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Math Help - Chaos Theory

  1. #1
    Junior Member JoeyCC's Avatar
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    Chaos Theory

    Lately I have been working with Fractal Art in photoshop and UF. I read into the subject and got interested, but I still have a few questions.

    For Example,

    • In simple terms, what is Chaos Theory?
    • How is Fractal Art (Geommetry) related to Chaos Theory?
    • What exactly is a Mandlebrot?
    This is it. If you're interested to see my productions, check www.violationcc.deviantart.com I have a few of my early works there.

    ~Kind Regards
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  2. #2
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    Quote Originally Posted by JoeyCC View Post
    Lately I have been working with Fractal Art in photoshop and UF. I read into the subject and got interested, but I still have a few questions.

    For Example,

    • In simple terms, what is Chaos Theory?
    • How is Fractal Art (Geommetry) related to Chaos Theory?
    • What exactly is a Mandlebrot?

    This is it. If you're interested to see my productions, check www.violationcc.deviantart.com I have a few of my early works there.

    ~Kind Regards
    Chaos theory is the theory of dynamical systems which are extremely sensitive to initial conditions. That covers a lot of ground! Dynamical systems are, quite simple, systems which behave transiently.

    The reason fractals are involved is because a fractical is indeed a dynamical system with initial condition sensitivity. Take the Koch Snowflake:

    Koch snowflake - Wikipedia, the free encyclopedia

    This is generated using an iteration of a general rule. Think of it as a programme which does this process any time it sees a straight line. The process is this:

    Whenever we come across a straight line segment:

    1. divide the line segment into three segments of equal length.
    2. draw an equilateral triangle that has the middle segment from step 1 as its base and points outward.
    3. remove the line segment that is the base of the triangle from step 2.

    The Koch Snowflake is what you get when you apply these iterations again and again to a shape that initially started off as an equilateral triangle. However, if our initial shape wasn't initially an equilateral triangle, then only 1 iteration would lead us to diverge from the koch snowflake, and the more iterations, the more we would diverge! So we can clearly see that the result of fractals is very sensitive to the initial conditions. Try it yourself. Take random shapes made of straight lines (they need not be equal in length!), and iterate the above process ^. You will see how it diverges from the Koch Snowflake.
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  3. #3
    MHF Contributor

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    Quote Originally Posted by JoeyCC View Post
    Lately I have been working with Fractal Art in photoshop and UF. I read into the subject and got interested, but I still have a few questions.

    For Example,

    • In simple terms, what is Chaos Theory?
    • How is Fractal Art (Geommetry) related to Chaos Theory?
    • What exactly is a Mandlebrot?
    This is it. If you're interested to see my productions, check www.violationcc.deviantart.com I have a few of my early works there.

    ~Kind Regards
    Chaos theory deals with certain "dynamical systems", either sequence in which the "next number" is determined by the current number or functions defined by differential equations, that exhibit "sensitive dependence on initial conditions": given any starting value there exist points arbitrarily close that would produce very different results.

    A fractal set of points is a set having fractional dimension (you have to be careful how you define "dimension", of course). In any dynamical system there may be "attractors", sets to which the sequences tend to converge no matter what the starting point. If the system is chaotic, those attractors tend to be fractal sets (and are called "strange attractors").

    "A Mandlebrot" is a person! Benoit Mandlebrot, who worked for IBM many years, wrote extensively on fractal sets as well as chaos theory. He developed what is now called the "Mandlebrot set" which is probably what you meant. If you think of points in the plane as being complex numbers, (x,y) corresponding to z= x+ iy, and look at the dynamical system defined by z_{n+1}= z_n^2+ c, the Mandlebrot set is the set of all values of c for which that sequence, starting at z_0= 0 converges.

    By the way, the Mandlebrot set can be thought of as an "index" to the Julia sets. The Julia set, J_c, is defined as the set of complex starting values z_0 such that the sequence z_{n+1}= z_n^2+ c converges. Notice that in the Mandlebrot case, we always start at z_0= 0 and change c while for a given Julia set, we fix c and change z_0. A Julia set, for given c, tends to look like the Mandlebrot set in the vicinity of c.

    By the way, Benoit Mandlebrot did all of this on very primitive computers (compared to what we have now). When I first started working with the Mandlebrot set (which was, of course, long after Mandlebrot did), I would start the program running and come back several hours later!

    Even worse, the French mathematician Gaston Julia, who developed the Julia sets, worked in the early part of the twentieth century, long before computers were even invented! He drew his sets by hand.
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