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Iteration, Feedback, and Change: Fractals

Published on Saturday, June 04, 2011 in , , , , ,

Wolfgang Beyer's Mandelbrot set renderingSince we've talked about artificial life, biological life, and even behavior and decisions, you might start getting the idea that the impressive developments of iteration, feedback, and change require a living entity to work.

If you take a closer look, you'll find not the case.

All you need is an object and a process that will act on that object. To allow for iteration, you simply need time and the potential for the process to be repeated over that time. The process involved with naturally provide the feedback, and the altered status is, of course, the change itself.

This is why so much of the development of complexity out of simplicity that we've seen can be explored with computers. Conway's Game of Life, and the prisoner's dilemma tournaments were both done entirely on computers. Neither computer simulation could be said to contain actual living elements, but the way the elements acted, it does effectively make us question our definition of life.

Take a look around you right now at all the man-made items you see. Consider their shapes, and you'll probably notice lots of straight lines, smooth curves, arcs, and easily-identifiable angles (45 degrees, 90 degrees. etc.). It's not too hard to realize that you could, with a little work, duplicate these shapes by working out their formulas.

Now go outside, and take a look at the shapes produced by nature, such as plants, animals, mountains, water flows, and more. The shapes in nature, while still retaining some design elements are a lot less organized. You may be thinking that you either couldn't develop a formula for these shapes, or that only one formula comes to mind: that of the Fibonacci Numbers and the Golden Mean:



The description I gave earlier involving objects, processes, and time makes it sound like these qualities could be examined in a more rigorous, mathematical way, especially considering that they're not feature unique to living beings.

The video above on Fibonacci numbers also suggests that there's a mathematic process at work in nature, but what is it? The tools to begin understanding shapes in nature, much like the tools of game theory, have only been developed recently.

It took the vision of the late Benoit Mandlebrot to develop this new field of math. To do so, he had to look beyond just the descriptions of the shapes themselves, and take the process that created them into consideration. By doing so, he developed a field called fractal geometry, explained in this NOVA documentary, Hunting the Hidden Dimension:



As you've seen, fractals have already proven themselves as an essential component of cellphones, and it doesn't seem like it will be long before, say, the fractal dimension of your blood flow and the true nature of your heartbeat will be as much a part of your medical record as your temperature. Who knows what other fields will be affected and how? We're just beginning to see the possibilities.

If you'd like to delve a little deeper into the nature of fractals, as opposed to just their effects, check out Arthur C. Clarke's documentary, Fractals: The Colors of Infinity.

Study of fractal geometry seems nice, and it's very impressive that it can be applied in such amazing ways to so many aspects of the real world. What is it, though, that makes them so applicable to the real world? In the next and final post in this series, we're going to see take a closer look at this strange connection to the real world, and what it means for us and our future.

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2 Response to Iteration, Feedback, and Change: Fractals

5:42 AM

I am really enjoying this series of posts, keep it up!

1:26 PM

Thanks, Matthew!