Forcing nonperiodicity with a single tile

It is easy to create nonperiodic tesselations of the plane composed of one or a few types of tiles. In most cases, however, the tiles employed can also be used to create simpler, periodic patterns. It is much more difficult to find shapes, or “prototiles,” that can fill space only by making a nonperiodic structure. We say that such sets are aperiodic, or that they “force” nonperiodicity, and there are many open questions about what types of structure can be forced and the prototiles required. In this article we discuss recent progress on the fundamental problem of forcing nonperiodicity using a single prototile, jokingly called an einstein (a German pun on “one stone”). A new example we found [1] shows one way in which an einstein can work and highlights several issues that arise in posing the problem precisely. One motivating factor in the search for an einstein comes from condensed matter physics. Local rules for how tiles fit together may represent the energetics of a physical system, which could support self–assembly into an ordered but nonperiodic structure. The discovery of icosahedral and decagonal phases of metallic alloys, in which the atomic structure shares the essential structure of the Penrose tilings, has opened our eyes to the fact that nonperiodic materials can indeed form spontaneously [2, 3]. In materials physics applications, where the tiles may represent clusters of many atoms or larger building blocks, the tiles can have complex shapes or markings that determine how they may be joined. Finding a single shape that can do the job may make the physical realization of such a material easier. The first example showing that it is possible to force nonperiodicity was Berger’s set of 20, 426 distinct prototiles [4]. Aperiodic sets with just two prototiles were subsequently discovered, the most famous being the Penrose tiles [5], nicely described by Martin Gardner [6]. Candidates with einstein–like features have been presented before, but there is no precise definition of the einstein problem, and several candidates that could be argued to qualify have not passed the consensus “I know it when I see it” test. There are several issues involved, including the specification of what counts as nonperiodic, what characteristics make for a valid prototile, and what form the local rules must take. We recently showed that the prototile in Figure 1 is an einstein and determined a number of remarkable properties of the tilings it forces. Ref. [1] contains two proofs of the forced nonperiodicity along with derivations of several intriguing properties of the tiling (including a surprising connection to the regular paperfolding sequence [7]). In working out the properties of the forced limit– periodic structure and searching for different ways of encoding the information about how the tiles must fit together, we were led to a series of questions about