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        riddles >> general problem-solving / chatting / whatever >> Brouwer's Amazing Fixed Point Theorem
        
(Message started by: william wu on May 11^th, 2004, 11:26pm)

Title: Brouwer's Amazing Fixed Point Theorem
Post by william wu on May 11^th, 2004, 11:26pm

A neat theorem in math is Brouwer's Fixed Point Theorem:

Let f : S [to] S be a continuous function from a non-empty, compact, convex set S [subset] [bbr]ⁿ into itself. Then there is an x* [in] S such that x* = f(x*). In other words, x* is a fixed point of the function f.

Armed with the theorem, one can make some pretty impressive statements describing some the theorem's consequences. Here are some:

Take a map of the city in which you live. Now lay the map down on the floor. There exists at least one point on the map which tells the location of the corresponding point below it on the floor.

Take two identical sheets of paper. Crumple one of them and put it atop the other. Then there exists at least one point on the top sheet that is directly above the corresponding point on the bottom sheet.

In the morning you see a circular puddle of water covered with oil. A gentle breeze blows all day, shifting the oil about the puddle's surface, but not breaking the oil film. When you return to the puddle in the evening, there exists at least one molecule of oil which lies exactly where you saw it in the morning.

Feel free to add more gems if you know some ... the more whimsical the better :)

Title: Re: Brouwer's Amazing Fixed Point Theorem
Post by towr on May 12^th, 2004, 12:46am

When you stir a (infinitely divisible) liquid, at least one 'molecule' returns to it's original place periodically.

Title: Re: Brouwer's Amazing Fixed Point Theorem
Post by Benoit_Mandelbrot on May 17^th, 2004, 9:04am

No matter where you point, your pointing line intersects at least one star.

Title: Re: Brouwer's Amazing Fixed Point Theorem
Post by towr on May 17^th, 2004, 9:09am

euhm.. I don't think that follows..
Not even if there are an infinite number of stars (that depends on the distribution)

Title: Re: Brouwer's Amazing Fixed Point Theorem
Post by Sir Col on May 17^th, 2004, 3:08pm

on 05/11/04 at 23:26:58, william wu wrote:

In the morning you see a circular puddle of water covered with oil. A gentle breeze blows all day, shifting the oil about the puddle's surface, but not breaking the oil film. When you return to the puddle in the evening, there exists at least one molecule of oil which lies exactly where you saw it in the morning.

As the number of molecules is finite and cannot represent every point on a continuous surface, won't this result fail? In fact, couldn't it be argued that no molecule can ever return to its original position?

Here's one for you...

Flying across the Atlantic, I wait for the attractive flight attendant to arrive and suggest that her inside leg represents the flight path. For the sake of demonstrating a fascinating fact I run my finger up her leg and explain that at some my finger will be at a point that coincides with our position on the ground below.

"Impressed" by this, she explains that each point on my face represents the surface of the earth and, by repeatedly punching me, at some point she will, metaphorically, obliterate the forsaken place from whence I came!

Battered and bruised I respectfully tell her that she hasn't quite grasped "Brouwer's Fixed Point Theorem". I ask if, when the plane comes into land, she would like me to demonstrate the theorem by modelling our path of decent with the aid of her breasts.

Who says that mathematics can't be fun?! ;D

Title: Re: Brouwer's Amazing Fixed Point Theorem
Post by TenaliRaman on May 17^th, 2004, 6:31pm

OMG LOL!!!!!!!!!!
:D :D