Self-assembly of the discrete Sierpinski carpet and related fractals

📝 Abstract

It is well known that the discrete Sierpinski triangle can be defined as the nonzero residues modulo 2 of Pascal’s triangle, and that from this definition one can easily construct a tileset with which the discrete Sierpinski triangle self-assembles in Winfree’s tile assembly model. In this paper we introduce an infinite class of discrete self-similar fractals that are defined by the residues modulo a prime p of the entries in a two-dimensional matrix obtained from a simple recursive equation. We prove that every fractal in this class self-assembles using a uniformly constructed tileset. As a special case we show that the discrete Sierpinski carpet self-assembles using a set of 30 tiles.

💡 Analysis

It is well known that the discrete Sierpinski triangle can be defined as the nonzero residues modulo 2 of Pascal’s triangle, and that from this definition one can easily construct a tileset with which the discrete Sierpinski triangle self-assembles in Winfree’s tile assembly model. In this paper we introduce an infinite class of discrete self-similar fractals that are defined by the residues modulo a prime p of the entries in a two-dimensional matrix obtained from a simple recursive equation. We prove that every fractal in this class self-assembles using a uniformly constructed tileset. As a special case we show that the discrete Sierpinski carpet self-assembles using a set of 30 tiles.

📄 Content

Self-assembly of the discrete Sierpinski carpet and related fractals (Preliminary version) Steven M. Kautz Department of Computer Science Iowa State University Ames, IA 50014 U.S.A. smkautz@cs.iastate.edu James I. Lathrop Department of Computer Science Iowa State University Ames, IA 50014 U.S.A. jil@cs.iastate.edu Abstract It is well known that the discrete Sierpinski triangle can be defined as the nonzero residues modulo 2 of Pascal’s triangle, and that from this definition one can easily construct a tileset with which the discrete Sierpinski triangle self-assembles in Winfree’s tile assembly model. In this paper we introduce an infinite class of discrete self-similar fractals that are defined by the residues modulo a prime p of the entries in a two- dimensional matrix obtained from a simple recursive equation. We prove that every fractal in this class self-assembles using a uniformly constructed tileset. As a special case we show that the discrete Sierpinski carpet self-assembles using a set of 30 tiles. 1 Introduction A model for self-assembly is a computing paradigm in which many small components interact locally, without external direction, to assemble themselves into a larger structure. Wang [8, 9]. first investigated the self-assembly of patterns in the plane from a finite set of square tiles. In Wang’s model, a tile is a square with a label on each edge that determines which other tiles in the set can lie adjacent to it in the final structure. The Tile Assembly Model of Winfree [10], later revised by Rothemund and Winfree [6, 5], refines the Wang model to provide an abstraction for the physical self-assembly of DNA molecules. We introduce some formal notation for the Tile Assembly Model in the next section. Briefly, a tile is a square with a label on each edge, which we represent as a string, but in addition each edge has a integer bonding strength of 0, 1, or 2, represented in Figure 1 by a dashed line, a solid line, or a double line, respectively. A tile may also have a label in the center 1 arXiv:0901.3189v1 [cs.OH] 21 Jan 2009 C G A T T A G C C G A T G G T C G A T T A G C G C T A G G C A T A T G G C C A T A C T A G A T G C T A XYZ ABC Z123 ATILE Figure 1: Winfree tile model. for informational purposes. Tiles are assumed not to rotate. Two tiles can potentially lie adjacent to each other only if the adjacent edges have the same label and the same bonding strength. Intuitively, the bonding strength and edge label model the bonding strength and “sticky ends” of a specially constructed DNA molecule, as illustrated in Figure 1 A tile system is assumed to start with an infinite supply of a finite number of tile types. A set of initial tiles, the seed assembly, is placed in the discrete plane. Self-assembly proceeds nondeterministically as new tiles bond to the existing assembly. The ability of tiles to bond is controlled by a system parameter called the temperature. In this paper we are concerned with temperature 2 systems, which means that a tile may bond to an existing assembly only if the sum of the bonding strengths of the edges in the tile that abut the assembly is at least 2. In this process, tiles may cooperate to create a planar structure, or (by appropriate inter- pretation of the labels) perform a computation. Winfree [10] and others [6, 5, 2, 1] have shown that such systems can perform computations such as counting and addition, and that in fact the model is universal: given an arbitrary Turing machine, there is a tile set for which each row of the resulting assembly is the result of a computation step of the Turing machine. It is also possible to use a finite tile set to generate infinite planar structures such as discrete fractals. The latter was made famous when Papadakis, Rothemund and Winfree [4] performed an experiment in which actual DNA molecules were used to self-assemble a portion of the discrete Sierpinski triangle. The discrete Sierpinski triangle has been used extensively as a test structure for in DNA self-assembly [10]. One reason for this is that it self-assembles using a simple set of only 7 tiles [10]. More generally, however, fractal structures are of interest because “Structures that self-assemble in naturally occurring biological systems are often fractals of low dimension, by which we mean that they are usefully modeled as fractals and that their fractal dimensions are less than the dimension of the space or surface that they occupy. The advantages of such fractal geometries for materials transport, heat exchange, information processing, and robustness imply that structures engineered by nanoscale self-assembly in the near future 2 will also often be fractals of low dimension.” [2] It is then natural to ask what other discrete fractals, other than the ubiquitous Sierpinski triangle, can self-assemble with a relatively small set of tile types in this model. In this paper we introduce an infinite class of self-similar discrete fractals, all of which self- assemble in Winfree’s model

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