## 1.1 Wacław Sierpinski

The period between the two world wars was not only the rebirth of Poland, it was also a time of very intensive development of the Polish school of mathematics. The main centers of development of this school in the interwar period were Warsaw, Lwów and Cracow. The leading mathematicians of that time often traveled between these cities, inspiring each other, exchanging knowledge and experience. And when we talk about leading mathematicians, it is impossible not to mention Wacław Sierpiński, who received his master’s degree in Warsaw, his doctorate in Cracow, and his habilitation in Lwów.

Could there be a better date of birth for a mathematician than March, 14th? On this day in 1882 in Warsaw, our hero was born. He went to the Imperial University of Warsaw for his studies, where, under the supervision of Georgy Voronoy (yes, it’s that Voronoy from Voronoy diagram – a partition of a plane into \(n\) cells, defined by \(n\) seeds. Each point of the plane is assigned to the cells whose seed is closest to it), he became interested in number theory, defended his degree as a candidate of sciences (the equivalent of today’s master’s degree) and began teaching mathematics at a junior high school.

He did not work there for long. As a very active person, he participated in school strikes, which caused him to lose his job. So he moved to Cracow, and at the Jagiellonian University, he quickly earned his doctorate, studying the sums of series \(\sum_{m^2+n^2 \leq x}f(m^2+n^2)\). Three years later, he had received habilitation at Lwów University. During that period and for the rest of his life, he traveled extensively to excellent mathematical centers around the world, including studying in Göttingen, where he met the mathematician Constantin Carathéodory (who worked for a time in Breslau, today’s Wrocław) and Hugo Steinhaus (about whom we write more in Chapter Three).

He was a great organizer, taking part in many initiatives. For example, in 1920, he together with Zygmunt Janiszewski (a mathematician, author of, among other, *Guide for the self-study*, Polish: *Poradnik dla samouków*, introducing mathematics at the university level) and Stefan Mazurkiewicz founded Fundamenta Mathematicae – the first journal in the world devoted to mathematical logic, set theory, and their applications. The journal continues to this day and is now published by the Institute of Mathematics of the Polish Academy of Sciences.

Sierpiński quickly became known as a great and very versatile mathematician. When in 1920, Jan Kowalewski (a mathematician and cryptologist, Lieutenant Colonel of the Polish Army) was forming a unit to break Soviet ciphers, he hired Wacław Sierpiński, Stefan Mazurkiewicz, and Stanisław Leśniewski (a philosopher, logician, and mathematician) to decrypt Soviet dispatches more efficiently.
The team was so effective that it was later credited with a significant contribution to the success of the Polish Army during the Polish-Soviet War, including the memorable Battle of Warsaw (a decisive battle of the Polish-Soviet War). The Cipher Bureau at that time was a very innovative venture; it is enough to say that it was established 20 years before the famous Bletchley Park (*Bletchley Park, also known as Station X is an estate in England located about 80 km from London. During World War II, it was home to a team of British cryptologists from the Government Code and Cypher School. They oversaw reading ciphertexts created on German Enigma machines, Lorenz machines and others*). Efficient radio listening, made possible by electronics engineers from the polytechnics, combined with effective cryptographic analysis, made possible by mathematicians from the Universities of Warsaw and Lwów, gave military commanders the necessary data on the enemy troops’ plans.

There were even anecdotes that the Cipher Bureau could decrypt the dispatches faster than the intended recipients. The successes of this team laid the foundations for the Cipher Bureau in Poznan, which was made famous a decade later by the Enigma breakers Marian Rejewski, Jerzy Różycki, and Henryk Zygalski – the three mathematicians and cryptologists who broke the Enigma cipher in 1932 significantly contributed to the Allies’ victory in World War II.

Over the years, Wacław Sierpiński taught students in junior high schools, lectured at universities, and wrote textbooks. There were many editions of the well-known textbook on arithmetic and geometry, which he wrote together with Stefan Banach and Włodzimierz Stożek. Because of his teaching activities, he became president of the Society of High School and University Teachers (Polish: Towarzystwo Nauczycieli Szkół Średnich i Wyższych). He was also the doctoral advisor for many prominent mathematicians, like Jerzy Spława-Neyman (a mathematician and statistician, creator of the concept of a confidence interval, author of the Neyman-Pearson lemma, fundamental to the construction of statistical tests), Otto Nikodym, Kazimierz Kuratowski (a mathematician, author of many interesting results in set theory and measure theory but also author of the theorem concerning the characterization of planar graphs), or Alfred Tarski – an outstanding logician, working, for example, on formalizing the concept of truth.

Sierpiński had numerous talents, but his interests were mainly related to conducting scientific research. He wrote a total of 113 papers, so it is impossible to summarize his many results briefly. His passion was studying the concept of infinity, whether in number theory, mathematical analysis, set theory, or topology. In 1915, at the age of 33, he proposed a method for constructing an interesting figure, later called the Sierpiński triangle. It is created as the result of an infinite sequence of specific operations. It is one of the most famous fractals today, although Benoit Mandelbrot (a Warsaw-born mathematician, recognized as a father of fractal geometry) introduced the concept of fractal only 60 years later, in 1975.

His tombstone can be found at Powązki Cemetery in Warsaw, and it bears the inscription *Investigator of infinity*.