For some, tiles are rarely thought of unless it’s time to renovate the house, but for mathematicians they present many puzzles – and a clever team has just solved a particularly tricky question. Researchers have identified a shape that was previously only theoretical: a 13-sided configuration called “the hat” that can tile a surface without repeating itself.
The hat is what is called an aperiodic monotile, which means that a single shape can cover a surface without any translational symmetry, or without its pattern ever repeating. The famous Penrose Tile are an example of an aperiodic tiling, where the pattern is aperiodic but uses two different shapes.
The hat tiling uses only one shape, an “einstein”, which is German for “a stone”, making the pattern an aperiodic monotile. The 13-sided hat is a polykite shape, consisting of eight kites connected by their edges. The existence of an aperiodic monotile was purely theoretical until a research team led by mathematician David Smith and his colleagues proved its existence in a preprinted paper released this month.
“You are literally looking for one in a million things. You filter out the boring 999,999 and then you have something weird, and that deserves further exploration,” co-author Chaim Goodman-Strauss, a mathematician at the National Museum of Mathematics, told new scientist. “And then, by hand, you start looking at them and trying to figure them out, and start pulling the structure out. This is where a computer would be worthless because a human had to be involved in building proof that a human could understand.
For mathematicians, the discovery seems to answer a long-standing question in the field of geometry. But for the rest of us, it might just represent a brilliant new option for bathroom tiles.