Symmetry

Modular forms are one of the most beautiful and mysterious objects in mathematics. What are they?

The proof establishes new conditions that cause connected oscillators to sway in sync.

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

A new phase of matter called a “time crystal” plays with our expectations of thermodynamics. The physicist Vedika Khemani talks with Steven Strogatz about its surprising quantum behavior.

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.

The surprisingly simple tile is the first single, connected tile that can fill the entire plane in a pattern that never repeats — and can’t be made to fill it in a repeating way.

Using “refreshingly old” tools, mathematicians resolved a 50-year-old conjecture about how to categorize important functions called modular forms, with consequences for number theory and theoretical physics.

Mathematicians predicted that if they imposed enough restrictions on how a shape might tile space, they could force a periodic pattern to emerge. But they were wrong.