Sierpiński, Wacław

Wacław Sierpiński, (born March 14, 1882, Warsaw, Russian Empire [now in Poland]—died October 21, 1969, Warsaw), leading figure in point-set topology and one of the founding fathers of the Polish school of mathematics, which flourished between World Wars I and II.

Sierpiński graduated from Warsaw University in 1904, and in 1908 he became the first person anywhere to lecture on set theory. During World War I it became clear that an independent Polish state might emerge, and Sierpiński, with Zygmunt Janiszewski and Stefan Mazurkiewicz, planned the future shape of the Polish mathematical community: it would be centred in Warsaw and Lvov, and, because resources for books and journals would be scarce, research would be concentrated in set theory, point-set topology, the theory of real functions, and logic. Janiszewski died in 1920, but Sierpiński and Mazurkiewicz successfully saw the plan through. At the time it seemed a narrow and even risky choice of topics, but it proved highly fruitful, and a stream of fundamental work in these areas came out of Poland until the intellectual life of the country was destroyed by the Nazis and the invading Soviet forces.

Sierpiński’s own work in set theory and topology was extensive, amounting to over 600 research papers, and toward the end of his life he added a further 100 papers on number theory. He expended much effort on giving a topological characterization of the continuum (the set of real numbers) and in this way discovered many examples of topological spaces with unexpected properties, of which the Sierpiński gasket is the most famous. The Sierpiński gasket is defined as follows: Take a solid equilateral triangle, divide it into four congruent equilateral triangles, and remove the middle triangle; then do the same with each of the three remaining triangles; and so on (see figure). The resulting fractal is self-similar (small parts of it are scale copies of the whole thing); also, it has an area of zero, a fractional dimension (between a one-dimensional line and a two-dimensional plane figure), and a boundary of infinite length. A similar construction starting with a square produces the Sierpiński carpet, which is also self-similar. Good approximations of these and other fractals have been used to produce compact multiband radio antennas.