Geometry

A new proof breaks a decades-long drought of progress on the problem of estimating the size of triangles created by cramming points into a square.

The telescope conjecture gave mathematicians a handle on ways to map one sphere to another. Now that it has been disproved, the universe of shapes has exploded.

A new result shows how much of the plane can be colored by points that are never exactly one unit apart.

A new proof marks major progress toward solving the Kakeya conjecture, a deceptively simple question that underpins a tower of conjectures.

Transcendental numbers include famous examples like e and π, but it took mathematicians centuries to understand them.

By treating molecules as geometric tessellations, scientists devised a new way to forecast how 2D materials might self-assemble.

Simple math can help explain the complexities of the newly discovered aperiodic monotile.

In all bilaterally symmetrical animals, from humans down to simple worms, nerves cross from one side of the body to the opposite side of the brain. Geometry may explain why.

So-called “higher symmetries” are illuminating everything from particle decays to the behavior of complex quantum systems.