Polyominoes Simulating Arbitrary-Neighborhood Zippers and Tilings

TL;DR

This paper introduces a method to simulate complex neighborhood tilings using simple polyominoes, bridging classical tiling theory with practical self-assembling systems and enabling the replication of arbitrary neighborhood interactions.

Contribution

It demonstrates that any valid path in an arbitrary neighborhood can be simulated by a simple ribbon of microtiles, extending to traditional tilings and preserving key properties.

Findings

Any valid path in an arbitrary neighborhood can be simulated by a simple ribbon.

Arbitrary neighborhood tilings can be represented by simple-neighborhood tilings.

The construction preserves essential properties of the original tilings.

Abstract

This paper provides a bridge between the classical tiling theory and the complex neighborhood self-assembling situations that exist in practice. The neighborhood of a position in the plane is the set of coordinates which are considered adjacent to it. 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 validly tiled path 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 sequence of tiles on the plane, in which successive tiles stick along their adjacent edge. 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.