# Fractals as solution to optimization problem?

What's the scientific reason for fractals being present in nature at such a large scale? Is it perhaps the solution of an optimization problem? For example, would the fractal based shape of certain plants, be the the solution evolution has given to the problem "what is the shape which could capture the largest amount of light from the sun in a given environment"? If yes, do you have other examples?

Would highly appreciate some book references which are on topic (i.e. not fractals per se but WHY fractals are out there).

Thanks.

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What evidence is there that fractals are common in nature? The examples I have seen strike me as mainly wishful thinking, much as the supposed ubiquity of the golden ratio in art is spread despite being debunked repeatedly. Biologically relevant physics is not scale-invariant except in very restricted contexts. –  Douglas Zare Feb 27 '12 at 0:18
Fractals provide very useful and general models in nature, undoubtedly... but I wouldn't say that a cabbage, a lightning, a rock fracture, a fern leaf are fractals. In general, I think "mathematical objects present in nature" is somehow a misleading expression. Mathematical objects are categories that we use to represent objects of the world, and the correspondence is by no means one-to-one, in either direction. Depending on our scopes, the sea surface can "be" a plane, but it may be a spherical cap as well, and so on. –  Pietro Majer Feb 27 '12 at 7:41
But surely it can't simply be some sort of coincidence. There has to be an explanation to this (leaving aside that in nature they're merely approximations to the abstract mathematical object). Wikipedia presents a brief list of fractals, Mandelbrot has written on the fractal geometry of nature. Are you saying that it's overly exaggerated? I believe some suggested that because of the simple repetitive rules for building a fractal they could stand for the very mechanisms which biological cells use to build organic structure. I'm not as mathematically aware as you guys are, so I thought I'd ask. –  mndc Feb 27 '12 at 10:36
I would think (though I am far from an expert) that self-similar structure would arise in nature not because it was optimal in any sense, but rather because it was random. The prototypical example would be Brownian motion. A process should be $C^\alpha$ and so have structure on all scales. Indeed, on "average" it should look the same on all scales. Maybe I'm wrong, but I imagine the "fractals" one sees in nature look more like Brownian motion and less like rigid (and strictly self-similar) objects like the Koch snow-flake. –  Rbega Mar 23 '12 at 4:39

standard book reference:

B. Mandelbrot, THE FRACTAL GEOMETRY OF NATURE

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With regard to coastlines see Sapoval et al., "Self-stabilised fractality of sea-coasts through damped erosion," arXiv:cond-mat/0311509.

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An influential paper argued that the fractal dimension of the circulatory system minimized loss of energy in its transport.

West GB, Brown JH, Enquist BJ.
A general model for the origin of allometric scaling laws in biology.
Science. 1997 Apr 4;276(5309):122-6.

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This kind of recalls a passage from Structural Stability and Morphogenesis by Rene Thom, p. 296:

In the last analysis, whence can life on our planet come but from the continuous flux of energy from the sun? The solar photons arriving in contact with the soil and seas are immediately stopped, and their energy abruptly degraded into heat; in this way the discontinuity of the earth and water surface is also a shock wave, a cliff down which the negentropy of the sun's rays falls. Now, life can be considered as some kind of underground erosion of this cliff, smoothing out the discontinuity; a plant, for example is nothing but an upheaval of the earth toward the light, and the ramified structure of its stem and root is the same as that found when a stream of water erodes a cliff and produces a mound of debris. Plastids, veritable photon traps, are the miniscule orifices where this subterranean circulation begins. The energy stored in the noble form of chemical energy begins its slow decline. It flows underneath the cliff like a fluid, and its circulation echoes the inverted pyramid of the ecology of living beings. Each living species is a structurally stable singularity, a chreod of this circulation. As in hydrodynamics the energy of a turbulant regime flows from lower-frequency oscillations toward higher frequencies, finishing in thermal chaos, so in life those with slower metabolism (plants) are the prey of the faster-metabolizing (animals).

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