## The Fixed Point Theorem

© Copyright 2003, Jim Loy

Question #1: I climbed a mountain, following a trail, in six hours (noon to 6 PM). I camped on top overnight. Then at noon the next day, I started descending. The descent was easier, and I made much better time. After an hour, I noticed that my compass was missing, and I turned around and ascended a short distance, where I found my compass. I sat on a rock to admire the view. Then I descended the rest of the way. The entire descent took four hours (noon to 4 PM). I thought that I remembered that there was a place on the trail where I was at the same place at the same time on both days. Can you tell if I was right?

Answer #1: Yes, there must be one "fixed point" on my trip. Here is a graph showing my height on the mountain versus time. Regardless of the changes in rate of travel, it should be obvious that the two paths must cross at one point, at least.

The problem was stated so that one path could not go around the end of the other path. This situation is ignored by Euclid, but modern geometry text books give this postulate, or something like it: If two points are on either side of another line, then a line drawn through the two points intersects the other line. This is true of curved line segments as well as infinite straight lines, as long as we are careful to prevent a line from going around the end of the other line.

The above situation is an example of the fixed point theorem, which states (according to MathWorld): "If g is a continuous function g(x) is an element of [a,b] for all x elements of [a,b], then g has a fixed point in [a,b]." Further, a fixed point is "a point which does not change upon application of a map, system of differential equations, etc." Above we are actually applying a map. The fixed point theorem has applications in almost all branches of mathematics.

Question #2: I have two maps of the USA (not the same scale). I crumple one of them up into a loose ball and place it on top of the other map entirely within the borders of the USA on the flat map. I wonder if there is any point on the crumpled map (that represents the same place in the USA on both maps) that is directly over it's twin on the flat map. What do you think?

Answer #2: Of course the point we are wondering about is a fixed point. Yes there must be at least one point on both maps that remains fixed (directly above or under its twin on the other map).