分形的多手严格自组装
Strict Self-Assembly of Fractals using Multiple Hands
摘要：
在本文中，我们考虑的问题，严格的自组装的无限分形瓷砖自组装。特别是，我们提供了瓷砖组装算法的离散谢尔宾斯基三角形和离散谢尔宾斯基地毯内的一类模型，我们称之为\emph{$h$-handed assembly model}（$h$ -HAM），它概括了2-HAM，以允许高达$h$组件在一个单一的组装步骤相结合。尽管有大量的考虑，还没有纯粹的生长自组装模型已被证明严格组装一个无限的分形没有显着修改的分形形状。在本文中，我们不仅实现了这一点，但在谢尔宾斯基地毯的情况下，能够实现它的2-HAM，在文献中研究得最好的瓷砖组装模型之一。我们的具体结果如下。我们为Sierpinski三角形提供了一个$6$ -HAM构造，它工作在比例因子1，30种瓦片类型，并以\emph{near perfect}的方式组装分形，其中所有中间组件都是递归分形的有限大小的迭代。我们进一步以比例因子3和990瓦片类型在$3$ -HAM内组装谢尔宾斯基三角形。对于Sierpinski地毯，我们给出了一个$2$ -HAM结果，该结果在比例因子3下工作，使用了1，216种瓷砖类型。我们进一步分析表明，aTAM无法严格组装本文中考虑的非树谢尔宾斯基三角形，并且需要多只手才能近乎完美地组装谢尔宾斯基三角形和谢尔宾斯基地毯。
英文摘要：
In this paper we consider the problem of the strict self-assembly of infinite fractals within tile self-assembly. In particular, we provide tile assembly algorithms for the assembly of the discrete Sierpinski triangle and the discrete Sierpinski carpet within a class of models we term the \emph{$h$-handed assembly model} ($h$-HAM), which generalizes the 2-HAM to allow up to $h$ assemblies to combine in a single assembly step. Despite substantial consideration, no purely growth self-assembly model has yet been shown to strictly assemble an infinite fractal without significant modification to the fractal shape. In this paper we not only achieve this, but in the case of the Sierpinski carpet are able to achieve it within the 2-HAM, one of the most well studied tile assembly models in the literature. Our specific results are as follows. We provide a $6$-HAM construction for the Sierpinski triangle that works at scale factor 1, 30 tile types, and assembles the fractal in a \emph{near perfect} way in which all intermediate assemblies are finite-sized iterations of the recursive fractal. We further assemble the Sierpinski triangle within the $3$-HAM at scale factor 3 and 990 tile types. For the Sierpinski carpet, we present a $2$-HAM result that works at scale factor 3 and uses 1,216 tile types. We further include analysis showing that the aTAM is incapable of strictly assembling the non-tree Sierpinski triangle considered in this paper, and that multiple hands are needed for the near-perfect assembly of the Sierpinski triangle and the Sierpinski carpet.
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