分层生长是自组装离散自相似分形的必要和（有时）充分条件
Hierarchical Growth is Necessary and (Sometimes) Sufficient to Self-Assemble Discrete Self-Similar Fractals
摘要：
在本文中，我们证明，在抽象的瓷砖组装模型（aTAM），一个基于增生的模型，只允许一个单一的瓷砖附加到一个不断增长的组件在每一步，有没有瓷砖组装系统能够自组装的离散自相似分形称为“H”和“U”分形。然后，我们表明，在一个相关的模型，它允许分层自组装，2-Handed组装模型（2 HAM），确实存在一个瓷砖组装系统，自组装的“U”分形和猜想，同样适用于“H”分形。这是离散自相似分形的第一个例子，它在2 HAM中自组装，但不在aTAM中，提供了模型的直接比较，并更好地理解了分层组装的力量。
英文摘要：
In this paper, we prove that in the abstract Tile Assembly Model (aTAM), an accretion-based model which only allows for a single tile to attach to a growing assembly at each step, there are no tile assembly systems capable of self-assembling the discrete self-similar fractals known as the "H" and "U" fractals. We then show that in a related model which allows for hierarchical self-assembly, the 2-Handed Assembly Model (2HAM), there does exist a tile assembly systems which self-assembles the "U" fractal and conjecture that the same holds for the "H" fractal. This is the first example of discrete self similar fractals which self-assemble in the 2HAM but not in the aTAM, providing a direct comparison of the models and greater understanding of the power of hierarchical assembly.
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