ABSTRACT
Rare are the articles or books that analyze the connection between the mathematical beings defined and manipulated by the geometricians, and the "natural" fractals identified in the real world. Reluctance to embark on this analysis appears to have caused much confusion in the literature; the term "fractal", when applied to natural systems, often means different things to different people, creating unnecessary difficulties in communication. In particular, K. J. Falconer may be consulted in order to get up to speed; they introduce the mathematical concepts and the notation. The rapid overview of mathematical "monsters" would not be complete without a brief reference to an interesting family of continuous, nowhere differentiable functions. They are occasionally referred to as "Weierstrass-like" but, for historical reasons, it seems more appropriate to call them "Bolzano-Weierstrass-like". Perhaps the most acute realization that the mathematical monsters have properties very close to those routinely found in nature was made by geographers.
