Self-assembly with Geometric Tiles

In this work we propose a generalization of Winfree's abstract Tile Assembly Model (aTAM) in which tile types are assigned rigid shapes, or geometries, along each tile face. We examine the number of distinct tile types needed to assemble shapes within this model, the temperature required for efficient assembly, and the problem of designing compact geometric faces to meet given compatibility specifications. We pose the following question: can complex geometric tile faces arbitrarily reduce the number of distinct tile types to assemble shapes? Within the most basic generalization of the aTAM, we show that the answer is no. For almost all n at least $\Omega(\sqrt{\log n})$ tile types are required to uniquely assemble an n×n square, regardless of how much complexity is pumped into the face of each tile type. However, we show for all n we can achieve a matching $O(\sqrt{\log n})$ tile types, beating the known lower bound of Θ(logn / loglogn) that holds for almost all n within the aTAM. Further, our result holds at temperature τ=1. Our next result considers a geometric tile model that is a generalization of the 2-handed abstract tile assembly model in which tile aggregates must move together through obstacle free paths within the plane. Within this model we present a novel construction that harnesses the collision free path requirement to allow for the unique assembly of any n×n square with a sleek O(loglogn) distinct tile types. This construction is of interest in that it is the first tile self-assembly result to harness collision free planar translation to increase efficiency, whereas previous work has simply used the planarity restriction as a desireable quality that could be achieved at reduced efficiency. This surprisingly low tile type result further emphasizes a fundamental open question: Is it possible to assemble n×n squares with O(1) distinct tile types? Essentially, how far can the trade off between the number of distinct tile types required for an assembly and the complexity of each tile type itself be taken?