From the Russian Empire to Königsberg
Hermann Minkowski was born on June 22, 1864, in Aleksotas, a town on the edge of Kovno (today Kaunas, Lithuania), then part of the Russian Empire. His parents, Lewin Boruch Minkowski and Rachel Taubmann, were Jewish; his father was a merchant who helped fund the building of the choral synagogue in Kovno. Hermann was the third son in a gifted family. His eldest brother Max ran the family business and collected art; his brother Oskar became a celebrated physician whose experiments on the pancreas were a milestone in the understanding of diabetes.
In 1872, when Hermann was eight, the family left the Russian Empire, where Jews faced mounting restrictions and persecution, and settled in Königsberg in East Prussia. There Lewin Minkowski built a business in rag export and, later, in mechanical tin toys. Because of this unusual path, Minkowski’s nationality is described in different sources as German, Russian, Polish or Lithuanian-German. He grew up and made his career in Germany.
A teenage prodigy and a lifelong friendship
Minkowski’s gift showed early. As a gymnasium pupil in Königsberg he was already reading Gauss, Dirichlet and Dedekind. He entered the University of Königsberg in April 1880, spending three semesters in Berlin, and there formed the friendship that shaped his life: with David Hilbert, a fellow student two years his senior, and with the young lecturer Adolf Hurwitz. The three took long walks together, day after day, roaming through mathematics, walks Hilbert later recalled with deep affection.
In 1881 the Paris Academy of Sciences set as the problem for its 1883 Grand Prix the number of ways an integer can be written as a sum of five squares. Minkowski, still a teenager, rebuilt the theory of quadratic forms from scratch and produced a solution of great elegance. The English mathematician Henry Smith had published an outline of a proof years earlier, and in April 1883 the Academy divided the prize between the eighteen-year-old Minkowski and the elderly, recently deceased Smith. Some English mathematicians protested; the Academy did not change its decision. Minkowski took his doctorate at Königsberg in 1885 under Ferdinand von Lindemann.
The geometry of numbers
Minkowski taught at Bonn from 1887, returned to Königsberg in 1894, and in 1896 moved to the Federal Polytechnic in Zurich (today ETH Zurich). In 1902 Hilbert, by then the leading figure at Göttingen, arranged for a chair to be created especially for him, and Minkowski spent the rest of his life there. He married Auguste Adler in Strasbourg in 1897; they had two daughters, Lily and Ruth.
His most original contribution to pure mathematics was what he called the geometry of numbers. The idea was to attack problems about whole numbers by picturing them as points in a regular grid, a lattice, and then reasoning geometrically about convex shapes laid over that grid. His central result, now known as Minkowski’s theorem, guarantees that a sufficiently large symmetric convex region must contain a lattice point other than the origin. The first part of his book Geometrie der Zahlen appeared in 1896, and his Diophantische Approximationen followed in 1907. The approach opened whole fields of study on convex bodies, packing problems and the arithmetic of number fields.
Minkowski was also a builder of the international mathematical community. When the first International Congress of Mathematicians met in Zurich in 1897, he sat on its organising committee, helped choose the speakers and chaired the section on arithmetic and algebra. He served as one of the secretaries of the 1900 congress in Paris, spoke on the geometry of numbers at Heidelberg in 1904, and represented Göttingen at the 1908 congress in Rome, a figure at the centre of the mathematical world at the moment it was becoming truly international.
Einstein’s teacher
At Zurich, Minkowski’s students included a young physics student named Albert Einstein. Einstein took several of his courses but was, by most accounts, an irregular attender, preferring to study physics on his own. Later, when Einstein’s 1905 paper on special relativity made his name, Minkowski is said to have recalled his surprise, remembering the student as a “lazy dog” who never bothered much about mathematics, a secondhand recollection, but one that captures the distance between the two men at the time.
At Göttingen, Minkowski turned increasingly to mathematical physics. With Hilbert he took part in a 1905 seminar on the electron theory of matter and studied the work of Hendrik Lorentz, Henri Poincaré and his former student. By 1907 he had concluded that the new physics was, at bottom, a statement about geometry.
