An amateur mathematician has solved a long-standing geometry mystery. David Smith, a “shape hobbyist” from England, was tinkering with cardboard cut-outs of a 13-sided polygon that vaguely resembled a fedora, when he realised he could be looking at an elusive ‘einstein’ shape.

The ‘einstein’ (‘one stone’) shape is a two-dimensional tile that can cover an infinite surface without overlaps or gaps, creating a pattern that never repeats. While individual tiles or subdesigns in the “hat” tiling may recur, the tiled surface is like the shape version of the number pi—its sequence never permanently repeats.

Smith emailed a mathematician acquaintance, who completed a proof backing the discovery. (Meanwhile, Smith discovered a second ‘einstein’, dubbed the “turtle”. That work has yet to be peer reviewed.)

In the 1970s, Nobel Prize winner Roger Penrose discovered a set of two tiles, called darts and kites, that can create this ‘aperiodic’ tiling. But whether this type of tessellation could be achieved with a single shape remained unknown, until now.

Some have questioned whether the “hat” and “turtle” truly count as single tiles, since the aperiodic pattern relies on mirror images. So far the consensus is that yes, they count as sought-after einsteins, with the non-mirror-image version simply a frontier to be explored.