# The Mathematical Monthly Calendar: Wacław Sierpiński (1882–1969): Master of fractals

## Wacław Sierpiński (1882–1969): master of fractals

In 1912, his investigations into a sequence of curves, which are named Sierpinski curves in his honor, attracted particular attention: a closed line is drawn according to a recursively defined rule, which becomes more and more refined step by step and apparently always more fills the surrounding square.

Wacław Sierpiński grew up as the son of a doctor in Warsaw – the capital of the Kingdom of Poland since 1815 (Congress of Vienna). However, this was not independent, but was ruled in personal union by the Russian Tsar - more precisely, suppressed: The aim was the Russification of the country; while the education of the Polish population should remain as low as possible. At the "University of the Tsar," as the University of Warsaw had been called since 1869, teaching was only in Russian; all professors were exclusively Russian.

Despite these difficulties, the gifted Wacław Sierpiński managed to enroll at the University of Warsaw at the age of 18 in mathematics and physics. In 1903 his contribution to number theory was proposed for an award and was to appear in the journal "Isvestia"; this involves estimating the number n of points with integer coordinates in a circle with radius \(r) around the origin. In 1837, Carl Friedrich Gauss proved that constants \(C) and \(k) exist such that \(| n – r^2| < Cr^k), where \(k\leq 1). Sierpiński can improve this estimate to \(k \leq \frac{2}{3}) (as of today: \(k \leq \frac{7}{11})).

But then he withdraws his approval for the publication because he does not want his first scientific work to be published in Russian. Despite refusing a compulsory Russian language test, he gets his university degree and starts working as a teacher of mathematics and physics at a girls' school in Warsaw. After this school is closed due to a strike, he moves to Kraków (then: Austria-Hungary) to do his doctorate there under Stanisław Zaremba (1863–1942).

1908 he was appointed professor in Lwów (Lemberg) in Galicia (today: Ukraine), which at that time also belonged to Austria-Hungary. The information from Thaddeus Banachiewicz (1882-1954) from Göttingen that Georg Cantor (1845-1918) had succeeded in describing points in the plane with a single coordinate (Cantor's diagonal method) prompted him to take a closer look at set theory, and he composes numerous writings.

When World War broke out, he was on Russian territory and he was being interned. The Russian mathematician Nikolai Luzin (1883-1950) manages to free him from internment and gives him the opportunity to work at the University of Moscow. This began a fruitful collaboration between the two mathematicians, which continued when Sierpiński took on a professorship at the University of Warsaw in 1919. There he founded the journal "Fundamenta Mathematicae" together with Zygmunt Janiszewski (1888–1920).

Both university teachers are committed to a "Polish School of Mathematics", a system of cooperation among Polish mathematicians, which also includes the further training of mathematics teachers. In addition to Warsaw as the national center of mathematics, another one was established in Lwów in 1929 under Stefan Banach (1892–1945, famous for "Banach's Fixed Point Theorem").

Sierpiński enjoys a worldwide reputation as a mathematician and receives numerous honors from universities in Europe and overseas. After the German occupation of Poland in World War II, he disguised himself as a simple employee of the Warsaw city administration, but secretly teaches at the "Underground University". During the Warsaw Uprising in 1944, the German occupying forces destroyed all of the university's library holdings; his house and all personal records are also burned. More than half of the academic staff of the mathematical faculties in Poland perish, so the reconstruction of Polish universities after World War II becomes very difficult. By the time he died, Sierpiński had written an incredible 724 scientific articles and 50 books, including many school books.

In number theory, Sierpiński deals, among other things, with the distribution of the digits of irrational numbers, a problem that has not yet been solved. Note: It is not possible to prove that the digits 0, 1, …, 9 occur with equal frequency in a decimal number by examining the distribution of the first 100 million digits, for example. Conversely, however, it is easier to construct a number that has an equal distribution of digits in a number system, for example 0, (1)(10)(11)(100)(101)(110)(111)… 2 or 0, (1)(2)(10)(11)(12)(20)(21)(22)… 3 and so on.

The Sierpinski conjecture is also unsolved: Are there infinitely many odd natural numbers \(k) such that the sequence \(k \cdot 2^n + 1 ) contains nothing but composite terms (i.e. no prime numbers)? It is assumed that 78557 is the smallest Sierpinski number (that is, for all smaller odd factors \(k) at some point there will be members of the sequence that are prime numbers).

In 1912, his investigations into a sequence of curves, which are named Sierpinski curves in his honour, attracted particular attention: According to a recursively defined rule, a closed line is drawn that extends more and more refined from step to step and seems to fill out the surrounding square more and more. The line becomes infinitely long, but the enclosed area (colored gray here) is only half the size of the area of the frame square!

In the sequence of Sierpinski triangles you start with a triangle with area \(A) and perimeter \(u), in which you enter the central triangle; the middle of the resulting four congruent triangles is removed (colored gray), and a middle triangle is added to the remaining ones, and so on. The resulting triangles are self-similar, that is, each triangular subfigure occurs itself in the sequence of triangles. The associated sequence of the areas \(A_n) of the removed surface sections is calculated using the formula \(A_n=\frac{A}{3}\cdot \sum\limits_{k=1} ^{n}) (left(frac{3}{4} right)^k) converges to A.

The sequence of the perimeters \(u_n), on the other hand, grows beyond all limits:

\(u_n=\frac{u}{3}\cdot \sum\limits_{k=1} ^{n}) (left(frac{3}{2} right) ^k)

The Sierpinski carpet is constructed analogously: A square is divided into nine squares of equal size and the middle sub-square is removed. In the sequence of the three-dimensional Sierpinski tetrahedron an octahedron is cut out of each of the tetrahedrons; the volume of the resulting body is halved from step to step, the volume sequence therefore converges to zero; on the other hand, the interface remains the same from step to step!