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Many of the deepest and most important areas of mathematics have arisen from questions about extremes - shortest lines, smallest areas, densest packings, fewest colours. Mathematicians have been grappling with such issues for centuries, and some go back thousands of years. The isoperimetric problem, for example - which asks for the shortest path enclosing a given area - dates back to the mythological founding of the city of Carthage. By contrast, it was only in 2017 that the densest ways to pack identical spheres into a space of 24 dimensions was finally proved.
Many of these problems are more than mere thought experiments. The origins of the Travelling Salesperson Problem - find the shortest route that visits a given set of cities - are self-explanatory. The Plateau problem, about the geometry of soap bubbles, now has applications as diverse as cosmology and biological development.
Reaching for the Extreme tells the stories of these and other similar problems: their historical roots, the struggles to solve them, and the uses that can be made of the results, when such uses exist.
Ian Stewart is Professor Emeritus of Mathematics at the University of Warwick. He is the author of the bestseller Professor Stewart's Cabinet of Mathematical Curiosities, as well as What's the Use?, Do Dice Play God?, Significant Figures, Incredible Numbers, Seventeen Equations that Changed the World, Professor Stewart's Casebook of Mathematical Mysteries and Calculating the Cosmos. He is a Fellow of the Royal Society.
