# Hermann Minkowski

# Hermann Minkowski

**1864-1909**

**Russian-German Mathematician**

Hermann Minkowski established the framework for modern functional analysis, expanded the understanding of quadratic forms, developed the geometry of numbers, and even contributed to Albert Einstein's theory of relativity. His achievements are all the more remarkable in light of his relatively short career.

Born on June 22, 1864, in the town of Alexotas (then in the Russian Empire and now, under the name of Kaunas, a part of Lithuania), Minkowski was the son of a German-Jewish rag merchant. In 1872, when Minkowski was eight years old, the family returned to Germany, settling in the town of Königsberg. (Ironically, Königsberg, renamed Kaliningrad, is now part of Russia.)

Minkowski's brother Oskar (1858-1931) would later become famous as the physiologist who discovered the link between diabetes and the pancreas, and Minkowski himself showed a prodigious talent for mathematics as a student at the University of Königsberg. He was only 17 years old when he set about to win a prize offered by the Paris Académie Royale des Sciences for a proof describing the number of representations of an integer as a sum of five squares of integers. Both Minkowski and the British mathematician H. J. Smith (1826-1883), who had first approached the problem in 1867, won the prize, though many observers since then have maintained that Minkowski's proof was superior in its use of more natural and general definitions.

After receiving his doctorate from Königsberg, where he began a lifelong friendship with fellow mathematician David Hilbert (1862-1943), Minkowski took a position at the University of Bonn. He remained there until 1894, then returned to Königsberg to teach for two years before moving to the University of Zurich.

Among Minkowski's most significant achievements was the geometry of numbers, which he introduced in 1889. The geometry of numbers involved the application of geometrical concepts to volume, which made it possible to prove a variety of theorems without recourse to numerical calculations. Starting with quadratic forms, he extended the method's application to ellipsoids and a variety of convex shapes, such as cylinders. From this development emerged studies in packing efficiency, that is, the most efficient packing of a space given different shapes, a task which has applications in a number of scientific fields.

Hilbert had arranged for the University of Göttingen to create for Minkowski a new professorship, which he took in 1902. There, his most distinguished pupil was Albert Einstein (1879-1955), who took more courses with Minkowski than with any other professor. This fact, however, did not indicate a personal liking between the two men. Indeed, Minkowski, who took an interest in relativity theory himself, viewed his own approach not as an augmentation of Einstein's but as an entirely independent method that he considered superior because of his greater understanding of mathematics.

In 1905, the same year that Einstein produced his first important work in relativity theory, Minkowski undertook a study of electrodynamics in which he compared the subatomic theories of both Einstein and Hendrik Lorentz (1853-1928). It was Minkowski, not Einstein, who first realized that both men's theories required an understanding of four-dimensional, non-Euclidean space.

Einstein later adopted aspects of Minkowski's findings into his general theory of relativity. By the time he published it, however, Minkowski was long gone, having died from a ruptured appendix on January 12, 1909, in Göttingen. He was just 44 years old.

**JUDSON KNIGHT**

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**Hermann Minkowski**