Dynamical Fractal Dimension: Direct and Inverse Problems

This paper presents a new method to determine the fractal dimension of plane curves. Two main problems are described, namely the direct and the inverse problem. It is shown that the periods of harmonic oscillators obtained by folding wire like structures with the same geometry as the terms of a fractal sequence provide the necessary information to determine the fractal characteristics of that sequence. More interesting is the determination of the fractal properties of a given curve from a sequence of samples cut off the original curve. Building up harmonic oscillators with these samples it is shown that fractal characteristics can be identified. The dynamical response depends on the initial conditions providing complementary information about the fractal characteristics of the object. The last section deals with a simple random fractal and confirms the use of different initial conditions to characterize completely the fractal sequence.