The problem of tiling a plane has fascinated builders and mathematicians alike since time immemorial. At first glance, the task is straightforward: squares, triangles, hexagons all do the trick producing well known periodic structures. Ditto any number of irregular shapes and combinations of them.
A much trickier question is to ask which shapes can tile a plane in a pattern that does not repeat. In 1962, the mathematician Robert Berger discovered the first set of tiles that did the trick. This set consisted of 20,426 shapes: not an easy set to tile your bathroom with.
With a warm regard for home improvers, Berger later reduced the set to 104 shapes and others have since reduced the number further. Today, the most famous are the Penrose aperiodic tiles, discovered in the early 1970s, which can cover a plane using only two shapes: kites and darts.
The problem of finding a single tile that can do the job is called the einstein problem; nothing to do with the great man but from the German for one– “ein”–and for tile–“stein”. But the search for an einstein has proven fruitless. Until now.
Today, Joshua Socolar and Joan Taylor at Duke University announce that they have solved the einstein problem and in the process they’ve discovered an entirely new way to approach the problem.
The tile they’ve discovered is essentially a modified hexagon shape. But they use a couple of tricks to achieve the result. First, they allow themselves to use a tile and its mirror image to tile a plane in an aperiodic fashion.
Obviously, some tilers may feel that this is bending the rules a little, so Socolar and Taylor go on to show that the mirror image is not necessary if the tile is allowed a 3D shape (see below).
“The tile presented here is the only known example of an aperiodic tile,” they say.
That’s an impressive result. After Penrose revealed his aperiodic tilings, physicists pointed out that certain crystals adopted similar patterns. It’ll be interesting to see whether nature has discovered Socolar and Taylor’s solution too.
Of course, the work leaves a substantial problem open: is it possible to tile a plane with a nonrepeating pattern using a single 2D tile?
I imagine Taylor and Socolar are puzzling over a bathroom wall at this very moment.
Ref: arxiv.org/abs/1003.4279: An Aperiodic Hexagonal Tile