Fractal and nonfractal properties of triadic Koch curve

Fractal geometry is being used in diverse research areas and it is proving to be an increasingly useful tool. Since there is a growing interest in the applications of fractal geometry in many branches of science and art, questions about its methodology, underlying principles and meaningful use, become more and more current. The present paper deals with the conceptual and methodological aspects of fractal geometry. By means of the fractal analysis and calculus we discuss some basic concepts of fractal geometry using as an example the triadic Koch curve. We present a system of parametric equations for that fractal, and derive its capacity dimension and two main inverse-power laws. Since little evidence is available on the properties of the Koch surface, our main endeavour is directed toward investigating its characteristics. Starting from an experimentally stated hypothesis that interior of some natural objects is solid we support this hypothesis theoretically on the Koch surface. We show that the areas of that surface converge to a finite number whose value we calculated. Using the capacity dimension we demonstrate that the surface area of the limit Koch curve is structureless so that it does not belong to the "first category"' of fractals. (c) 2006 Elsevier Ltd. All rights reserved.