Polya

George Pólya (1887-1985)

George Pólya was born in Budapest on 13 December 1887. His father Jakab (who died in 1897) had been born Jakab Pollák, of Jewish parents, and with a surname which suggested Polish origin. It is likely that ancestors had emigrated from Poland to Hungary, where a lesser degree of anti-Semitism existed. However Jakab converted to Catholicism believing that this would help him advancing in a career and changed his name to the more Hungarian Pólya. George’s mother had also been of Jewish background with similar history. Her paternal grandfather, Max Deutsch, had in fact converted to Presbyterianism and worshipped with Greek Orthodox Romanians.

George’s father Jakab had been a solicitor with a great mind, but one who was prepared to pursue a case in which he believed with no fees. He was not financially successful despite the time he lived in being considered a golden age for Hungary.

As a student George attended a state run high school with a good academic reputation. He was physically strong and participated in various sports. His school had a strong emphasis on learning from memory, a technique which he found tedious at the time but later found useful. He was not particularly interested in mathematics in the younger years. Whereas he knew about the Eötvös Competition and apparently wrote it he also apparently failed to hand in his paper.

He graduated from Marko Street Gymnasium in 1905, ranking among the top four students and earning a scholarship to the University of Budapest, which he entered in 1905. He commenced studying law, emulating his father, but found this study boring and changed to language and literature. He had become particularly interested in Latin and Hungarian, where he had had good teachers. He also began studying physics, mathematics and philosophy. His development was greatly influenced by the legendary mathematician Lipót Fejér, a man also of wit and humour, who also taught Riesz, Szegö and Erdös. Fejér had discovered his theorem on the arithmetic mean of Fourier Series at the age of 20.

Pólya soon concentrated his studies on mathematics and in 1910 finished his doctorate studies, except for his dissertation. He took a year in Vienna and returned to Budapest in 1911-12 to give his doctoral dissertation and met Gábor Szegö, seven years younger, who was to become one of his major collaborators.

In the fall of 1912 he went to Göttingen for postdoctoral study and met David Hilbert, Richard Courant, Felix Klein and Hermann Weyl. In 1913 he was offered a position in Frankfurt, but was discouraged from staying in Germany and turned the job down after being told he was a “bloody Jew” by a ruffian on a train and went on to the University of Paris for further postdoctoral work.

In 1914 he took up a position at the Eidgenössische Technische Hochschule (ETH) in Zurich, an institution which boasted the names of the physicists Röentgen and Einstein (1900) among its graduates. This position was arranged by the mathematician Adolf Hurwitz (1859 to 1919), who had studied at various times under Kronecker, Klein and Weierstrass and was the other great influence on Pólya. The ETH was next door, and closely associated with, the University of Zurich and Polya had joint teaching rights with the University.

In 1914 Pólya was called up by Hungary to fight in the war, but by this time he had adopted Russell’s pacifism and refused to go. The fear that he might be arrested for being unpatriotic meant that he did not return to his native country until after World War II. In Zurich he met his future wife, Stella Weber. They married in 1918 and were still together 67 years later when Pólya died. They had no children.

Inspired by walks in the woods near Zurich, Pólya in 1912 published one of his major results, the solution of the random walk problem. In this problem one walks in an infinite rectangular grid system, at each node having an equal probability of walking to each of the adjoining nodes on his next leg. Pólya was able to show that in the two dimensional case it was almost certain (but with probability 1) that one would eventually return to the original position, but one would almost never (with probabilty 0) return to the origin in the case of three or more dimensions.

Pólya was interested in chemical structure, which led him in 1924 to publishing the classification of seventeen plane-symmetry groups, a result which was later to inspire the Dutch artist M.C. Escher.

In 1924 he spent a year in England, working with G.H. Hardy and J.E. Littlewood at Oxford and Cambridge. This collaboration led to publication in 1934 of the book Inequalities, which included a new proof by Pólya of the AM-GM inequality based on the Maclaurin expansion of the exponential function.

In 1925 Pólya, with Szegö, published arguably one of his most influential books, Aufgaben und Lehrsätze aus der Analysis, volumes 19 and 20 of the series Die Grundlehren der Mathematischen Wissenschaften published by J. Springer, Berlin. A whole generation - the generation of Erdös, Szekeres and their circle, and later, learned their mathematics not so much from the lectures they attended but by trying to solve the problems of this book one after another and debating their solutions with each other. Problem solving as a method of teaching and learning may never have been practiced on such a scale, and with such success, before (or since).

One of Pólya’s most famous results, the Pólya Enumeration Theorem, was published in 1937. This also arose from his interest in chemical structure and looking at possible configurations of the benzene ring and other figures with 6 vertices. Generalising a theorem by Burnside in Group Theory, Pólya showed how one can determine the number of different assignments of atoms, or colours, to vertices as sides of geometrical figures.

A special case of this is the Necklace Theorem, which shows how many necklaces of n beads can be constructed with k colours available, assuming there is an infinite supply of each colour.

In the case where n is prime and necklaces are regarded as unchanged by rotation, the number of configurations is k+(kⁿ-k)/n. Thus if there are 5 beads and 3 colours the number of necklaces is 3+(3⁵-3)/5=3+240/5=3+48=51.

In the case where n is composite there is also a formula but it is a little more complicated. However more details of this and another version of the theorem can be found here.

In 1940 the Pólyas became increasingly concerned, with George’s Jewish background, of the possibility of a German invasion of Switzerland, and decided to leave for the United States.

He was offered a research position by his old collaborator, Gábor Szegö, now at Stanford, but he did not initially accept it, going instead to Brown University. In 1942 he did move to Stanford, however, where he stayed until his retirement from teaching in 1953. After 1953 he stayed at Stanford, living at Palo Alto until his death, as a Professor Emeritus.

In 1945, Pólya published one of his most famous books, How to Solve It. Then in 1951 he published, with Gábor Szegö, Isoperimetric Inequalities in Mathematical Physics.

Gábor Szegö had been a winner of the Eötvös Competition in 1912 and Pólya saw the value in competitions. In 1946 Pólya and Szegö founded the Stanford University Competitive Examination in Mathematics. In its first year 322 students from 60 schools in California entered. The competition grew to having typically 1200 students from 150 schools in 3 western states. However the competition was terminated in 1965 when Stanford shifted its emphasis to postgraduate study. Pólya however continued his activity in this area by publishing problem material in books and journals.

Pólya was particularly interested in the high school curriculum and was concerned about the new maths curriculum. He eventually saw the curriculum change back to basics and was not happy with the way this happened either.

In 1954 he published the two volume book Mathematics and Plausible Reasoning and in 1962 and 1965 a further two volume set entitled Mathematical Discovery.

From his retirement in 1953 Pólya took an active interest in improving the standard of teaching and took steps to establish, with NSF funding, eight-week Summer Institutes for mathematics teachers, first at the college level (1953-1960), then later for teachers of high school and eventually moving the Institutes to Switzerland.

George Pólya died on 7 September 1985.

References

1. George Pólya, Master of Discovery, 1887-1985, Harold and Loretta Taylor, Dale Seymour Publications, 1993.
2. The Random Walks of George Pólya, Gerald L Alexanderson, MAA, 2000

With thanks to a useful suggestion from Laci Kovács.

Written by Peter Taylor, June 2000. Adapted slightly January 2008 with help from Kevin McAvaney.

This T Shirt, which celebrates the Necklace Theorem, is available from the AMT.