wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> hard >> Temperature Antipodes
        
(Message started by: william wu on Nov 13^th, 2003, 6:18pm)

Title: Temperature Antipodes
Post by william wu on Nov 13^th, 2003, 6:18pm

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

Title: Re: Temperature Antipodes
Post by Icarus on Nov 13^th, 2003, 6:41pm

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

Hint: [hide]Brouwer might have something to say on the subject.[/hide]

Title: Re: Temperature Antipodes
Post by Eigenray on Nov 13^th, 2003, 7:58pm

You don't need anything more than [hide]the intermediate value theorem[/hide] for the problem as stated. If you want two points with the same temperature and pressure, then you can try asking [hide]Borsuk[/hide] or [hide]Ulam[/hide], but I'm not sure how [hide]Brouwer[/hide] can help you out directly.

Title: Re: Temperature Antipodes
Post by Icarus on Nov 13^th, 2003, 8:29pm

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 [hide]intermediate value theorem[/hide] 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.