# Self-Assembly of 4-sided Fractals in the Two-handed Tile Assembly Model

## TL;DR

This paper demonstrates the first 2HAM systems capable of finitely self-assembling discrete self-similar fractals, including the Sierpiński carpet, at scale factor 1, and explores limitations for certain 3-sided fractals.

## Contribution

It introduces novel 2HAM systems that finitely self-assemble 4-sided fractals at scale factor 1, expanding understanding of self-assembly capabilities.

## Key findings

- Finitely self-assembled the discrete Sierpiński carpet in 2HAM.
- Generalized the system to assemble any 4-sided fractal.
- Proved some 3-sided fractals cannot be finitely self-assembled.

## Abstract

We consider the self-assembly of fractals in one of the most well-studied models of tile based self-assembling systems known as the Two-handed Tile Assembly Model (2HAM). In particular, we focus our attention on a class of fractals called discrete self-similar fractals (a class of fractals that includes the discrete Sierpi\'nski carpet). We present a 2HAM system that finitely self-assembles the discrete Sierpi\'nski carpet with scale factor 1. Moreover, the 2HAM system that we give lends itself to being generalized and we describe how this system can be modified to obtain a 2HAM system that finitely self-assembles one of any fractal from an infinite set of fractals which we call 4-sided fractals. The 2HAM systems we give in this paper are the first examples of systems that finitely self-assemble discrete self-similar fractals at scale factor 1 in a purely growth model of self-assembly. Finally, we show that there exists a 3-sided fractal (which is not a tree fractal) that cannot be finitely self-assembled by any 2HAM system.