Sept 2010 - Fractals

Science Heresy - September 2010

Fractals and the Myth of Scalability

Since the early eighties the mathematical ideas of chaos theory and and the closely related field of fractals have captured the popular imagination. Chaos theory has found some application in fluid dynamics and related fields while fractals have proved popular as a method of creating digital imagery. Images created in this way may resemble natural objects such as clouds and coastlines. Intriguing patterns which are purely abstract in appearance may be generated which have a fascinating elegance and mystery. The best known of these is the Mandelbrot Set.

The essential feature of a fractal object is that of self-similarity or scalability, that is, that fractals are by definition similar on every scale however large or small. There is no way in which a person examining a fractal can tell whether they are looking at something very large or very small or somewhere in between; the fractal has the same "look" on every scale. This idea of similarity precedes the discovery of fractals. It was, for example, used in the analysis of surface gravity waves on liquids by the Russian mathematician Sergei Kitaigorodoskii as early as 1961. By assuming that ocean waves must be similar on every scale he was able to draw some interesting conclusions about the physics of ocean waves, in particular, about their power spectra.

Over the last 40 years or so a mystique has grown up around scalability and fractals. These are very elegant and seductive concepts but are they in fact valid descriptions of nature and natural objects? A moment's reflection will show that they are not. Far from being commonly fractal, natural objects are in fact never fractal. They may on occasion be fractal-like or self-similar over a range of scales but they must be self-similar on every scale in order to be truly fractal. Only mathematical objects can have this property. We know by experience that there is always some scale at which the self-similarity of natural objects breaks down. The "coastline" of a rocky headland in (perfectly still) water may be fractal-like over a range of scales from millimetres to tens of meters but on a spatial scale of hundreds of meters beaches become evident and these are not self-similar (there are no micro-beaches). Below the scale of millimetres the crystalline structure of the rocks themselves gives them a very different appearance from the same rocks on a larger scale.

The same is true of waves. Small waves do not look the same as large waves much to the chagrin of movie makers who wish to use wave tanks to film scenes of stormy seas. There is a lower limit to the size of gravity waves. At scales of centimetres and less, capillary waves are the dominant wave form on the surface of water because surface tension is more important than gravity at these scales. There are no gravity waves shorter than a few centimetres in wavelength and only gravity waves are seen to "break". Even when small gravity waves do break, their appearance is very different from large breaking waves because of the way that surface tension and viscosity control the bubble size in the foam.

All this may appear to be academic but it has had serious practical consequences. Because of Kitaigoroskii's profound but inaccurate insights into the nature of ocean waves, an entire generation of ocean wave specialists (who were nearly all mathematicians) believed that ocean wave "wind sea" power spectra must follow a fifth power law. They believed this despite considerable experimental evidence to the contrary. These idealized spectra were then used by engineers to compute design stresses on offshore structures. Because they used the fifth power law spectrum of theory rather than the fourth power law spectrum actually observed, these stresses were overestimated for big (long) waves and underestimated for small (short) waves. The nett result has been that all offshore structures are overdesigned. Oil rigs are very expensive things to build, so that this simple misconception may have cost the offshore oil industry hundreds of millions of dollars during the decades that the assumption of perfect scalability has been the "correct" view of ocean surface waves.

Another idea foisted on us by the mathematical idealogues is that images of fractals are somehow more pleasing to the eye than are non-fractals. No evidence is ever presented for this view which is after all, a testable hypothesis. Artists and graphic designers could be asked to assess the aesthetics of various patterns in double blind tests. The image at the top of this page shows part of the Mandlebrot set. The image below is a photograph of sunlight defracted by waves in shallow water. The upper image is a fractal. The lower image is not fractal because it is the outcome of both gravity waves and capillary waves at a scale at which they are both of roughly equal importance. The lower image is pleasant to look at nontheless whereas fractals tend to have a spooky, alien quality that is not always pleasing to the human eye.

September 2010