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 Sierpiński carpet). We present a 2HAM system that finitely self-assembles the discrete Sierpiński 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.

中文翻译：

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双手砖组装模型中四边形分形的自组装

我们在研究最深入的基于瓷砖的自组装系统模型之一（称为两手瓷砖组装模型（2HAM））中考虑分形的自组装。特别是，我们将注意力集中在称为离散自相似分形的一类分形上（包括离散Sierpiński地毯的一类分形）。我们介绍了一个2HAM系统，该系统以比例因子1有限地自我组装了离散的Sierpiński地毯。此外，我们给出的2HAM系统很适合推广，我们描述了如何修改该系统以获得有限自我的2HAM系统。从一组无限的分形（我们称为4边形分形）中组合任何一个分形。我们在本文中给出的2HAM系统是在纯自增长模型中按比例因子1有限自组装离散自相似分形的系统的第一个示例。最后，我们表明存在一个3边形分形（不是树形分形），该分形不能由任何2HAM系统有限地自我组装。