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Sphere Packing Solved in Higher Dimensions

The Ukrainian mathematician Maryna Viazovska has solved the centuries-old sphere-packing problem in dimensions eight and 24.

insights puzzle

Solution: ‘The Problem With Dancing Shapes’

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

insights puzzle

The Problem With Dancing Shapes

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


A Life in Games

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.


A Proof That Some Spaces Can’t Be Cut

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


Scientists Conjure Curves From Flatness

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.

2014 Fields Medal and Nevanlinna Prize Winners

A Tenacious Explorer of Abstract Surfaces

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.

Photograph of Saturn and its moons.

A New Tool to Help Mathematicians Pack

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.