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Not a very satisfactory definition, as it misses out the all-important term "self-similar" From: <quote> A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales. The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales. A plot of the quantity on a log-log graph versus scale then gives a straight line, whose slope is said to be the fractal dimension. </quote>

The bit about fractional dimensions needs to be added, because this extends the idea of dimensions quite considerably. --Felix 06:43, 26 April 2007 (EDT)

"The fractal nature of some plants has convinced many people that God must have designed the natural world" makes no sense whatsoever as an argument. If anything, I would have thought that it would have substantiated the case for the universe NOT being designed, since what occcurs in one place is simply replicated in another - this time on different scales. In any case, plants do NOT have a fractal structure. Some (not all) are self-similar on two (or possibly three) scales. Fractals are infinitely self-similar. At best this statement is an irrelevance, at worst it completely discredits the article . It should be removed.--Felix 09:15, 26 April 2007 (EDT)

Fractals and Intelligent Design

Since this is meant to be an encyclopedia article, I agree that the statement "The fractal nature of some plants has convinced many people that God must have designed the natural world" should be backed up with a source, or removed because it's just an opinion. As for the statement "Evolutionary biologists have yet to propose a satisfactory explanation for the spontaneous appearance of complex patterns in living organisms. They cannot adequately explain the evolutionary advantage of an endlessly repeating pattern and how it comes about from elementary biological or chemical processes.", the reference to the fern is a not-so-elusive answer. Fractals are expressions of simple rules that can result in complex-looking patterns when carried out, but are still based on simple rules being repeated iteratively. In the case of the fern, this particular pattern of fractal branching, repeated over and over, results in an efficient spread of foliage that balances the capture of light and air in shady settings like forests where ferns have thrived. --DinsdaleP 10:55, 2 March 2009 (EST)

But why would nature do this? Why would it know that fractals are efficient for photosynthesis? Why didn't develop big leaves? No satisfactory explanation has been offered that I've seen. -Foxtrot 11:09, 2 March 2009 (EST)
Nature doesn't "know" how to do anything. The observed rules and principles of chemistry and physics are what come into play in cases like this. I'm not botanist, but here's my guess where ferns are concerned. Many plants grow in proportion to the available resources to sustain them, and fern's pattern of small fronds that grow based on the repeating of a simple fractal pattern is more easily scalable than one which depends on fewer, bigger leaves. A five-inch fern in the forest will have many small fronds and leaves working to make it food, while a 5-inch maple seedling will have just two or three full-size leaves, so while each leaf produces more food than a set of fern leaves, the consequence of losing even one of those leaves is devastating to the maple. That's why through natural selection ferns thrive in shady forests, while broad-leaved plants thrive in more open spaces.
To address your questions at a more basic level, we are still learning more about the connections between inorganic and organic chemistry, and how complexity could arise without a designer's intervention. We don't question, for example, that salt and other minerals arrange their crystals in very precise lattices as they form. There are rules of chemical bonding that govern this, and experiments over time have shown beyond doubt that this is an intrinsic property of these elements, and not of supernatural influence. We've known since the 1950's that simple chemicals, when mixed and exposed to electricity, can form compounds like amino acids on their own. Just last Fall, scientists in Montreal showed that chemistry can also explain the formation of something even more complex like a ribosome. Here's the key except from this article:
"But proving that chemicals can spontaneously form simple amino acids did not prove that spontaneous action could create more complex mechanisms. "In the absence of such explanations, some people could imagine unseen forces at work when such complex structures emerge in nature," said Steinberg. Steinberg was able to show otherwise. He found the ribosome was put together using relatively simple structural rules, a bit like a three-dimensional puzzle. For critics who ask why spontaneous formation didn't lead to something other than the ribosome, Steinberg used mathematical models to show there was no other possibility. The ribosome simply wouldn't hold together if it were constructed any other way."
In short, questions about how life can form on its own without a designer or supernatural intervention are being answered every day, and our collective understanding of the big picture grows as those questions are answered. Every time someone states that "Nobody can explain 'X'", what they should be stating instead is that "Nobody can explain 'X' yet " --DinsdaleP 12:02, 2 March 2009 (EST)
The experiment you cite has potentially serious structural flaws: is it truly a random process causing spontaneous complexity or was the experiment itself designed in the precise way needed to create this degree of complexity? What safeguards can you have against this in a human-designed (and thus artificial) experiment?
Also, you have not answered my original question about the shape. A fractal is complex, having all the self-similarity. To produce it, the fern would have to divide, bisect, and scale its own operations. What would be the source of such power? These are not simple shapes to come up with. In contrast, a circle (or oval) is a relatively simple shape. To scale it you just have to evenly push out the border, expanding outwards. If all our plantlife were like tree trunks or the circular fungi that grow on logs, I would find your arguments more plausible. But the wide complexity of shapes and structures found in nature cannot have such a simple principle as natural selection to explain them! It cannot have another explanation than that of being designed by an Intelligent Being. -Foxtrot 16:41, 2 March 2009 (EST)
I haven't read the original papers related to the experiment, so I can't comment on your questions regarding it. As for the fractal question, that's the beauty of many fractal formulas - a relatively simple expression can yield complex results simply by iterating it enough times. Take a look at this page regarding fractal fern patterns, and you'll see that subtle variations in the formula parameters yield different outcomes. If these patterns were encoded in DNA, it's easy to see how it wouldn't take much of a genetic variation to result in diversity over time. Also, check out this page, which lists various patterns that can be generated by the fractint freeware program, with links to sample output. If you've never played with fractint before, you're in for a treat. --DinsdaleP 17:07, 2 March 2009 (EST)

Thanks for the pretty pictorial representations of recreational mathematics. I'd like to keep to the issue at hand, though. Do fractals arise in nature as a result of God's Design or can they somehow occur from an evolutionary process? Your answer in the previous paragraph seems to be "If these patterns were encoded in DNA, it's easy to see how it wouldn't take much of a genetic variation to result in diversity over time." Do I see someone hedging their response with a hypothesis? If these patterns were encoded in DNA, then they would be encoded in the encoding for life, and they would indeed reproduce. But what (or, Who?) puts them in the DNA in the first place? How are these patterns encoded in that double helix? How did the DNA get to finding that infinite pattern in the first place, one which requires you to reproduce the smaller parts similarly to the whole? How did natural selection induce DNA that directs the plant to infinitely fractalize, if you don't mind the neologism? It seems so complicated to mechanize without an intelligent being there to initiate it. Your links to the other sites actually support me here: we had to have intelligent beings (people) enter in the parameters and generators in order to create the fantastic fractals. I could do it, but could some amino acids come up with that by themselves spontaneously? -Foxtrot 19:15, 2 March 2009 (EST)

I had referenced the Canadian research above that showed how something considered too complex to form by chance, a ribosome, not only could, but did so because of well-understood principles of chemistry. Science is continuously uncovering new findings that increase our understanding of the building blocks of life, and while the answer to many important questions is still "we don't know yet", more and more of those questions are answered each year. The important thing is that when a natural explanation for something is not yet known, that doesn't imply that the explanation must be supernatural. --DinsdaleP 19:45, 2 March 2009 (EST)