- Person
- Hermann Minkowski
- Lived
- 1864–1909
- Field
- Mathematics
- From
- German
The story
Number theory, the study of whole numbers, had long been a field of ingenious algebraic arguments. Hermann Minkowski came to it through quadratic forms, expressions such as sums of squares in several variables. At 18 he shared the Paris Academy's Grand Prix for work on them, and his 1885 doctoral thesis at Königsberg continued that research.
His great new idea was to use pictures. He imagined whole numbers as the points of a regular grid, or lattice, and asked what happens when convex shapes are placed over that grid. The first example of the method appeared in the lecture he gave in 1887 to qualify for a post at the University of Bonn, on spatial intuition and the minima of quadratic forms, although that lecture was not published until 1991.
His central result, now called Minkowski's theorem, was proved in 1889. It says that any convex region that is symmetric about the origin and large enough, with volume greater than 2 to the power n in n dimensions, must contain a grid point other than the centre. That simple geometric fact turned out to settle hard questions about Diophantine approximation and algebraic numbers. One famous consequence, known as Minkowski's bound, shows that the class number of an algebraic number field is always finite.
The first 240 pages of his book Geometrie der Zahlen (Geometry of Numbers) appeared in 1896, and the complete work was published in 1910, after his death; it was reprinted in 1953 and 1968. In 1907 he published Diophantische Approximationen, an elementary introduction to the geometry of numbers and its uses in number theory. He also spoke on the subject at the International Congress of Mathematicians in Heidelberg in 1904.
MacTutor calls the geometry of numbers his most original achievement. The work led on to the study of convex bodies and to packing problems, the question of how shapes can be fitted inside a given space. Today Minkowski's theorem, which gives an upper bound on the length of the shortest non-zero vector in a lattice, has applications in lattice-based cryptography as well as in number theory.
Why it matters
The geometry of numbers became a lasting branch of mathematics, and lattice theory today underpins areas from coding to modern cryptography.
About Hermann Minkowski
Mathematician who taught the young Einstein and in 1908 fused space and time into a single four-dimensional “spacetime”, the geometry on which relativity still rests.
More about Hermann Minkowski
- Minkowski helped launch the world's maths congressHermann Minkowski helped organise the first International Congress of Mathematicians in Zurich in 1897, chairing its arithmetic and algebra section.
- Minkowski rewrote electrodynamics in four dimensionsAt Göttingen Hermann Minkowski worked out a four-dimensional treatment of electrodynamics, recasting the laws of electricity and magnetism for moving bodies.
- Hermann Minkowski joined space and time into spacetimeIn a 1908 lecture Hermann Minkowski, once Einstein's teacher, showed that relativity is best understood in a single four-dimensional 'spacetime', the language of modern physics.
- Minkowski suggested Hilbert's famous list of problemsIn a January 1900 letter, Hermann Minkowski urged David Hilbert to sketch the problems future mathematicians should tackle, inspiring Hilbert's celebrated 23 problems.
- Minkowski's maths guides robots and machine toolsThe Minkowski sum, a way of adding shapes named after Hermann Minkowski, is used in robot motion planning, collision detection and machining.
Sources
- Hermann Minkowski MacTutor History of Mathematics, University of St Andrews
- Hermann Minkowski Wikipedia
- Minkowski's theorem Wikipedia
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