarus
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 Re: Temperature Antipodes   « Reply #1 on: Nov 13th, 2003, 6:41pm » Quote Modify

Seen it; did it; bought the T-shirt. Paul has been preceded here - this is a classic topology problem.

Hint: Brouwer might have something to say on the subject.
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Eigenray
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 Re: Temperature Antipodes   « Reply #2 on: Nov 13th, 2003, 7:58pm » Quote Modify

You don't need anything more than the intermediate value theorem for the problem as stated.  If you want two points with the same temperature and pressure, then you can try asking Borsuk or Ulam, but I'm not sure how Brouwer can help you out directly.
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Icarus
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 Re: Temperature Antipodes   « Reply #3 on: Nov 13th, 2003, 8:29pm » Quote Modify

Not directly, but this can be approached by the same sort of procedure as used in the fixed point theorem.

However, you are right that the intermediate value theorem provides a very nice and quick proof. In fact, with a little more effort you can prove a stronger result:

There is a closed curve incircling the earth such that for every point not on the curve, it and its antipode are on opposite sides of the curve (I'm sure there is a name for such curves, but I'm not sure what it is), and such that the temperature of every point on the curve is equal to the temperature at the antipode.
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"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
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