- Person
- Hermann Minkowski
- Lived
- 1864–1909
- Field
- Mathematics
- From
- German
The story
Hermann Minkowski, born in 1864 to a Jewish family in Aleksotas in the Russian Empire, spent much of his career on the geometry of numbers, a field he created by studying how convex shapes sit over a regular grid of points. MacTutor notes that this work led him on to the study of convex bodies and to packing problems, the question of how shapes can be fitted inside a given space.
Among the tools that came out of this work is a simple way of adding two shapes. Take every point of one shape and add it to every point of the other; the set of all the results is a new shape. This operation is now called Minkowski addition, or the Minkowski sum. Subtracting in the same way gives the Minkowski difference. The sums also appear in the Brunn–Minkowski theorem, an important inequality about the volumes of shapes.
What began as pure mathematics turned out to be highly practical. In robot motion planning, engineers use Minkowski sums to compute the configuration space, the set of all positions an object can occupy among obstacles without touching them. In physics engines for games and simulations, Minkowski differences are used with the GJK algorithm to detect collisions between convex shapes. In numerically controlled machining, the Minkowski sum of a cutting tool and its path gives the exact shape of the cut in the material.
The idea reaches further still. In image processing the Minkowski sum and difference are known as dilation and erosion, and Minkowski addition plays a central role in mathematical morphology, the analysis of shapes in images. The 3D modelling program OpenSCAD uses Minkowski sums to outline one shape with another, and they are used in aggregation theory, for example in economics, when individual units are described as sets.
Minkowski died of appendicitis in Göttingen in 1909, aged 44, long before computers existed. His geometric ideas found a second life in the machines of the digital age, where they help robots, factories and software handle the physical world safely and precisely.
Why it matters
An idea from early twentieth-century number theory now helps robots, factories and software handle the physical world safely.
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.
- Minkowski invented the geometry of numbersHermann Minkowski created the 'geometry of numbers', solving problems about whole numbers by picturing them as points in a grid, published from 1896.
- 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.
Sources
- Minkowski addition Wikipedia
- Hermann Minkowski MacTutor History of Mathematics, University of St Andrews
Another act 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.