A Small Aperiodic Set of Tiles

We give a simple set of two tiles that can only tile aperiodically | that is no tiling with these tiles is invariant under any in nite cyclic group of isometries. Although general constructions for producing aperiodic sets of tiles are nally appearing, simple aperiodic sets are fairly rare. This set is among the smallest sets ever found. A tiling is non-periodic if there is no in nite cyclic group of isometries leaving the tiling invariant. In E, this is equivalent to requiring that no translation leaves the tiling invariant. A set of tiles is aperiodic if it is possible to completely tile the plane with comgruent copies of the tiles, but only non-periodically. For example, a pair of unit squares, one black and one white, is not an aperiodic set of tiles: it is possible to tile non-periodically with black and white squares but they can tile periodically as well. Here we give a new, simple example of a set of aperiodic tiles, the T (trilobite) and C (cross) ( gure 1); in any tiling with these tiles, we will require that the \tips" of the tiles meet as pictured at right. (A local condition such as this is a \matching rule"). Two variations of the tiles are given at the end of this paper. These tiles are among the simplest ever found, and are related to a a family of aperiodic sets of 2 tiles in each En, n 3 [10]. The reader may wish to examine a photocopy of the appendix with a pair of scissors. cross tips must meet like so: i.e.