The case for patterns that never repeat themselves

For most of the twentieth century the working assumption inside crystallography was almost embarrassing in its plainness: if a shape tiles the plane, it tiles it on a grid. Translate the unit cell, repeat. The grid was the contract. Then in the spring of 1974, a young relativist at Oxford named Roger Penrose drew two quadrilaterals — a fat one he called a kite, a thin one he called a dart — and proved that two shapes were enough to fill the plane without ever repeating themselves.

A pattern with five-fold symmetry

What makes the tiling unsettling is not its complexity but its restraint. There are only two pieces. The matching rules forbid them from ever lining up into a periodic lattice, yet they fit together perfectly, forever, in every direction. Spin the result by seventy-two degrees and the figure repeats itself rotationally — a five-fold symmetry that, by the older theorems, ought to have been impossible on flat paper.

Thirty-three years later, two physicists found the same pattern carved into a fourteenth-century Persian shrine. The architects who placed it there did not have a proof. They had a vocabulary called Girih, ten geometric tiles, and a patient eye. The mathematics had been there all along, waiting for a notation to catch up to it.