A random multifractal tilling

Abstract

We develop a multifractal random tilling that fills the square. The multifractal is formed by an arrangement of rectangular blocks of different sizes, areas and number of neighbors. The overall feature of the tilling is an heterogeneous and anisotropic random self-affine object. The multifractal is constructed by an algorithm that makes successive sections of the square. At each n-step there is a random choice of a parameter ρ i related to the section ratio. For the case of random choice between ρ 1 and ρ 2 we find analytically the full spectrum of fractal dimensions.

Semi-multifractal optimization algorithm

“…The antibody of type A produces four different clones containing antibodies B and C, while the antibody of type B produces two different clones containing antibodies A and D. The antibody of type C produces three different clones: one clone containing two antibodies of type B and two clones containing antibodies A and D. The antibody of type D produces one clone containing two antibodies of type A. Similar divisions are presented in studies (Corso et al 2004;Pereira et al 2005;Corso and Lucena 2005) as a deterministic and a random multifractal tilling.…”

“…A similar approach is found in the work (Pereira et al 2005). The difference consists in the fact that the number of objects created at a specific scale is unpredictable.…”

“…A method of the solution space division proposed in Song and Qian (2006) creates a fractal. Different ways of space division are realized in the study presented in Pereira et al (2005). This division has not an evolutionary character and creates a deterministic multifractal.…”

Percolation Properties and Universality Class of a Multifractal Random Tiling

“…In Ref. 4, it is proven that in the limit n → ∞, the RM T is in fact a multifractal and its spectrum of fractal dimensions is determined analytically. In order to develop an algorithm that simplifies the study of percolating properties, we choose section ratios of the form r/(s + r) for r and s integers.…”

“…Inspired by the modeling of natural phenomena, which always shows a random ingredient, a random version for this model was developed. 4 This is called the random multifractal tiling (RM T ). In the present work, we explore percolation as a critical phenomenon on the RM T .…”