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A fractal algorithm like Mandelbrot is self-similar in all size scales. This is not the case in nature. A tree is fractal in the sense that each branch is similar to the tree as a whole. But that is only true at a certain range of scales, about 0.1 - 10 meter. The forest does not look like one tree, nor does the texture at millimeter scale or the cellular structure at microscopic scale. At those scales the tree may or not be fractal, but at least it is not fractal in the same way as on the 1 meter size. Nature breaks the fractal regime after only a couple or so self-similar scales.

Are these transitions between fractal regimes studied by biology?

Are there any ideas around about how different fractal regimes might be connected with emergent phenomena? At very small scales there exists no tree, but only elementary quantum particles/waves. Macromolecules and cells and the tree and the forest each emerge as a phenomena on different size scales. I wonder if this emergence is matched by the different fractal regimes at those scales?

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I think that you are mixing up the causality between fractals and emergent phenomena.

There are LOTS of different types of emergent phenomena, and only some of them are connected to fractals. Many are instead connected to other physical phenomena like phase transitions (e.g., traffic jams) or feedback loops (e.g. dune formation).

Biology does indeed make use of emergent phenomena at many different scales. This is, however, only what we should expect from an evolutionary process that is building patterns through trial and error selection. Critically, emergence is the formation of complex patterns from relatively simple local interactions, and these patterns are often highly robust due to the nature of the physics driving them.

This means that emergent phenomena are an evolutionary "shortcut" for achieving many kinds of useful structure. Fractal-generating programs, in particular, are a one very simple way of encoding large-scale space-filling patterns such as tree branches or blood vessels (other space-filling patterns, like hair follicles or shell patterns use different mechanisms). And remember, they typically run "inversely" in biology, starting small and growing rather than starting large and subdividing---thus the natural truncation of iterations by the limit in growing scale.

It is thus unsurprising that organisms would first "discover" fractals rather than something more akin to a human-style blueprints for encoding such phenomena, since the repeated local rule of a fractal is so much less complex. It's just that we discovered fractals later and they are less intuitive to us, so they look more remarkable to human thinkers.

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Fractals are by far more ubiquituous than it may appear at a first glance, spanning essentially all the levels of organisation of living organisms; including vascular and neural networks in animals and plants, metabolic networks in bacteria, and the food networks characterizing whole communities. (See, e.g., the answers to this question for more discussion and the relevant links.) All these networks are self-similar, and one cannot really claim that there are transitions between different fractal regimes (at least, such claim is explicitly excluded by one of the postulates of the cited theory.)

That the existence of these fractal networks is an example of an emergent phenomenon (in strict physical sense, as, e.g., described by Anderson), does not cause any doubt. What is less clear is whether 1) these networks are the result a single symmetry-breaking/emergence event that modern organisms inherited from their most recent common ancestor or 2) whether organisms naturally evolve toward fractal structures as a means of increasing their fitness (the theory states that fractals lead to the highest energy efficiency).

The last paragraph in the OP suggests that the question might be posed in more general sense, of the emergence phenomena in biology. Indeed, if we start with atoms, one can treat evolution as a sequence of emergent phenomena in their strict physical definition. To give just a few steps:

  • emergence of simple molecules and chemical reactions between them
  • emergence of self-replicating polymers
  • acquisition of cellular membrane
  • emergence of multicellular organisms

This question gives links to some more or less readable discussions. Yet, it is fair to say that in terms of exact theory we have not much evolved since Anderson (although a fe succinct results were obtained, e.g., regarding teh tehrmodynamics of self-replicating polymers: here and here.)

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