Mathematics

The concept of dimension seems simple enough, but mathematicians struggled for centuries to precisely define and understand it.

Michael Freedman’s momentous 1981 proof of the four-dimensional Poincaré conjecture was on the verge of being lost. The editors of a new book are trying to save it.

One of the strangest results in mathematics explains how it’s possible to turn one sphere into two identical copies, simply by rearranging its pieces.

How readers used their geometry skills to survive a dangerous puzzle.

Researchers are turning to the mathematics of higher-order interactions to better model the complex connections within their data.

The mechanism behind leopard spots and zebra stripes also appears to explain the patterned growth of a bismuth crystal, extending Alan Turing’s 1952 idea to the atomic scale.

Crows recently demonstrated an understanding of the concept of zero. It’s only the latest evidence of animals’ talents for numerical abstraction — which may still differ from our own grasp of numbers.

A pair of researchers has shown that trying to classify groups of numbers called “torsion-free abelian groups” is as hard as it can possibly be.

By focusing on relationships between solutions to polynomial equations, rather than the exact solutions themselves, Évariste Galois changed the course of modern mathematics.