The abstract Tile-Assembly Model (aTAM) was initially introduced as a simple model for DNA-based self-assembly, where synthetic strands of DNA are used not as an information storage medium, but rather a material for nano-scale construction. Since then, it has been shown that the aTAM, and variant models thereof, exhibit rich computational dynamics, Turing completeness, and intrinsic universality, a geometric notion of simulation wherein one aTAM system is able to simulate every other aTAM system not just symbolically, but also geometrically. An intrinsically universal system is able to simulate all other systems within some class so that $m\times m$ blocks of tiles behave in all ways like individual tiles in the system to be simulated. In this paper, we explore the notion of a quine in the aTAM with respect to intrinsic universality. Typically a quine refers to a program which does nothing but print its own description with respect to a Turing universal machine which may interpret that description. In this context, we replace the notion of machine with that of an aTAM system and the notion of Turing universality with that of intrinsic universality. Curiously, we find that doing so results in a counterexample to a long-standing conjecture in the theory of tile-assembly, namely that discrete self-similar fractals (DSSFs), fractal shapes generated via substitution tiling, cannot be strictly self-assembled. We find that by growing an aTAM quine, a tile system which intrinsically simulates itself, DSSF structure is naturally exhibited. This paper describes the construction of such a quine and even shows that essentially any desired fractal dimension between 1 and 2 may be achieved.

翻译：抽象Tile组装模型（aTAM）最初是作为基于DNA自组装的简化模型提出的，其中合成DNA链并非作为信息存储介质，而是用作纳米尺度构建的材料。此后研究表明，aTAM及其变体模型展现出丰富的计算动力学特性、图灵完备性以及内禀通用性——这是一种几何模拟概念，使得某个aTAM系统不仅能符号化地模拟其他所有aTAM系统，还能实现几何结构的精确模拟。内禀通用系统能够模拟某类别中的所有其他系统，其$m\times m$区块在各方面表现得如同被模拟系统中的单个Tile。本文在aTAM框架内结合内禀通用性探讨了Quine（自产生程序）的概念。传统Quine指在可解释描述的图灵通用机器上仅输出自身描述的程序。在此背景下，我们将“机器”替换为aTAM系统，将“图灵通用性”替换为“内禀通用性”。有趣的是，这一转换推翻了Tile组装理论中长期存在的猜想：离散自相似分形（通过替换铺砌生成的分形结构）无法被严格自组装。我们发现通过生长aTAM Quine（即能内禀模拟自身的Tile系统），DSSF结构会自然呈现。本文详细描述了此类Quine的构造方法，并证明其可实现1到2之间任意分形维度的生成。