Reaching for the Extreme: How the Quest for the Biggest, Fewest, and Weirdest Makes Math

From bestselling author and mathematician Ian Stewart, the fascinating story of the extreme problems that have driven math forward from antiquity to today

"Stewart has a genius for explanation."—New Scientist

Many of the deepest and most important areas of mathematics have emerged from questions about extremes—the shortest path between two points on a curved surface, the smallest area spanning a wire, or the fewest colors needed to make a map. Mathematicians have been pushing restlessly toward extremes for thousands of years. The isoperimetric problem, for example—which asks for the shortest route enclosing a given area—can be traced to ancient Carthage. By contrast, it was only in 2017 that the densest ways to pack identical spheres into a 24-dimensional space was proven. In Reaching for the Extreme, bestselling author Ian Stewart, one of the world’s most popular writers on mathematics, presents a dazzling, wide-ranging tour of math’s outer limits.

Stewart tells the stories of sixteen superlative problems—their history, the struggles to solve them, and the uses of some of the results. From the biggest number to the smallest, the fastest fall to the weirdest symmetry, and the best fold to the shortest proof, these questions are either pure thought experiments or are motivated by real-world challenges. The Plateau problem, about the geometry of soap bubbles, led to the notion of a minimal surface—now used in cosmology, biology, and other fields. Meanwhile, the 2023 discovery of a single tile shape that covers the infinite plane without repeating the same pattern has no application—yet.

Reaching for the Extreme illuminates how mathematicians drive knowledge forward by reaching for the edges and solving some of the world’s most fascinating problems.