Optimal self-assembly of finite shapes at temperature 1 in 3D

TL;DR

This paper demonstrates that any finite, connected 2D shape can be uniquely assembled in a nearly 3D temperature 1 tile assembly model with optimal complexity, closely simulating 2D temperature 2 results.

Contribution

It introduces a minimal 3D tile assembly construction that achieves optimal shape complexity at temperature 1, extending 2D temperature 2 results to a nearly 3D setting.

Findings

Achieves optimal tile complexity for shape assembly

Constructs a nearly 3D assembly in only two planes

Simulates 2D temperature 2 assembly in 3D at temperature 1

Abstract

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 a scaled-up version of $X$ at temperature 1 in 3D 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).