Geometry

Physicists recently mapped the hidden shape that underlies the quantum behaviors of a crystal, using a new method that’s expected to become ubiquitous.

Geometry may have its origins thousands of years ago in ancient land surveying, but it has also had a surprising impact on modern physics. In the latest episode of The Joy of Why, Yang-Hui He explores geometry’s evolution and its future potential through AI.

A new proof represents the culmination of a 65-year-old story about anomalous shapes in special dimensions.

A powerful mathematical technique is used to model melting ice and other phenomena. But it has long been imperiled by certain “nightmare scenarios.” A new proof has removed that obstacle.

The deceptively simple Kakeya conjecture has bedeviled mathematicians for 50 years. A new proof of the conjecture in three dimensions illuminates a whole crop of related problems.

A new proof extends the work of the late Maryam Mirzakhani, cementing her legacy as a pioneer of alien mathematical realms.

Physicists and marine biologists built a quantitative framework that predicts how coral polyps collectively construct a variety of coral shapes.

A new proof reveals the answer to the decades-old “moving sofa” problem. It highlights how even the simplest optimization problems can have counterintuitive answers.

A new proof marks the first progress in decades on important cases of the so-called kissing problem. Getting there meant doing away with traditional approaches.