What is the Mandelbrot Set 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 allows you to investigate the Mandelbrot Set, the worlds most famous fractal. In this activity you will investigate the Mandelbrot set associated with the formula f(Z) = Z2 + C where Z and C are complex numbers in the form of ordered pairs. Definitions:

  • prisonervalue of z in the formula f(z)=z2 + c where at each iteration the resulting value becomes smaller and smaller, approaching zero
  • escapeevalue of z in the formula 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 the Mandelbrot set -- the world's most famous fractal -- we need to use the fact that Julia sets for a particular C in the function above are either fractal dust or are one connected piece.

    For C=(0.5, 0.5) we get the fractal dust Julia set:

    Here's one with connected Julia set; C = (0,0).

    Mandelbrot came up with the idea of plotting the Cs that had connected Julia sets, and found that the boundary was very interesting -- and Fractal!

    The black points are the Cs with connected Julia sets, and the colored points are Cs with fractal dust Julia sets. Mandelbrot proved a person could tell if the Julia set would be fractal dust or not depending on whether the starting point (0,0) was a prisoner (connected) or escapee (dust). So the colors come from how quickly the starting point (0,0) orbit gets out of the circle of radius 2.


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