Scaled Tree Fractals Do not Strictly Self-assemble

Barth, Kimberly; Furcy, David; Summers, Scott M.; Totzke, Paul

doi:10.1007/978-3-319-08123-6_3

Cited by 13 publications

(43 citation statements)

References 11 publications

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“…By definition of the Sierpinski triangle, there are three substages of each stage that correspond to the three points in the generator, i.e. (0, 0), (1,0), and (0, 1). The nondeterministic binding of one of the initiator types initiates an assembly sequence which grows either a tooth or gap on the assembly to which it is attached.…”

Section: Discrete Self-similar Fractalsmentioning

confidence: 99%

“…Because of this, and the complex aperiodic nature of fractals, they are a natural target of study during the development of artificial self-assembling systems. As one of the first mathematical abstractions of self-assembling systems, Winfree's abstract Tile Assembly Model (aTAM) [18] has been the platform for several results showing the impossibility of self-assembling discrete self-similar fractals such as the Sierpinski triangle 1 [13] and similar fractals [1], and also for designing systems which can approximate them [13,14,17]. In a more generalized model called the 2-Handed Assembly Model [4] (2HAM, a.k.a.…”

Section: Introductionmentioning

confidence: 99%

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Hierarchical self-assembly of fractals with signal-passing tiles

et al. 2017

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In this paper, we present high-level overviews of tile-based self-assembling systems capable of producing complex, infinite, aperiodic structures known as discrete self-similar fractals. Fractals have a variety of interesting mathematical and structural properties, and by utilizing the bottom-up growth paradigm of self-assembly to create them we not only learn important techniques for building such complex structures, we also gain insight into how similar structural complexity arises in natural self-assembling systems. Our results fundamentally leverage hierarchical assembly processes, and use as our building blocks square "tile" components which are capable of activating and deactivating their binding "glues" a constant number of times each, based only on local interactions. We provide the first constructions capable of building arbitrary discrete self-similar fractals at scale factor 1, and many at temperature 1 (i.e. "non-cooperatively"), including the Sierpinski triangle.

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Section: Discrete Self-similar Fractalsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Hierarchical self-assembly of fractals with signal-passing tiles

et al. 2017

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“…It is used to fool any claimed non-cooperative simulation of cooperative binding. This lemma has since found use elsewhere and indeed has been generalized in various ways [49][50][51][52][53]. An interesting aspect of the negative result is that it holds for 3D noncooperative systems; they too cannot simulate arbitrary tile assembly systems.…”

Section: A Complexity Theory For Self-assembly: the Abstract Tile Assmentioning

confidence: 99%

Intrinsic universality and the computational power of self-assembly

Woods

2015

Phil. Trans. R. Soc. A.

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Molecular self-assembly, the formation of large structures by small pieces of matter sticking together according to simple local interactions, is a ubiquitous phenomenon. A challenging engineering goal is to design a few molecules so that large numbers of them can self-assemble into desired complicated target objects. Indeed, we would like to understand the ultimate capabilities and limitations of this bottom-up fabrication process. We look to theoretical models of algorithmic self-assembly, where small square tiles stick together according to simple local rules in order to carry out a crystal growth process. In this survey, we focus on the use of simulation between such models to classify and separate their computational and expressive powers. Roughly speaking, one model simulates another if they grow the same structures, via the same dynamical growth processes. Our journey begins with the result that there is a single intrinsically universal tile set that, with appropriate initialization and spatial scaling, simulates any instance of Winfree's abstract Tile Assembly Model. This universal tile set exhibits something stronger than Turing universality: it captures the geometry and dynamics of any simulated system in a very direct way. From there we find that there is no such tile set in the more restrictive non-cooperative model, proving it weaker than the full Tile Assembly Model. In the two-handed model, where large structures can bind together in one step, we encounter an infinite set of infinite hierarchies of strictly increasing simulation power. Towards the end of our trip, we find one tile to rule them all: a single rotatable flipable polygonal tile that simulates any tile assembly system. We find another tile that aperiodically tiles the plane (but with small gaps). These and other recent results show that simulation is giving rise to a kind of computational complexity theory for self-assembly. It seems this could be the beginning of a much longer journey, so directions for future work are suggested.

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“…Aperiodic structures are theoretically fundamental to the concept of Turing universal computation as well as embodied in many mathematical and natural systems as fractals. In fact, the complex aperiodic structure of fractals, as well as their pervasiveness in nature, have inspired much previous work on the self-assembly of fractal structures [7,20], especially discrete self-similar fractals (DSSF's) [1,8,12,13,17,19,20]. In a tribute to their complex structure, previous work has shown the impossibility of self-assembly of several DSSF's in the aTAM and 2HAM [1,12,13,15,17,19,20] yet there have also been results showing some models and systems in which their self-assembly is possible [3,8,10,11].…”

Section: Introductionmentioning

confidence: 99%

Hierarchical growth is necessary and (sometimes) sufficient to self-assemble discrete self-similar fractals

et al. 2019

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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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Scaled Tree Fractals Do not Strictly Self-assemble

Abstract: In this paper, we show that any scaled-up version of any discrete self-similar tree fractal does not strictly self-assemble, at any temperature, in Winfree's abstract Tile Assembly Model.

Cited by 13 publications

References 11 publications

Hierarchical self-assembly of fractals with signal-passing tiles

Hierarchical self-assembly of fractals with signal-passing tiles

Intrinsic universality and the computational power of self-assembly

Hierarchical growth is necessary and (sometimes) sufficient to self-assemble discrete self-similar fractals

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