A Random Multifractal Tilling

TL;DR

This paper introduces a novel multifractal tiling method that creates a heterogeneous, anisotropic, self-affine pattern filling a square, with an analytical spectrum of fractal dimensions for certain random parameters.

Contribution

It develops a new algorithm for constructing a multifractal tiling with random parameters, providing analytical results for its fractal spectrum.

Findings

Analytical derivation of the full spectrum of fractal dimensions.

Construction of a multifractal tiling filling a square.

Demonstration of heterogeneity and anisotropy in the pattern.

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 $\rho_i$ related to the section ratio. For the case of random choice between $\rho_1$ and $\rho_2$ we find analytically the full spectrum of fractal dimensions.