双手瓦片组装模型中四边形的自组装
Self-Assembly of 4-sided Fractals in the Two-handed Tile Assembly Model
摘要：
我们考虑了基于瓷砖的自组装系统中研究最充分的模型之一（称为双手瓷砖组装模型（2 HAM））中的分形自组装。特别是，我们把注意力集中在一类分形称为离散自相似分形（一类分形，包括离散Sierpienski地毯）。我们提出了一个2HAM系统，自组装的离散Sierpienski地毯与比例因子1。此外，2HAM系统，我们给自己被推广，我们描述了如何可以修改这个系统，以获得一个2HAM系统，该系统可以自组装的任何分形从一个无限集的分形，我们称之为4边分形。本文给出的2HAM系统是在自组装的纯生长模型中以比例因子1自组装离散自相似分形的系统的第一个例子。最后，我们证明了存在一个3边分形（这不是一个树分形），不能被任何2HAM系统自组装。
英文摘要：
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.
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