I am infinitely fascinated by so called fixed points. My fascination does not stem from any particular mathematical result or fact, but rather from the beauty of the concept itself. Even though it is an extremely useful concept both in mathematics and computer science (it is for example used in the analysis of programming languages), it has an intrinsic value and character that I am deeply infatuated with.
It is a good example of a mathematical concept from "higher" or more "advanced" mathematics that is abstract and powerful, yet not particularly hard to understand. In the video above, I try to explain in simple terms what a fixed point is. Have a look!
Imagine that you have a hammer, a nail and a wall. We can view the act of hammering the nail into the wall as the application of a mathematical function. Eventually, the nail is all the way in, and then no further hammering has any effect. Or you can hush students in a classroom; eventually everybody (hopefully) will be quiet, and the hushing has no effect anymore. Then, you have reached the so-called fixed point.
Or imagine that you have two identical sheets of graph paper. Take one of the sheets and crumble it up into a ball. If you now place this ball on top of the other sheet, there must be a point that is placed directly above itself. This means that there is a point on the crumbled up sheet that has a corresponding point on the sheet below. This point is also called a fixed point. No matter how you crumble up the paper, there will always exist at least one fixed point like this.
Do fixed points like this always exist? In the case of the nail and the graph papers, the answer is yes. This is also the case in many other similar situations.
In mathematics and computer science, there are many exciting results about fixed points, and these are usually called fixed point theorems. In general, a theorem is a true mathematical assertion for which one has found a mathematical proof. Fixed point theorems usually say that a particular mathematical function has a fixed point under certain conditions.
One of the most famous fixed point theorems is Brouwer's fixed-point theorem, named after the Dutch mathematician and philosopher L.E.J. Brouwer (1881–1966). In simplified terms, this theorem states that if you stir around in a cup of coffee, there will always be at least one point that has not moved relative to its starting point.
Let us do an experiment with a rectangle like this.
We now make a copy of the rectangle that is slightly smaller. Below, we have made a rectangle that is exactly 61.80339% (because why not?) as large as the original, and we have placed this on top of the original.
We now have a situation where there has to be a fixed point! This is the point that is "on top of itself" and that has not moved relative to the starting point. If we imagine that the original rectangle consists of points or coordinates, and that all of these are also in the smaller rectangle, then at least one of these points must be exactly above its corresponding point on the rectangle below.
We can give a visual proof of the existence of the fixed point by continuing this process of taking smaller and smaller copies in the same manner. The fixed point is located exactly at the place where all the pictures disappear inwards. Can you see where the fixed point is located?
We can also do the same with a screenshot from the video. First, we define a transformation of the image in the same way as above:
By repeating this transformation many times, we get the following picture!
Mathematics is fundamentally about ideas, patterns and concepts – not only formulas, computations, and rules, like so many mistakenly believe. One such concept is the fixed point of a mathematical function.
A related concept is that of infinity. You may have noticed that in order to actually reach the fixed point in the transformations above, we would have to repeat the process infinitely many times? This is a very common phenomenon in mathematics, and it has lead to notions such like convergence, limit and many others.
It is worthwhile to ponder concepts like these, and the concept fixed point is one of the most beautiful in all of mathematics. What is your favorite fixed point?