Optimal program-size complexity for self-assembly at temperature 1 in 3D

TL;DR

This paper demonstrates that in a 3D self-assembly model at temperature 1, it is possible to construct an N x N square with optimal tile complexity of O(log N / log log N), even with minimal 3D constraints.

Contribution

It introduces a 3D temperature 1 optimal encoding construction that achieves minimal tile complexity for square assembly, answering an open question.

Findings

Achieves optimal tile complexity for N x N squares at temperature 1

Works with tiles restricted to two planes in 3D space

Develops a general 3D encoding construction of independent interest

Abstract

Working in a three-dimensional variant of Winfree's abstract Tile Assembly Model, we show that, for all $N \in \mathbb{N}$, there is a tile set that uniquely self-assembles into an $N \times N$ square shape at temperature 1 with optimal program-size complexity of $O(\log N / \log \log N)$ (the program-size complexity, also known as tile complexity, of a shape is the minimum number of unique tile types required to uniquely self-assemble it). Moreover, our construction is "just barely" 3D in the sense that it works even when the placement of tiles is restricted to the $z = 0$ and $z = 1$ planes. This result affirmatively answers an open question from Cook, Fu, Schweller (SODA 2011). To achieve this result, we develop a general 3D temperature 1 optimal encoding construction, reminiscent of the 2D temperature 2 optimal encoding construction of Soloveichik and Winfree (SICOMP 2007), and perhaps of independent interest.