Fun Tasks
# Universal Math Law

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Today, we will discuss one theorem that can explain a variety of phenomena in the world around us, namely the formation of galaxies, the formation of countries, and even blood clotting! We will discuss a logically necessary process, a kind of mathematical entropy.

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Recall that we use the phrase "entropy increases" when discussing irreversible changes in physical systems (an example of an irreversible process is mixing coffee and milk).

Imagine that in a certain group of people, each has no more than four friends. In addition, let each member of our group wear a colored T-shirt. An example of such a group is shown below:

This figure is called a graph. In our graph, lines connect friends; for example, Eva is friends with John and Linda. As often happens in society, members of our company try to comply with fashion, namely, at each subsequent moment in time they put on a T-shirt of the color most often found among the personâs friends (including him or herself). So, among Lindaâs friends, two people have red T-shirts. So if Linda wants to change the color of her T-shirt, she will change it to red:

Let's see what happens to our company if we repeatedly select a random person in it, asking him or her to change the color of the T-shirt to the one that is the most popular in his or her neighborhood. Using a computer, let us implement 10, 100, and 1000 changes:

What do we see? The first thing you can pay attention to is that the coloring of our system stops changing over time. Secondly, in the final state, two large groups of familiar people with T-shirts of the same color (purple and red) were formed in our system.

You will find a rigorous formulation and proof of these results in a math article written by me and my student. The names of the corresponding branches of mathematics that cover these topics are dynamical systems and cellular automata. You can learn these and many other theories by signing up for a trial lesson with tutors who teach Graph and Set Theory.

Now the question arises: could it be that the formation of large groups of people with the same interests (musical, political, etc.) is due only to the fact that a person tends to adopt the features of his environment? What if the formation of galaxies or, conversely, coagulation that occurs in a test tube are not caused by the forces of physical attraction but logically follow from the exchange processes occurring in these systems? This is what happens to the system above; local changes cause the formation of big groups of people with the same color as their T-shirts. The reader is invited to think about the questions above!