Geometry

After just a few months of work, a complete newcomer to the world of sphere packing has solved one of its biggest open problems.

A tetrahedron is the simplest Platonic solid. Mathematicians have now made one that’s stable only on one side, confirming a decades-old conjecture.

From brain folds to insect architecture, L. Mahadevan explains how complex biological forms and behaviors emerge through the interplay of physical forces, environment and embodiment.

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.