The roughness and general shape of fractal curves can be varied almost at will, simply by altering the mathematical formulæ used to define them. Roughness is described by a number called `fractal-dimension', with lower numbers representing smoother curves. The notion of fractal-dimension can be used to quantify the roughness of many naturally-occurring or man-made phenomena, such as soil surfaces, or plastic food-wrap, or the variability of the exchange rate of the Australian dollar. In the last of these examples, a graph of tick-by-tick (transaction-by-transaction) currency-exchange data, against time on the horizontal axis, may be analyzed to determine its fractal properties and quantify its roughness. Exchange-rate predictions made during periods of lower roughness are usually more reliable; so it is of practical importance to be able to quantify the `erraticism' (as it is sometimes called) of the trading environment.
In the first two examples, surface roughness plays a major role in determining physical properties. A rougher soil surface has greater surface area exposed to the atmosphere, so more readily absorbs gases from the air--therefore ``rougher is better''. A sheet of plastic food-wrap offers less opportunity for micro-organisms to attach themselves to its smoother surface. This assists in preserving freshness--so ``smoother is better''.
Rainfall reduces the roughness of soil surfaces. Its effect may be quantified by analyzing measurements of surface height both before and after rain. Different manufacturing processes produce plastic food wrap with different levels of roughness. The process that gives the smoothest surface may be determined by estimating fractal-dimension. I have worked on both problems in collaboration with the CSIRO's Division of Mathematics and Statistics. Let me say a little about the plastic food-wrap problem.
Data describing the surface of the food-wrap are obtained by coating a small part of it with a very thin layer of gold, just a few molecules thick, then recording the surface height in a scanning, tunnelling electron-microscope. These data are modelled mathematically, in terms of a theoretical abstraction known as a Gaussian `random field', which is a special sort of random surface. Mathematical properties of these random fields can be used to determine the way in which measurements of correlation between surface-heights at different places should be used to estimate the fractal-dimension. They also help quantify the reliability of these estimates. There are several different fractal-dimension estimates that can be employed. Mathematical analysis can be used to determine which is the most accurate. Increased intuition from by a mathematical analysis also helps suggest ways of modifying estimators to produce even greater accuracy.