Searching For Einstein

No, not Einstein. We’re searching for einstein. Literally “ein Stein,” which translated from German means “one stone.” The one stone we’re looking for is a very special type of tile which, when repeated, can cover an infinite floor without leaving any gaps and without admitting any sort of pattern.

We call an arrangement of tiles that covers the plane without any gaps or overlaps a tiling, and a tiling is called non-periodic if it has no translational symmetry. That means, if I pick the tiling up and move it in any direction, I won’t be able to fit it back down on itself. A nice example of a non-periodic tiling where we allow two types of tiles is the Penrose tiling. And if we loosened our restrictions slightly to allow tiles which are not connected, Socolar and Taylor found such a tiling in 2010. So more formally, the search for einstein is the search for a single connected tile that tiles only non-periodically. Recently, in the quest for einstein, some interesting progress has been made.

First some basics. Let’s think about tilings that only use a single convex polygon, that is, a polygon whose angles all bulge out instead of in. If we allow patterns and periodicity, then it’s easy to imagine how you could achieve a non-overlapping gap-free tiling that with a square, triangular, or even hexagonal tile. Even though sometimes they can be in disguise.

We now know that these are the only three possible regular hexagonal tilings. They were first discovered by Karl Reinhardt in 1918. Image via Wikimedia Commons.

Things get a bit more interesting when we consider pentagons. In the early 1900’s Karl Reinhardt found five examples of families of pentagonal tilings. Several more were found by various people over the years, including 4 families which were found by housewife and mathematical enthusiast Marjorie Rice, who recently passed away. And just last year, as reported on this blog by Evelyn Lamb, another pentagonal tiling was found, bringing the total number of known families of pentagonal tilings to 15. For some fun teachable moments involving pentagonal tilings, check out mikesmathpage.

These are the 15 families of tilings with convex pentagons. Here colors indicate the number of edges touching each pentagon. Image courtesy of Wikimedia Commons.

In breaking news, mathematician Michaël Rao of France’s CNRS proved that these are precisely all of the convex tilings of the plane. There are just the 15 known families of pentagonal tilings, 3 hexagonal tilings, and all triangular and quadrilateral tilings. Of note, is that Rao’s work involves a computer assisted proof, which allowed him first to establish some bounds via theoretical methods and then do an exhaustive search. Rao’s conclusion: there are no convex polygons that admit only non-periodic tilings, that means, the einstein tile must not be a convex polygon.

This means, if we want to find einstein, we need to start looking at concave tiles.