Generalized Forcing in Aperiodic Tilings

An aperiodic tiling is one which uses a nite set of prototiles to tile space such that there is no translational symmetry within any one tiling. These tilings are of interest to researchers in the natural sciences because they can be used to model and understand quasicrystals, a recently-discovered type of matter which bridges the gap between glass, which has no regular structure, and crystals, which demonstrate translational symmetry and certain rotational symmetries. The connection between these two areas of research helps to create a practical motivation for the study of aperiodic tilings. We plan to study both theoretical and applied aspects of several types of aperiodic structures, in one, two, and three dimensions. Penrose tilings are tilings of the plane which use sets of two prototiles (either kites and darts, or thin rhombs and thick rhombs) whose edges are marked in order to indicate allowed arrangements of the tiles. There is a small number of legal vertex con gurations, and most of these force the placement of other tiles which may or may not be contiguous with the original ones. It is somewhat intuitive that in most cases the placement of a group of tiles forces the placement of some set of adjacent tiles. It is much more di cult to believe that many groups of tiles also force the placement of in nitely many non-adjacent tiles. This phenomenon has implications which may be important in designing and implementing an e cient data structure to explore and store information about aperiodic tilings. For the most part, we plan to study forced tiles in two-dimensions, although we believe that this theory may be generalized to higher dimensions. We will use musical sequences and Ammann bars, concepts established by Conway and Ammann, to develop an algorithm for predicting the placement of xed bars given an initial sequence of parallel bars. This, in turn, may be used to x the positions of tiles and clusters of tiles within any legal tiling. Our goal is to determine a method for predicting the position of in nitely many forced tiles given an initial cluster of tiles. These descriptions will be partly grammarbased in ations of musical sequences, and partly algebraic solutions, and will themselves be aperiodic.