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

Tile self-assembly in which tiles may bind in a non-cooperative fashion is often referred to as “temperature 1 self-assembly” or simply “non-cooperative self-assembly”. In this type of self-assembly, a tile may non-cooperatively bind to an assembly via (at least) one of its sides, unlike in cooperative self-assembly, in which some tiles may be required to bind on two or more sides. Cooperative self-assembly leads to highly non-trivial theoretical behavior but two-dimensional non-cooperative self-assembly is conjectured to be only capable of producing highly-regular shapes and patterns, which, in general, cannot be interpreted as complex computation. Remarkably, Cook et al. (Temperature 1 self-assembly: deterministic assembly in 3D and probabilistic assembly in 2D. In: Proceedings of the 22nd Annual ACM–SIAM Symposium on Discrete Algorithms 2011 ) showed that three-dimensional non-cooperative self-assembly is essentially as powerful as cooperative self-assembly, with respect to Turing universality and self-assembly of N × N squares. In this paper, we consider the non-cooperative self-assembly of just-barely 3D shapes. Working in a three-dimensional variant of Winfree’s abstract Tile Assembly Model, we show that, for an arbitrary finite, connected shape X ⊂ℤ^2 , there is a tile set that uniquely self-assembles at temperature 1 into a 3D representation of a scaled-up version of X with optimal program-size complexity, where 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 (SIAM J Comput 36(6):1544–1569, 2007 ).