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Thread: Fractals

  1. #1 Fractals 
    Forum Bachelors Degree Shaderwolf's Avatar
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    O.K. I don't have much experience with fractals, so.. I have a couple of questions.

    #1) what is their use?!?!?!
    #2) I saw one... and had a thought. take a equilateral triangle. That's your start. flip it upside down so that the points are sticking pot of the other triangle's sides. we'll pretend that you can't see any lines in the center of the picture. You have added some area. for each of the six little triangles, do the same thing. You have added less area. If you go forever, the triangles will be so small you can't see them. After a while though, there will be a huge amount of tiny triangles added each time. On what turn will the amount of area added each turn start to increase? We'll say that one side of the original triangle = 1 unit. How much area will there be? Is there a limit mathematically?


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  3. #2  
    Forum Professor serpicojr's Avatar
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    The figure you describe in number 2 is the Koch snowflake (http://en.wikipedia.org/wiki/Koch_snowflake). You are correct that the number of triangles added at each step grows very quickly--exponentially, in fact. However, their area decreases exponentially at a faster rate, so the result is that the area added at each step still decreases exponentially. And you can always sum up numbers which decrease exponentially (this is basically the geometric series). Thus the area is finite, and the Wikipedia article gives a formula for the total area of the object. Note, however, that side lengths decrease exponentially at a slower rate than the number of triangles, so the total side length is infinite! Pretty wacky.

    As for number 1, aside from being mathematically interesting, fractals can be useful in modeling natural phenomena (snowflakes, plants, mountain ranges, coastlines, and lightning, according again to the almighty Wikipedia) and image compression. The basic idea behind these applications is that fractals can generate very complex shapes with minimal information. Again, I defer to the Wikipedia article: http://en.wikipedia.org/wiki/Fractal#Applications


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    Moderator Moderator AlexP's Avatar
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    I actually could've explained the Koch snowflake thing and serpicojr beat me to it. Oh well. I'll live. Another thing similar to the Koch snowflake idea (just to make this post more meaningful), while not falling under the category of fractals, is Gabriel's horn. Take the graph of 1/x on the interval [1,infinity), and rotate the shape around the x-axis. It will have a volume of pi (like the finite area of the Koch snowflake), but an infinite surface area (like the infinite perimeter of the Koch snowflake).
    "There is a kind of lazy pleasure in useless and out-of-the-way erudition." -Jorge Luis Borges
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  5. #4 Re: Fractals 
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    Quote Originally Posted by Shaderwolf
    O.K. I don't have much experience with fractals, so.. I have a couple of questions.

    #1) what is their use?!?!?!
    How about looking at "Chaos and Fractals" by Peitgen. Lots of interesting ideas in that book. For example, the Lorenz Attractor is discussed. It's that Owl-eye shaped figure as one of the three icons of Chaos Theory. It has a profound geometry about it: trajectories never touch! Ever wonder if traveling back in time is possible? Wouldn't that be . . . touching a point in space time . . . again? Sometimes I imagine riding on a trajectory in that attractor and wondering what I'd see? Would it be an expanding universe all about me?

    Consider the Mandelbrot Set. That's an iterated map: the output is fed-back into the input as one of the most complicated structures in Mathematics emerges . . . What's DNA doing?

    How about the mammalian cortex? It's massively fed-back too (output of many neurons get fed-back to the input of other neurons in the same neural assembly? I wonder . . .
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