Roger Penrose Solved Aperiodic Tiling in 1974 — Then Iranian Architects Were Found to Have Done It in 1453

An aperiodic tiling is a way of covering the plane with copies of a small set of tiles such that the resulting pattern never repeats — slide it, and the alignment fails. The 1960s mathematical question was whether such tilings existed at all. Robert Berger answered yes in 1966, with a set of about 20,000 tiles. Through the late 1960s and early 1970s, the count was reduced. In 1974 the Oxford mathematical physicist Roger Penrose published a set of just two tiles — a "kite" and a "dart," or alternatively a fat and thin rhombus — that could tile the plane in an infinity of ways, but never periodically. The arrangement showed an apparent five-fold rotational symmetry that classical crystallography had said was impossible.

The pattern's afterlife crossed disciplines. In 1982, the materials scientist Dan Shechtman observed an aluminum-manganese alloy diffracting electrons in a sharp ten-fold pattern that looked exactly like what Penrose tilings predicted. Crystallographers initially refused to publish his result. Linus Pauling famously called him "a quasi-scientist." Shechtman won the 2011 Nobel Prize in Chemistry for the discovery, by which point quasicrystals had been found in nature in a Russian-meteorite Khatyrkite sample.

The oddest historical wrinkle came in 2007. The Harvard physicist Peter Lu and the Princeton physicist Paul Steinhardt analyzed Iranian girih tile-work — the geometric pattern tradition decorating Islamic architecture — and showed that the spandrels of the Darb-e Imam shrine in Isfahan, built in 1453, had been laid out as an effectively perfect Penrose tiling. The fifteenth-century Persian master tilers had cracked the math more than five hundred years before Penrose. They left no recorded explanation of how.