Some fractals are supposed to mimic natural objects aren't they? Like ferns and Koch Snowflakes. One of the principal ideas of fractals is that they have infinite self-similarity. But natural objects don't do this. But is this disparity a bad thing?

Natural objects do this, but only to an almost infinite point. They are limited by design to only be self-similar for so long before you get into an atomic mess.

There is a reason why living objects may choose fractal structure. It might simply be a conservation of genetic material (DNA, and the subsequent synthesis of protein). If you have separate genes that specify growth of a tree's trunk, its main branches, its smaller branches, the sticks that come off of those, the twigs that come off of those .. etc, you start to get into a mess. How many genes should be made and how do you control them all?

For trees, adopting a single physical rule of bifurcation at certain points reduces the control complexity of the system, while allowing for significant variation and adaptability of form.

Natural objects such as snowflakes and river systems also take on fractal shapes because the physical laws that define their shapes work on many different size scales. Of course, all bets are off once one gets down to molecular and atomic scales, where physical laws themselves are changing.

Halcyon&on has a good point about fractal structure being a possible method of DNA conservation, but that is not the only possible reason for the occurance of fractal-like structures in living organisms. Fractal structures can also have advantages in other terms such as surface area to volume ratios. The branching structures in the alveoli of the lungs and the capillary beds of the circulatory system are good examples of such fractals. Their repeated branching structure gives them a high surface area to volume area, which is excellent for O2, CO2, and other chemical exchanges. The same can be said for the branching of trees, which uses small amounts of tissue to create a large area for leaves, which we all know absorb the light that the trees need.

More area = more light = more energy for the trees.
Less volume = less cells to support = less effort.

The fractal-like structure of these organisms is probably not directly coded in the DNA of the organism. There are lots of mathematical ways to generate fractals, but I doubt one would find any analogue to them in the DNA. It seems most likely that the complex interactions of systems of proteins are what result in these fractal-like structures, and that this being advantageous, was a property evolution decided to hang on to.

I say fractal-like to refer to such structures, because fractals are not rigidly defined in the mathematical literature, and also because these structures generally do not have all of the properties attributed to fractals, at least not in the mathematical sense. Mathematical fractals are self-similar on infinite scales, and are often predictably self-similar. Natural examples of fractals on the other hand, tend to have only qualitative self-similarity which can extend over large or small scales. It's a tricky thing to define, but the concepts of chaos theory and the fractal nature of real world objects are almost definitely linked.

life as a fractal

Life is a fractal: a huge overall structure which replicates itself at a smaller size, then smaller, then smaller, down and down and down until the parts are almost impossible to see, but if you were able to look closer you could see that they have the same shape as the larger structure of which they are a part.

The goal of my life is to enjoy myself as much as possible, and right now I'm trying to relax while still doing something somewhat productive, and now my goal for this node is to make people think, and now it's to reveal something about me, and now it's to convey this concept as best I can, and now it's to complete this thought effectively, to finish this sentence, to finish this word, to type this letter. A series of goals, spiraling down tinier and tinier but still recognizable as having the same shape as that of which they are a part, if you zoom in far enough, so that life is just a series of nested and nested and nested goals, and in the end what it comes down to is that every moment of your life is a triumph; every infinitesimal instant, you have something to strive for, and every instant you're achieving something, be it activating the neurons in your brain to complete a thought or nodding your head to acknowledge the praise that your thought received.

Some things are so small you hardly even notice them; the search for a key on the keyboard is over almost before you start thinking about doing it, but it's a step, however small, on the way to a larger task. Letters combine to make words, words combine to make clauses, clauses make sentences, sentences make paragraphs, and so on, and when you're done you have an overall shape which depends on the smaller parts and reflects them, and your writings combine to reflect your mind which spins off these smaller but still beautiful copies of its contents, and meanwhile your cells form skin and bones and organs, all containing and reflecting the DNA that makes you who you are, and all these aspects of your being mirror each other and exhibit that similarity of structure that allow them to combine to create You.

And yet You are still only a tiny, infinitesimal part of an incomprehensibly larger Fractal, which if you could zoom out far enough you would see contains all of Creation, and you would see how beautiful it is, and you would see that in a way, if you look at it just right, in a way it looks like a huge version of You, and you would realize that no matter how small you feel, you are part of this beauty, and all this is tied up in who You are, and you are truly meant to be here. And you would understand that the fractal is not perfect, but that for all its flaws it is so amazing that you can only marvel at the good fortune that allowed you to be part of it.

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