Polyominoes simulating arbitrary-neighborhood zippers and tilings

This paper provides a bridge between the classical tiling theory and cellular automata on one side, and the complex neighborhood self-assembling situations that exist in practice, on the other side. A neighborhood N is a finite set of pairs (i, j) ∈ Z, indicating that the neighbors of a position (x, y) are the positions (x + i, y + j) for (i, j) ∈ N . This includes classical neighborhoods of size four, as well as arbitrarily complex neighborhoods. A generalized tile system consists of a set of tiles, a neighborhood, and a relation which dictates which are the “admissible” neighboring tiles of a given tile. Thus, in correctly formed assemblies, tiles are assigned positions of the plane in accordance to this relation. We prove that any path filled with tiles defined in a given but arbitrary neighborhood (a zipper) can be simulated by a simple “ribbon” of microtiles. A ribbon is a special kind of polyomino, consisting of a non-self-crossing rectilinear sequence of tiles on the plane, in which successive tiles are adjacent along an edge, and where each tile needs to match glues with only two other tiles: its predecessor and its successor on the path. Our constructions simulate each of the existing tiles by a polyomino of microtiles, whose shape is used to simulate the given tile and the communication of information between itself and its neighbors. The polyominoes can then be catenated together to simulate the entire complexneighborhood tiled path by a continuous two-tile-neighborhood ribbon. Finally, we extend this construction to the case of traditional tilings, proving that we can simulate arbitrary-neighborhood tilings by simple-neighborhood tilings, while preserving some of their essential properties.