Symmetry

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.

Two new studies suggest that certain tetrahedral arrangements of galaxies outnumber their mirror images, potentially reflecting details of the universe’s birth. But confirmation is needed.

A team of mathematicians has solved an important question about how solutions to polynomial equations relate to sophisticated geometric objects called Shimura varieties.

Physicists have been busy exploring how our universe might emerge like a hologram out of a two-dimensional sheet. New clues have come from the symmetries found on an infinitely distant “celestial sphere.”

The “gravitational memory effect” predicts that a passing gravitational wave should forever alter the structure of space-time. Physicists have linked the phenomenon to fundamental cosmic symmetries and a potential solution to the black hole information paradox.

In nonreciprocal systems, where Newton’s third law falls apart, “exceptional points” are helping researchers understand phase transitions and possibly other phenomena.

Watanabe invented a new way of distinguishing shapes on his way to solving the last open case of the Smale conjecture, a central question in topology about symmetries of the sphere.