A hobbyist's cardboard cutout finally solved a 50-year tiling mystery
For decades, mathematicians searched for a single tile shape that could cover an infinite flat surface edge-to-edge with no gaps, yet never settle into a repeating pattern, something achieved before only with sets of several different tiles, like Penrose's famous pair. Could one single shape, repeated endlessly, ever force that permanent non-repetition?
Reveal the answer
In March 2023, hobbyist David Smith, working with mathematicians Craig Kaplan, Joseph Myers and Chaim Goodman-Strauss, announced 'the hat', a 13-sided polygon that tiles the plane aperiodically entirely on its own, solving the long-standing 'einstein problem' (a pun on the German 'ein stein', one stone). They soon found an even simpler relative, nicknamed 'the spectre', that manages the same feat without needing any mirror-image copies.