Occurrence and uses of Fractals

Consider the Amazon river system. The box-counting method can be used to determine its fractal dimension, giving $D_b =1.85$. By comparison the fractal dimension of the Nile river is about $1.4$ [Takayasu, Section, 2.1.2]. The branching of a tree looks like that of a river and its fractal dimension lies between $1.3$ and $1.8$, with a mean of $1.5$. [D. Morse, et. Al., Nature, 314 (1985) 731]. The shape of lightning discharges is similar to that of rivers. The fractal dimension in this case is about $1.7$ [L. Niemeyer, et.al. Phys. Rev. Lett., 52 (1984), 1033]. Consider the diameter distribution of blood vessels in a bat's wings. If $N(r)$ Is the number of vessels thicker than $r$, then $N(r) \propto r^{-2.3}$, [Takayasu, Sect.2.2.1]. Given a time-varying signal, one may consider its power spectrum, that is, the Fourier transform of the intensity. The power spectrum indicates the relative magnitude of different frequency components in the signal. Empirically it is found that fluctuations (or noise) in many man-made and natural systems have a power spectrum of the form $f^{-a}$ where $a \sim 1$. This is generically referred to as a "$1/f$" law. Note that if $a=0$ then all frequencies would have the same magnitude (the fluctuations are random) and one has 'white' noise. For $a$ close to one, which is what is observed, the lower frequencies dominate and one has "pink noise". The $1/f$ noise has been observed in electrical circuits, voltage fluctuation of nerve cells, heart beats and even in music. This universal occurrence, which cannot be due to random fluctuations that give rise to white noise, demands a simple explanation but there is currently none (see however the later chapter on self-organised criticality). The $1/f$ noise is of course fractal because of its self-similarity. Finally, if $a \ge 2$ one has "brown noise" which for audio signals sounds dull and is not observed in nature. Thus it seems that the naturally occuring and interesting "$1/f$ noise" is poised between the randomness of white noise and the dullness of brown noise. Look at the pictures of branching air passages in the lung, the branching of blood vessels in the human body, or the folds on the surface of the brain.


Why should nature design such fractal structure ? Recall from the example of the Koch curve that one can accomodate long lengths in small areas. Thus nature apparently maximises functional efficiency while using minimum space by adopting fractal structures. Fractals images have been used for image creation in science-fiction movies (e.g. Star Wars) and also for data compression (If you are interested, check out the work of Lindenmayer and Barnsley and explore the related fractals using the FRACTINT software). In case you have not guessed it yet, Fig.(2.1) is not an image of a real fern but a computer simulation! Here is another simulation that looks natural:


The graphs shown below plot the actual S&P 500 stock index for one year, five year, and ten year spans. Do the time-series look self-similar to you ?


Now look at Mandelbrot's paper on finance [Mandelbrot, Scientific American Feb 1999, pg.73] . Some of the figures there are of real data but some have been artificially generated using fractal concepts. Can you tell them apart? These last two examples show that time series in finance are generated by complicated processes that are apparently self-similar on many scales.