Quasiperiodicity

Tompaidis studies the symplectic map. This map codifies dynamics as we understand it now. He studies quasiperiodicity, which was initially discovered in number theory. When Prof. Schweiger presented Jacobi´s periodicity discovery in the nineteenth century, my internal voices were telling me that this was important. All I could gather immediately in my mind was to ask if Prof. Schweiger knew about cellular automata, as presented in Wolfram´s book, "A New Kind of Science". He did not. At the end of the lecture Prof. Glen Van Brummelen from Quest University in Canada, told me that he had just heard Wolfram give a talk the previous week, on cellular automata. The word is slowly getting around that these mathematical objects are important. I did not pursue the issue with either scientist; only now it is downing on me that there is something deep here.

I am reading papers now on these topics and my ideas are coming together. I write here where I am right now.

Number Theory may likely be the oldest branch of mathematics where periods were found. Astronomy also gave our ancestors the sense of repetition. Every day the Sun is up there, this led us to think of day, month and year; when the other astronomical objects showed us their periods.

Then the crisis in Greek Mathematics with Pythagoras Academy presented to our minds the possibility of something else. Non-periodicity.

Jacobi pursued more periodicity in mathematics, and did seem to get confused when he did not find it. Now though, he was closer to the concerns of modern dynamicists. That was the bell that rang in my head very lightly, in Mexico City last week.

I will say now that Jacobi suffered a mild case of Aristotelean shock. Instead of pursuing the study of quasiperiodicity scientifically, as Tompaidis is doing. He just could not finish his thoughts.

Even last century Leon Bernstein at the Illinois Institute of Technology, kept on trying to get more and more periodicity, and did not focus instead on quasiperiodicity.

Penrose was one of the first mathematicians to recognize the deep importance of this mathematical phenomenon, when he invented his period five Tiling.

Now there are technical applications of this new branch of mathematics; the study of quasiperiodicity.