Fractals

In this paper, we reveal a remarkable connection between the Thue-Morse sequence and the Koch snowflake. Using turtle geometry and polygon maps, we realize the Thue-Morse sequence as the limit of polygonal curves in the plane. We then prove that a sequence of such curves converges to the Koch snowflake in the Hausdorff metric. In the final section we consider generalized Thue-Morse sequences and provide a characterization of those that encode curves converging to the Koch snowflake.

Two-dimensional (2-D) fractal lattices, that can be representative of fracture surface/aperture roughness or 2-D permeability distribution in a porous medium, are generated using fractional Brownian motion (fBm) for different grid sizes. Invasion percolation simulations are applied on these lattices to investigate the influences of the grid size on displacement. It is found that the selection of the number of grids for computer experimentation of displacement processes is critical. The fractal dimension of the invasion cluster decreases with increasing fractal dimension of the lattice and the number of grids of the lattice. In modeling the displacement processes in fractal porous media or a single fracture between two rough surfaces, consistent fractal behavior is expected any value of the number of grids above 128 × 128. Hence, one should be careful in selecting the number of points in measuring the permeability distribution in a 2-D porous medium or roughness of the fracture surfaces.

A new algorithm for the solution of the inverse problem concerning interpolatory iterated functions systems (IFS) is presented. It combines the advantages of the greedy least squares approach of Mazel and Hayes, and the self-affinity in the wavelet scalogram. Thus, the algorithm is computationally cheap and yields results with high accuracy.

We present a method of construction of vector valued bivariate fractal interpolation functions on random grids in ℝ². Examples and applications are also included.

The continuity of the map φ : c → dim_H(J_c) where J_c is the Julia set of the map f_c(z) = z² + c is discussed in this paper. Using the Hausdorff dimension, a partial order is defined on the Mandelbrot set M. Some properties of irrational points defined by this order are also studied.

It is a well-known fact that when visualizing an IFS-attractor through the chaos game, it is possible that the first points mapped will come closer to but stay visibly different from the attractor. This simple fact will be analyzed in more detail, through visualizations of different aspects of this convergence process. It will be shown that, in applying on every point in a 2D-plane the same sequence of mappings and coloring each point according to convergence distance, neighboring points form structures which resemble the attractor itself. Further, it is in this way possible to generate boundaries of the attractor that vary between small and coarse-grained. Using these results, it will be shown that it is possible to, starting with an IFS-attractor, construct fractals of which this IFS-attractor is a subset.

Tanpura is a multi-stringed accompanying drone instrument extensively used in classical music in India. The instrument is plucked by finger. We know that jitter, shimmer and complexity perturbations (CP) are found also in tanpura signals. The source of origin of these perturbations was reported to be related to some sort of nonlinearity associated with the strings and their mode of attachment.

The objective of the present study is to see in what way fractal-dimensional analysis may be helpful to relate the apparent nonlinearity and also if there is any relation of these dimensions with different part of the signals like attack time, quasi-steady state and the decay.

In the paper fractal dimension, D₀ and generalized dimensions, D₂ to D₁₀ and their dynamic behavior over time are studied for 15 tanpuras having four strings each, tuned to Pa (5th note), Sa (tonic), Sa (tonic) and Sa′ (lower octave tonic). The obeying of power law indicates nonlinearity in the production source system.

The presence of multifractality is studied through an examination of relationship between q and D_q and the functional relationship between D_qs.