Aperiodic Tiling

There are lots of ways to tile a 2D plane with a single shape - for example, with squares like you might see on a tile floor. This kind of tessellating tiling demonstrates periodic behaviour - they have a translational symmetry. Some tilings that use multiple shapes can demonstrate ‘aperiodic tiling’ where they can tile infinitely without ever repeating a full pattern. It was an open question whether it was possible to do this with just one tile. This question was called the 'Einstein problem' because 'ein Stein' means "one stone" in German.

In 2023, the first aperiodic monotile called “the hat” was discovered. This is a tile that can tile a plane without repeating its pattern, using the tile and its reflection. This discovery was shortly followed by a true aperiodic monotile - a slightly adjusted version of the hat called "the spectre". The spectre can tile the plane aperiodically just by itself.

This pattern is an aperiodic tiling of spectres. The numbers in the pattern correspond to a potential colouring of the tiles so that none of the tiles of the same colour share an edge, so you can choose whether you would like to colour the tiles.