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 system 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.

中文翻译：

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分层增长是必要的，并且（有时）足以自组装离散的自相似分形

在本文中，我们证明了在抽象的瓷砖装配模型（aTAM）中，这是一个基于增生的模型，该模型仅允许单个瓷砖在每个步骤上附着到正在增长的装配上，没有能够自我装配的瓷砖装配系统离散的自相似分形，称为“ H”和“ U”形。然后，我们表明，在允许分层自组装的相关模型2手组装模型（2HAM）中，确实存在一个瓦片组装系统，该系统可以对“ U”形分形和猜想进行自我组装，而对于这种情况，同样适用“ H”形。这是离散的自相似分形的第一个示例，该分形在2HAM中而不在aTAM中自组装，从而提供了模型的直接比较以及对分层组装的强大理解。