Showing papers on "Fractal dimension published in 1979"

Discussion paper: fractals, attractors, and the fractal dimension

Abstract: The evidence is that among the diagrams shown by the various speakers there is hardly one that is not either a confirmed or a suspected fractal, o r a t least related to a fractal. In the talks fractals are mostly referred to as “monster sets” or “strange sets,” and the like, with many of them being labeled “strange attractors.” In other circles, we hear some of them described as “exceptional sets,” “weird sets,” or “dragons.” It is, however, to be hoped that none of these terms is t o become entrenched in a technical sense. Since the Latin for “irregular and fragmented” isfructur, I coined “fractal” to denote them.* The recent recognition of their role in science has had several different and independent sources, of which the first are in Lorenz’s well-known paper of 1963 arid in my own papers.t Students of fractal attractors and the like will have to evaluate their diverse “degrees of monstrosity,” and study the relationships, if any, between their fractal and topological properties. Several alternative measures have been debated by mathematicians. Hausdorff and Besicovitch have advanced a notion of continuous dimension D, which became extremely well known to an extremely small number of very abstract-minded scholars. I claim this dimension is immensly intuitive and widely useful.* The impression that D could have no concrete application was due in part of the fact that its definition referred to the local properties of certain weird sets. However, the most interesting among these sets possess. the property of scaling, which expresses a very strong relationship between their local, global, and intermediate characteristics, and it implies that D is also capable of tackling numerically a certain important nontopological facet of overall geometric “form.” I t has been convenient to call D the fractal dimension. For any set (in a metric space), it can be shown that D 2 DT, where D is the fractal dimension and DT the topological dimension. As is fit for a dimension, the topological DT is always an integer, but “strangely” enough, we find more often than not that the fractal D is not an integer. For example, the fractal dimension of a surface is not less than 2, but it need not be exactly 2. A set is fractal if and only if D > DT.* It may be of interest that the fractal dimension of the Lorenz attractor (as estimated by Velarde on the basis of an argument due to Pomeau and Ibanez) is about