A Modification of the Penrose Aperiodic Tiling

From black and white linoleum on the kitchen floor to magnificent Islamic mosaic to the intricate prints of M.C. Escher, tilings are an essential component of the decorative arts, and the complex mathematical structure behind them has intrigued scientists, mathematicians, and enthusiasts for centuries. Johannes Kepler, most famous for his laws of planetary motion, was known to have been interested in tilings, and his work was used centuries later when Roger Penrose extended one of his four hundred year old sets of tiles to form the famous Penrose tiles in 1976. Penrose’s original aperiodic set contained six tiles, but his most well-known aperiodic set contains only two tiles. The first known aperiodic set of tiles was discovered by Hao Wang. It contained more than 20,000 square tiles (now referred to as Wang tiles) with different edge colorings that were assembled without allowing rotations or reflections. However, the existence of sets of tiles that admit non-periodic tilings of the plane (tilings that possess no translational symmetry) was disputed until Robert Berger proved the undecidability of the domino problem (for Wang tiles) in 1966. The question of the domino problem was essentially whether there existed an algorithm to determine whether a set of tiles admitted a tiling of the plane. With the proposed algorithm, an aperiodic tiling would cause the algorithm to continue forever, so implicit in the problem was the question of whether aperiodic tilings of the plane existed. When Berger proved the undecidability of the domino problem, he proved the existence of aperiodic tilings. These explorations were merely mathematical endeavors until 1984, when an experiment revealed a gaping hole in the classical theory of crystals. The result was a new theory of quasicrystals, which defined a class of crystalline solids that were capable of diffraction but did not possess translational symmetry. With the discovery of these structures, the mathematical study of aperiodic tilings became relevant to crystalography and physics. In order to study these quasicrystals mathematically, they are modeled on tilings. From these tilings an “r-discrete” equivalence relation and the associated groupoid can be constructed. From here, the groupoid can further be given a (locally compact) topological structure, which in turn produces a groupoid C*-algebra, in the sense described by J. Renault. Motivated by the “noncommutative geometry” program of A. Connes, J. Kellendonk later showed that when using the Penrose tilings to model quasicrystals, the elements of