Optimal Self-assembly of Finite Shapes at Temperature 1 in 3D

Working in a three-dimensional variant of Winfree’s abstract Tile Assembly Model, we show that, for an arbitrary finite, connected shape \(X \subset \mathbb {Z}^2\), there is a tile set that uniquely self-assembles into a 3D representation of X at temperature 1 with optimal program-size complexity (the program-size complexity, also known as tile complexity, of a shape is the minimum number of tile types required to uniquely self-assemble it). Moreover, our construction is “just barely” 3D in the sense that it only places tiles in the \(z = 0\) and \(z = 1\) planes. Our result is essentially a just-barely 3D temperature 1 simulation of a similar 2D temperature 2 result by Soloveichik and Winfree (SICOMP 2007).