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Assigning elements from a large collection to one of two categories can yield almost “magical” predictions about highly complicated problems without actually solving them.

In a geometrically designed social club, how do dancing, triangles and hexagons mix?

The mathematician John Horton Conway’s myriad accomplishments — including the Game of Life, sprouts and the surreal numbers — are the product of a mind at play.

Mathematicians have solved the century-old triangulation conjecture, a major problem in topology that asks whether all spaces can be subdivided into smaller units.

Researchers have found a set of rules for imbuing flat surfaces with curvature, enabling them to form a virtually unlimited range of three-dimensional structures.

Maryam Mirzakhani, who became the first woman Fields medalist for drawing deep connections between topology, geometry and dynamical systems, has died of cancer at the age of 40. This is our 2014 profile of her life and work.

Improvements in how densely spheres and other shapes can be packed together could lead to advances in materials science, deep space communication and theoretical physics.

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