SOLVING THE ‘EINSTEIN’ PROBLEM

I believe the most important mathematical breakthrough this century is the solution to the long-standing ‘einstein’ (one stone) problem. The einstein problem asks whether there is a shape that can tile an infinitely large horizontal surface so that the pattern never repeats. Brilliant minds had searched for decades for such shapes. Then in 2022, David Smith, a retired print technician and amateur maths enthusiast, began working with software and cardboard cut-outs at his home in Bridlington, Yorkshire. Smith had worked for years on tiling patterns and had a strong intuition his shape, nicknamed the ‘hat’, would both tile the surface and never repeat. He didn’t have the mathematical tools to prove his hunch, however, so he turned to the community of tiling enthusiasts and got help from Prof Craig Kaplan at the University of Waterloo, Canada; Prof Chaim Goodman-Strauss from the University of Arkansas; and software engineer Dr Joseph Myers from Cambridge. Together they came up with computer-based and analytic proofs for a whole family of shapes, and their preprint study was greeted with international acclaim in March 2023 – even FEATURE Listen to More or Less: Behind the Stats on BBC Sounds though the ‘hat’ occasionally needed to be flipped over to successfully tile the plane. But no sooner had the preprint of their work been released, than David came up with the ‘spectre’ – a chiral aperiodic monotile, which didn’t need flipping. Even more impressive, the ‘spectre’ was a member of a larger class of such tiles that allows the straight edges to be wavy. Again, his colleagues proved the truth of his intuition. This was a beautiful achievement, led by an extraordinary mathematical hobbyist.

RIGHT A tiling of ‘hats’ – this tiling can continue forever and there will never be a repeat in the pattern

BELOW Prof Craig Kaplan was one of the people who helped David Smith prove that his hat and spectre shapes solved the einstein problem