Self-Assembly of Shapes at Constant Scale using Repulsive Forces

TL;DR

This paper presents a novel self-assembly algorithm that constructs arbitrary shapes at a constant scale using a 2HAM model with positive and negative interactions, achieving optimal tile complexity and minimal waste.

Contribution

It introduces a shape construction method in an extended 2HAM model that operates at constant scale with optimal tile complexity and minimal garbage, unlike previous models requiring complex features.

Findings

Constructs arbitrary shapes with optimal tile complexity

Operates at constant scale factor

Produces only constant-size garbage assemblies

Abstract

The algorithmic self-assembly of shapes has been considered in several models of self-assembly. For the problem of \emph{shape construction}, we consider an extended version of the Two-Handed Tile Assembly Model (2HAM), which contains positive (attractive) and negative (repulsive) interactions. As a result, portions of an assembly can become unstable and detach. In this model, we utilize fuel-efficient computation to perform Turing machine simulations for the construction of the shape. In this paper, we show how an arbitrary shape can be constructed using an asymptotically optimal number of distinct tile types (based on the shape's Kolmogorov complexity). We achieve this at $O(1)$ scale factor in this straightforward model, whereas all previous results with sublinear scale factors utilize powerful self-assembly models containing features such as staging, tile deletion, chemical reaction networks, and tile activation/deactivation. Furthermore, the computation and construction in our result only creates constant-size garbage assemblies as a byproduct of assembling the shape.