"Fractals" and "chaos" have become increasingly popular in geology; however, the use of "fractal" methods is mostly limited to simple cases of selfsimilarity, often taken as the prototype of a scaling property if not mistaken as equivalent to a fractal as such. Here; a few principles of fractal and chaos theory are clarified, an overview of geoscience applications is given, and possible pitfalls are discussed. An example from seafloor topography relates fractal dimension, self-similarity, and multifractal cascade scaling to traditional geostatistical and statistical concepts. While the seafloor has neither self-similar nor cascade scaling behavior, methods developed in the course of "fractal analysis" provide ways to quantitatively describe variability in spatial structures across scales arid yield geologically meaningful results. Upon hearing the slogan "the appleman reigns between order and chaos" in the early 1980's and seeing colorful computer-generated pictures, one was simply fascinated by the strangely beautiful figure of the "appleman" that, when viewed through a magnifying glass, has lots of parts that, are smaller, and smaller, and smaller applemen. The "appleman" is the recurrent feature of the Mandelbrot set, a self-similar fractal, and in a certain sense, the universal fractal (e.g., see Peitgen and Saupe, 1988, p. 195 ff.). Soon the realm of the appleman expanded, made possible by increasing availability of fast, cheap computer power and increasingly sophisticated computer graphics. In its first phase of popularity, when the Bremen working group traveled with their computer graphics display seeking public recognition through exhibits in the foyers of savings banks, the fractal was generally considered to be a contribution to modern art (Peitgen and Richter, The Beauty of Fractals, 1986). While the very title of Mandelbrot's famous book, The Fractal Geometry of Nature (1983), proclaims the discovery of the proper geometry to describe nature, long hidden by principals of Euclidean geometry, the "fractal" did not appeal to Earth scientists for well over two decades after its rediscovery by Mandelbrot (1964, 1965, 1967, 1974, 1975).
Nigel Reading—“Dynamical Symmetries: Mathematical Synthesis between Chaos Theory (Complexity), Fractal Geometry, and the Golden Mean”
