Polyominoes Simulating Arbitrary-Neighborhood Zippers and Tilings

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.

💡 Research Summary

The paper bridges classical tiling theory with the more intricate neighborhood relations that arise in practical self‑assembly systems. In a generalized tile system a tile set T, a neighborhood N (any finite set of integer vectors) and a relation R⊆T×T×N specify which tiles may appear adjacent to each other. Traditional models restrict N to the four cardinal directions, but many physical processes involve longer‑range or asymmetric interactions.

A “zipper” is defined as a path of tiles that respects the relation R: each tile in the sequence is adjacent to its predecessor by exactly one vector from N. Zippers model linear growth patterns such as polymer chains or filamentous nanostructures. The authors’ first main result shows that any zipper, no matter how complex the underlying neighborhood, can be simulated by a “ribbon” – a non‑self‑crossing polyomino consisting of unit squares (microtiles) that attach only along shared edges, i.e., a standard 4‑direction tile system.

The construction proceeds in two stages. First, each original tile t∈T is replaced by a unique polyomino block B(t). The block contains a set of “attachment ports”, one for every vector v∈N that t may use. Ports are realized as specific glue labels on the edges of the microtiles forming the block. Second, the zipper’s order (t₁,…,t_k) is reproduced by laying the blocks in the same sequence, aligning the appropriate ports of B(t_i) and B(t_{i+1}). Because the internal geometry of each block forces the ports to line up exactly, the resulting ribbon respects the original relation R while using only the four orthogonal adjacencies.

The second major theorem extends this encoding from a single path to an entire tiling of the plane. The authors introduce “patches”, finite regions of the original tiling, each of which is converted into a collection of blocks whose boundary ports match those of neighboring patches. By tiling the plane with these converted patches, the global structure of the original N‑neighborhood tiling is preserved: connectivity, periodicity, and determinism remain intact.

Complexity analysis shows that the number of distinct microtiles required grows linearly with |T|·|N|, and the overall construction can be carried out in polynomial time. Thus the method is theoretically efficient and amenable to practical implementation.

To illustrate the theory, two concrete examples are presented. The first converts a system with an L‑shaped neighborhood into a ribbon that reproduces a spiral zipper; the second handles a hexagonal‑grid neighborhood, demonstrating that the periodic pattern of the original tiling survives the transformation. Both cases confirm that the simulated ribbon or converted tiling behaves indistinguishably from the source system with respect to the intended assembly rules.

In conclusion, the authors provide a systematic way to replace arbitrary‑neighborhood self‑assembly models with ordinary 4‑direction tile systems without losing essential structural properties. This opens the door to applying the rich toolbox of classical tiling theory—such as decidability results, aperiodicity constructions, and algorithmic synthesis—to more realistic self‑assembly scenarios. Future work is suggested in optimizing the size of the polyomino blocks, extending the approach to three dimensions, and integrating the constructions with experimental DNA‑ or protein‑based nanofabrication pipelines.