Summary

Fractals are shapes that are self-similar on multiple scales (spatial or temporal), and have (in general) fractional dimensions. Natural fractals are of course self-similar only over a limited range, and the similarity is often statistical rather than exact. Nature apparently chooses fractal structures to optimize functional efficiency given limited resources. (An interesting recent article is Ref[5]. Physical growth mechanisms, such as DLA, are reasonable explanations for some natural fractals, while for fractals in biological systems some algorithms have been proposed (such as those of Lindenmeyer and Barnsley) that give realistic pictures. Though natural fractals have intricate structures, it is remarkable that one is able to 'explain', at least qualitatively, the complex geometry of nature using computer simulation of models obeying simple rules. Thus in this case one might say that the emergence of complex patterns and structures can be understood as arising from simple underlying causes. However note that to capture the depth of self-similarity, one has to iterate the algorithms over many steps, and it is often not possible to guess what the final structure will look like until the numerous steps have been performed. Self-similarity implies that a system is scale-invariant, or equivalently, it means the absence of a charactersitic length scale. Scale-invariance will be revisited when we study phase transitions and self-organised criticality. Look out for fractals later in the course when we study chaos, phase transitions and self-organised criticality. In the meantime, read about some technological apllications of fractals at Ref.[6].