# Non-cooperatively assembling large structures: a 2D pumping lemma cannot be as powerful as its 1D counterpart

## TL;DR

This paper demonstrates the first asymptotically efficient constructions in 2D non-cooperative tile assembly, showing it can grow large assemblies with minimal tile types, contrasting previous limitations on its computational power.

## Contribution

It proves that 2D non-cooperative tile assembly can reliably produce large structures efficiently, challenging prior assumptions about its computational limitations.

## Key findings

- Achieved Omega(n log n) assembly growth with n tile types
- First asymptotically efficient construction in 2D non-cooperative model
- Contrasts with previous results showing limited computational power

## Abstract

We show the first asymptotically efficient constructions in the so-called "noncooperative planar tile assembly" model. Algorithmic self-assembly is the study of the local, distributed, asynchronous algorithms ran by molecules to self-organise, in particular during crystal growth. The general cooperative model, also called "temperature 2", uses synchronisation to simulate Turing machines, build shapes using the smallest possible amount of tile types, and other algorithmic tasks. However, in the non-cooperative ("temperature 1") model, the growth process is entirely asynchronous, and mostly relies on geometry. Even though the model looks like a generalisation of finite automata to two dimensions, its 3D generalisation is capable of performing arbitrary (Turing) computation, and of universal simulations, whereby a single 3D non-cooperative tileset can simulate the dynamics of all possible 3D non-cooperative systems, up to a constant scaling factor. However, it was shown that the original 2D non-cooperative model is not capable of universal simulations, and the question of its computational power is still widely open. Here, we show an unexpected result, namely that this model can reliably grow assemblies of size Omega(n log n) with only n tile types, which is the first asymptotically efficient positive construction.