What is the Julia Sets Activity?

Trigonometry Related Pages: 347-348 "An Application of Complex Numbers to Fractals"
College Algebra and Trigonometry Related Pages: 664-665 "An Application of Complex Numbers to Fractals""
Precalculus Related Pages: 585-586 "An Application of Complex Numbers to Fractals""

This activity allows you to see an applicaiton of complex numbers. The standard form of a complex number is (a+bi). Complex numbers can also be thought of as ordered pairs, instead of the standard form (a+bi) use the form (a,b) where the x-coordinate is the real part and the y-coordinate is the imaginary part.

This activity is a precursor to ultimately understanding the Mandelbrot Set, the worlds most famous fractal. In this activity you will investigate Julia Sets associated with the function f(z)=z2 + c where z and c are both complex numbers in the form of ordered pairs.

Definitions:
  • prisonervalue of z in the function f(z)=z2 + c where at each iteration the resulting value becomes smaller and smaller, approaching zero
  • escapeevalue of z in the function f(z)=z2 + c where at each iteration the resulting value becomes larger and larger, approaching infinity.
  • iterationrepeating a set of rules or steps over and over
  • Julia SetsThe set of all the points for a function of the form z2+c.

    To understand Julia sets and ultimately the Mandelbrot set -- the world's most famous fractal -- we need to be able to find the boundary between the prisoner and escapee sets for a particular value of c.

    An example of a function is: f(Z) = Z2 + (0.5, 0.5). If we start with the point (0, 0) and plug it in for Z

    f(0, 0) = (0, 0)2 + (0.5, 0.5) = (0, 0) + (0.5, o.5) = (0.5, 0.5)

    f(0.5, 0.5) = (0.5, 0.5)2 + (0.5, 0.5) = (0, 0.5) + (0.5, 0.5) = (0.5, 1)

    f(0.5, 1) = (0.5, 1)2 + (0.5, 0.5) = (-0.75, 1) + (0.5, 0.5) = (-0.25, 1.5)

    f(-0.75, 1.5) = (-0.75, 1.5)2 + (0.5, 0.5) = (-2.1875, -0.75) + (0.5, 0.5) = (-1.6875, -0.25)

    f(-1.6875, -0.25) = (-1.6875, -0.25)2 + (0.5, 0.5) = (2.7852, 0.84375) + (0.5, 0.5) = (3.2852, 1.34375)

    Here is a graph of these points all connected from one iteration to the next:

    This is in the escapee set for C=(0.5, 0.5). If we continue this process for many values of Z we can get a feel for the picture of the Julia set. Here is the Julia Set for for all values of Z in the function
    f(Z) = Z2 + (0.5, 0.5):

    Notice that there are no black areas on this image. Black denotes the prisoners and the other colors on the picture are the escapees, and the different colors denote how quickly the point's orbit jumped out of the circle of radius 2. There actually are prisoners but they are completely scattered around and isolated from each other. This forces the Julia set -- the points that neither are prisoners or escapees -- to be completely disconnected from each other. The .

    Here's another one with an obvious prisoner set; C = (0,0).

    This has a nice big connected set of prisoners -- giving a connected Julia set, the circle of radius 1. Mandelbrot found that these are the only two things that happen, either the Julia set is totally disconnected (fractal dust) or it is one piece.

    Here are a few of the more popular Julia sets:

    C = (-1, 0):

    C = (-.1, 0.8):

    C = (0.5, 0):

    C = (-0.8, 0.4)

    The first two are connected and the last two are fractal dust. Julia Sets are the basis for the world's most famous fractal, The Mandelbrot Set.

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