Self-assembly of patterns in the abstract tile assembly model

In the abstract Tile Assembly Model, self-assembling systems consisting of tiles of different colors can form structures on which colored patterns are "painted." We explore the complexity, in terms of the numbers of unique tile types required, of assembling various patterns. We first demonstrate how to efficiently self-assemble a set of simple patterns, then show tight bounds on the tile type complexity of self-assembling multi-colored patterns on the surfaces of square assemblies. Finally, we demonstrate an exponential gap in tile type complexity of self-assembling an infinite series of patterns between systems restricted to one plane versus those allowed two planes. This paper is an expansion over a version in the proceedings of the 21st International Conference on Unconventional Computation and Natural Computation (UCNC 2024), with several improved results and full details of all proofs.