[Reading Science] The 60-Year Math Puzzle 'Einstein' Solved

British Amateur Mathematician Proves Existence of 'Aperiodic Tiles',
One Shape Can Fill Plane Without Repeating Pattern

An amateur mathematician from the United Kingdom has finally discovered the 60-year-old mathematical puzzle known as the 'Einstein (ein + stein)'. Pavement blocks, wallpaper, bathroom tiles, and building marble usually feature simple patterns repeated over and over. Staring at them can hardly be anything but boring. To avoid this, multiple shapes are needed, which makes installation more complicated. Scientists, tired of this monotony, have spent the past 60 years exploring whether it is possible to create a shape that can fill an infinite plane without repeating patterns using only one shape, and they have finally found the answer.

According to the international academic journal Nature on the 1st (local time), British amateur mathematician David Smith and his research team proved the existence of a 13-sided 'aperiodic tile' that can fill a plane infinitely without repeated patterns using only one shape. Smith and his team had previously posted a similar research result on the preprint site arXiv on March 20. However, at that time, the aperiodic tiles discovered by Smith were not one but two. There were two tiles: a 13-sided tile resembling a 'hat' and its mirror image with left and right reversed, and both were necessary to arrange an infinite plane without pattern repetition.

Then, on the 29th of last month, Smith's research team finally announced new research results showing that by slightly modifying the shape of the existing 13-sided tile, it is possible to fill the plane infinitely without pattern repetition without flipping the tile. The team calls this shape they devised the 'Einstein' shape. This is a pun in German combining the name of the 20th-century genius physicist Einstein and the words for 'one' and 'stone' (ein + stein). It means that the entire wall can be covered without repeated patterns using a single shape.

In the paper published in March, the research team created a 'metatile' by connecting several of these 13-sided shapes, and then combined these to form a larger 'Supertile,' which can be repeated infinitely without repeating the same pattern. They also demonstrated that changing the size of each shape by shortening or lengthening the sides still allowed the plane to be filled without creating repeating patterns.

Since the 1960s, the search for aperiodic tiles has been a longstanding challenge in mathematics. In 1963, American mathematician Robert Berger proved that using 20,426 shapes, it is possible to decorate a plane without repeating patterns. Later, in 1974, Roger Penrose, a professor at the University of Oxford in the UK, proved that it is possible with only two shapes.