Geometry, dynamics and fractals

Abstract

Consider a collection of elastic wires folded according to a given pattern induced by a sequence of fractal plane curves. The folded wires can act as elastic springs. Therefore it is easy to build up a corresponding sequence of simple oscillators composed by the elastic springs clamped at one end and carrying a mass at the opposite end. The oscillation periods of the ordered sequence of these oscillators are related following a power law and therefore display a fractal structure. The periods of each oscillator clearly depend on the mechanical properties of the wire, on the mass at the end and on the boundary conditions. Therefore there are infinitely many possibilities to design a dynamical fractal sequence in opposition to the well defined fractal dimension of the underneath geometric sequence. Nevertheless the geometric fractal dimension of the primordial geometric curve is always related somehow to the dynamical fractal dimension characterizing the oscillation period sequence. It is important to emphasize that the dynamical fractal dimension of a given sequence built up after the geometry of a primordial one is not unique. This peculiarity introduces the possibility to have a broader information spectrum about the geometry which is otherwise impossible to achieve. This effect is clearly demonstrated for random fractals. The present paper deals with a particular family of curves, namely curves belonging to the Koch family. The method is tested for the simple Koch triadic and for random Koch curves. The method has also proved to be useful to identify the fractal dimension of a sequence given just one of its terms. Remarkable is the quality of information obtained with this technique based on very simple and basic concepts. Some of these aspects will be presented in this paper but much more, the authors believe, is still hidden behind the dynamic properties of fractal structures.