Patterns

It’s surprisingly difficult to prove one of the most basic properties of a number: whether it can be written as a fraction. A broad new method can help settle this ancient question.

After decades of uncertainty, a motley team of programmers has proved precisely how complicated simple computer programs can get.

Two mathematicians have shown that origami can, in principle, be used to perform any possible computation.

John Conway’s Game of Life, a famous cellular automaton, has been found to have periodic patterns of every possible length.

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

The mechanism behind leopard spots and zebra stripes also appears to explain the patterned growth of a bismuth crystal, extending Alan Turing’s 1952 idea to the atomic scale.

A glass sponge found deep in the Pacific shows a remarkable ability to withstand compression and bending, on top of the sponge’s other unusual properties.

Two mathematicians have proved the first leg of Paul Erdős’ all-time favorite problem about number patterns.

Randomness would seem to make a mathematical statement harder to prove. In fact, it often does the opposite.