Here is one of the coolest things we know about the world we live in:
Brouwer's Fixed Point Theorem (Plane Statement)
Every continuous function f from a closed disk to itself has at least one fixed point.
Now what does all this mean?
Well the (Plane Statement) Means that this particular statement of Brouwer's Theorem only applies in the plane.
Plane, for the purposes of this discussion, means a sheet of flat paper that we can draw on.
A continuous function is basically a drawing we can make on that sheet of paper if we can draw the curve of the
function without ever lifting the pencil from the paper. The definition of a function here is a bit weak for the purposes of the theorem, but it is serviceable.
A closed disc is basically a circle in the plane; closed just means the start and end points are the same, and disc means circle.
Final point, the function goes from inside the disc, back inside itself. This basically means that if we plug in values from the first drawing, we'll get a second drawing completely contained inside the circle.
These are all the requirements of the problem.
Ready for the translation now?
Easy. Take a sheet of paper and lay it on a larger sheet of paper. Draw out the outline of the smaller sheet of paper on the larger sheet, and then crumple up the smaller sheet and put it back inside the larger sheet anywhere you like as long as its within the outline you drew earlier. Somewhere in that crumpled up ball of paper there is at least one point that is exactly above the point on the larger sheet where it started. That is, somewhere in that ball is a point that didn't move anywhere at all relative to the its horizontal position with the larger sheet of paper.
Some of you may protest that using rectangular sheets of paper or square sheets of paper won't result in the same thing. I ask you, if you give me any size sheet of any shape of paper you want, can't I draw an even bigger circle around that sheet of paper? Well, yes I can. So any 2 dimensional shape will do, since they all can be contained within an infinitely large closed disc in the plane.
Wicked no?
6/27/2010, 01:56