The union of space and time
On September 21, 1908, Minkowski addressed the Assembly of German Natural Scientists and Physicians in Cologne. His lecture, “Space and Time” began with one of the most quoted passages in the history of physics: space by itself and time by itself were “doomed to fade away into mere shadows” and only a union of the two would remain real. What Einstein had expressed as rules about clocks, rods and moving observers, Minkowski recast as the geometry of a single four-dimensional continuum, spacetime.
In Minkowski’s picture, every event has a place and a time, and the separation between two events is measured by an “interval” that combines distance and time in a way all observers agree on, even when they disagree about lengths and durations taken separately. He drew the first spacetime diagrams, with each object’s history traced as a “world line” and showed that the strange effects of relativity, slowed clocks, contracted lengths, follow naturally from this geometry. He also produced a four-dimensional treatment of the equations of electrodynamics in moving bodies.
How spacetime changed physics
Einstein was not immediately impressed. He reportedly dismissed the new formalism as “superfluous learnedness.” But when he began the long struggle to extend relativity to gravity, he found he could not do without it. General relativity, completed in 1915, describes gravity as the curvature of spacetime, a curved version of the very geometry Minkowski had introduced. By 1912 Einstein was writing to Arnold Sommerfeld that he had gained a great respect for mathematics, and historians credit Minkowski and Hilbert with changing his view of its role in physics.
Today “Minkowski spacetime” is the stage on which special relativity, particle physics and quantum field theory are written; every physics student learns to draw his diagrams and light cones. His purely mathematical work has had a practical afterlife too. Minkowski’s theorem underpins the theory of lattices, which is central to modern lattice-based cryptography; the “Minkowski sum” of shapes is a standard tool in robotics and computer graphics for working out how objects can move without colliding; and the “Minkowski distance” is a familiar measure in statistics and machine learning.
A Jewish family in German science
Minkowski’s story belongs to a generation of Jewish families who escaped the restrictions of the Russian Empire and found opportunity in the universities of Imperial Germany. He and his brother Oskar both rose to the top of their professions, and the two brothers share a grave. Minkowski’s path to the most prestigious mathematical department in Germany ran through friendship: it was Hilbert who used his influence at Göttingen to have a chair created for him.
The family’s later history was shaped by the century that followed. In 1919 Minkowski’s daughter Lily married the electrical engineer and inventor Reinhold Rudenberg. In 1936, after the Nazis came to power, Rudenberg left Germany for England, and in 1939 he became a professor at Harvard, one of the many German-Jewish scientists whose emigration carried the culture Minkowski had helped build across the Atlantic. Minkowski himself did not live to see any of it.
A friend remembered
Minkowski was remembered as modest, warm and witty, and as a man of exceptional loyalty. He played a quiet but important part in the history of mathematics behind the scenes: in a letter of January 5, 1900, he advised Hilbert to use his forthcoming lecture at the International Congress of Mathematicians in Paris to sketch the problems future mathematicians should tackle. Hilbert followed the suggestion, and his list of 23 problems shaped the agenda of twentieth-century mathematics.
Hilbert’s own words after Minkowski’s death are the best testament to the man. Since their student years, he wrote, Minkowski had been his best and most dependable friend; science had seemed to them “a garden full of flowers” in which they enjoyed looking for hidden pathways together, and he called his friend “a rare gift from heaven.”
Sudden death and legacy
Just months after the Cologne lecture, in January 1909, Minkowski was struck down by a ruptured appendix. He died in Göttingen on January 12, 1909, at the age of forty-four. Max Born, who had come to Göttingen to work with him, delivered an obituary address on behalf of the mathematics students. His collected papers were published in 1911, edited by Hilbert.
He left behind a body of work cut short at its height, and one idea so powerful that it became the natural language of modern physics. A lunar crater and the asteroid 12493 Minkowski bear his name, but his true monument is a word every physicist uses without thinking: spacetime.
Sources & further reading
Hermann Minkowski, Wikipedia · Hermann Minkowski, MacTutor History of Mathematics (St Andrews) · Minkowski, “Space and Time” (Raum und Zeit), Wikisource translations · Researched from the sources above and checked against them. Spotted an error? Tell us.
A life of tikkun olam. This is part of tikkun olam, the Jewish idea of repairing the world, and every one of us can add to it.