Last weekend, we went up to Baguio, the summer capital of the Philippines, to escape the summer heat. As we were going up the long and winding road to Baguio, I find myself frequently looking at my watch, impatient to get over the long trip. As I looked at the time, I also took note of the place where we were during that time. Suddenly, a question popped in my head: “Will it be possible that we’d be in this same place at this time when we return to Manila?”. We can visualize this problem using the diagram below:
In the figure above, the blue dot is the location along the way from Manila to Baguio which we occupied where the clock time was the same when going up as when going down. It turns out that such a place exists. However, where it is located is arbitrary. To see this, suppose that we leave Manila at 6 am and arrive in Baguio at 6 pm. (On the average, it takes about 5-6 hours to reach Baguio from Manila. In this case, we will be taking our time enjoying the scenery). On the day that we go back, we will leave Baguio at 6 am and arrive in Manila at 6 pm. To find the place and time in question, use 2 cars, one starting in Manila going to Baguio and the other car starting from Baguio and going to Manila. When they meet each other along the way, that’s precisely the place and time we are looking!.
The problem above is an illustration of the power of Topology, a branch of mathematics that studies the properties of mathematical objects that are invariant under stretching without tearing or gluing. For a topologist, there is no difference between a doughnut and a cup because we can always deform the doughnut to look like a cup.
The most obvious topological property of the doughnut is the single hole it has. Any transformation that stretches or deforms it will always retain that hole. Which means it is always possible to find a transformation that will transform your doughnut to a cup. Of course, don’t eat the cup!
Topology is sometimes called “Rubber Sheet Geometry” because it studies objects and its properties that are invariant under stretching. In the diagram below, we start with a flat sheet of rubber with a straight line in it. Stretch this rubber in any which way. You will notice that the line, even though it will not look straight anymore, will still be one piece of line and not two. In other words, the line retains its property of connectedness. It will still be in one piece and not suddenly become two pieces.
Brouwer’s Fixed Point Theorem
The Baguio trip example I gave is an instance of what is known as the Brouwer’s Fixed Point Theorem. In its simplest form, it states that any continuous function from a closed interval to itself has a fixed point. Let be the closed interval in 1 dimension. This means that and lie also in . Furthermore, whatever number is mapped to, it is always greater than or equal to . Also, whatever number is mapped to, it is always less than or equal to .
Define the function . We can easily see that and . Since is a continuous function, then there is a point such that (by the Intermediate Value Theorem), that is, . The number is called a fixed point of .
For example, consider the cosine function defined in ,
then by Brouwer’s Fixed Point Theorem, there is a point such that
As can be seen from the diagram above, this point is the intersection of the graph of the cosine function and the line and is equal to 0.73909.
The Baguio trip has filled me with fond memories of my university days especially on Topology. I’ll be sharing them in my subsequent posts.