Showing papers on "n-flake published in 2012"
TL;DR: In particular, Sierpinski gasket and von Koch flake are explicitly obtained by simplex transformations as mentioned in this paper. But this is not the case in the case of fractal lattices, which are defined by iterative maps on a simplex.
Abstract: A fractal lattice is defined by iterative maps on a simplex. In particular, Sierpinski gasket and von Koch flake are explicitly obtained by simplex transformations.
14 citations
TL;DR: In this article, the authors proposed a fractal dimension for the Cantor sect and showed that fractal dimensions can be used for fractal box counting, which is an important technique.
Abstract: Keywords: Cantor sect; fractal; fractal dimension; box-counting dimension DOI: http://dx.doi.org/10.3329/bjsir.v46i4.9598 BJSIR 2011; 46(4): 499-506
5 citations
TL;DR: In this article, a uniform method for estimating fractal characteristics of systems satisfying some type of scaling principle is proposed, based on representing such systems as generating Bethe-Cayley tree graphs.
Abstract: We propose a uniform method for estimating fractal characteristics of systems satisfying some type of scaling principle. This method is based on representing such systems as generating Bethe-Cayley tree graphs. These graphs appear from the formalism of the group bundle of Fibonacci-Penrose inverse semigroups. We consistently consider the standard schemes of Cantor and Koch in the new method. We prove the fractal property of the proper Fibonacci system, which has neither a negative nor a positive redundancy type. We illustrate the Fibonacci fractal by an original procedure and in the coordinate representation. The golden ratio and specific inversion property intrinsic to the Fibonacci system underlie the Fibonacci fractal. This property is reflected in the structure of the Fibonacci generator.
4 citations
TL;DR: Borders are tight, even when restricted to supersets of the Sierpinski triangle, by presenting a strict self-assembly that adds communication fibers to the fractal structure without disturbing it.
Abstract: The Tile Assembly Model is a Turing universal model that Winfree introduced in order to study the nanoscale self-assembly of complex DNA crystals. Winfree exhibited a self-assembly that tiles the first quadrant of the Cartesian plane with specially labeled tiles appearing at exactly the positions of points in the Sierpinski triangle. More recently, Lathrop, Lutz, and Summers proved that the Sierpinski triangle cannot self-assemble in the “strict” sense in which tiles are not allowed to appear at positions outside the target structure. Here we investigate the strict self-assembly of sets that approximate the Sierpinski triangle. We show that every set that does strictly self-assemble disagrees with the Sierpinski triangle on a set with fractal dimension at least that of the Sierpinski triangle (≈1.585), and that no subset of the Sierpinski triangle with fractal dimension greater than 1 strictly self-assembles. We show that our bounds are tight, even when restricted to supersets of the Sierpinski triangle, by presenting a strict self-assembly that adds communication fibers to the fractal structure without disturbing it. To verify this strict self-assembly we develop a generalization of the local determinism method of Soloveichik and Winfree.