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Sierpinski triangle

About: Sierpinski triangle is a research topic. Over the lifetime, 1965 publications have been published within this topic receiving 26478 citations.


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TL;DR: This work reports the molecular realization, using two-dimensional self-assembly of DNA tiles, of a cellular automaton whose update rule computes the binary function XOR and thus fabricates a fractal pattern—a Sierpinski triangle—as it grows.
Abstract: Algorithms and information, fundamental to technological and biological organization, are also an essential aspect of many elementary physical phenomena, such as molecular self-assembly. Here we report the molecular realization, using two-dimensional self-assembly of DNA tiles, of a cellular automaton whose update rule computes the binary function XOR and thus fabricates a fractal pattern—a Sierpinski triangle—as it grows. To achieve this, abstract tiles were translated into DNA tiles based on double-crossover motifs. Serving as input for the computation, long single-stranded DNA molecules were used to nucleate growth of tiles into algorithmic crystals. For both of two independent molecular realizations, atomic force microscopy revealed recognizable Sierpinski triangles containing 100–200 correct tiles. Error rates during assembly appear to range from 1% to 10%. Although imperfect, the growth of Sierpinski triangles demonstrates all the necessary mechanisms for the molecular implementation of arbitrary cellular automata. This shows that engineered DNA self-assembly can be treated as a Turing-universal biomolecular system, capable of implementing any desired algorithm for computation or construction tasks.

916 citations

Journal ArticleDOI
TL;DR: The notion of spectral dimensionality of a self-similar (fractal) structure is recalled and its value for the family of Sierpinski gaskets derived via a scaling argument is derived in this paper.
Abstract: The notion of spectral dimensionality of a self-similar (fractal) structure is recalled, and its value for the family of Sierpinski gaskets derived via a scaling argument. Various random walk properties such as the probability of closed walks and the mean number of visited sites are shown to be governed by this spectral dimension. It is suggested that the number SN of distinct sites visited during an N-step random walk on an infinite cluster at percolation threshold varies asymptotically as : SN ∼ N2/3, in any dimension.

740 citations

Journal ArticleDOI
TL;DR: In this paper, the Sierpinski gasket has been used to construct a Brownian motion, a diffusion process characterized by local isotropy and homogeneity properties, and it is shown that the process has a continuous symmetric transition density, p
Abstract: We construct a “Brownian motion” taking values in the Sierpinski gasket, a fractal subset of ℝ2, and study its properties. This is a diffusion process characterized by local isotropy and homogeneity properties. We show, for example, that the process has a continuous symmetric transition density, p t(x,y), with respect to an appropriate Hausdorff measure and obtain estimates on p t(x,y).

539 citations

Journal ArticleDOI
TL;DR: The main object of as discussed by the authors is the renormalization of difference operators on post-critically finite (p.c.f. for short) self-similar sets, which are large enough to include finitely ramified selfsimilar sets.
Abstract: The main object of this paper is the Laplace operator on a class of fractals. First, we establish the concept of the renormalization of difference operators on post critically finite (p.c.f. for short) self-similar sets, which are large enough to include finitely ramified self-similar sets, and extend the results for Sierpinski gasket given in [10] to this class. Under each invariant operator for renormalization, the Laplace operator, Green function, Dirichlet form, and Neumann derivatives are explicitly constructed as the natural limits of those on finite pre-self-similar sets which approximate the p.c.f. self-similar sets. Also harmonic functions are shown to be finite dimensional, and they are characterized by the solution of an infinite system of finite difference equations

399 citations

Journal ArticleDOI
TL;DR: In this article, a fractal model of soil texture and pore structure is proposed based on the concept of fractal geometry, which is used for the Sierpinski carpet pore size distribution.
Abstract: Numerous empirical models exist for soil water retention and unsaturated hydraulic conductivity data. It has generally been recognized that the empirical fitting coefficients in these models are somehow related to soil texture. However, the fact that they are empirical means that elaborate laboratory experiments must be performed for each soil to obtain values for the parameters. Moreover, empirical models do not shed insight into the fundamental physical principles that govern the processes of unsaturated flow and drainage. We propose a physical conceptual model for soil texture and pore structure that is based on the concept of fractal geometry. The motivation for a fractal model of soil texture is that some particle size distributions in granular soils have already been shown to display self-similar scaling that is typical of fractal objects. Hence it is reasonable to expect that pore size distributions may also display fractal scaling properties. The paradigm that we use for the soil pore size distribution is the Sierpinski carpet, which is a fractal that contains self similar “holes” (or pores) over a wide range of scales. We evaluate the water retention properties of regular and random Sierpinski carpets and relate these properties directly to the Brooks and Corey (or Campbell) empirical water retention model. We relate the water retention curves directly to the fractal dimension of the Sierpinski carpet and show that the fractal dimension strongly controls the water retention properties of the Sierpinski carpet “soil”. Higher fractal dimensions are shown to mimic clay-type soils, with very slow dewatering characteristics and relatively low fractal dimensions are shown to mimic a sandy soil with relatively rapid dewatering characteristics. Our fractal model of soil water retention removes the empirical fitting parameters from the soil water retention models and provides parameters (fractal dimension) which are intrinsic to the nature of the fractal porous structure. The relative permeability functions of Burdine and Mualem are also shown to be fractal directly from fractal water retention results.

354 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202362
2022112
202181
202092
2019100
2018103