Mandelbrot Pic
This project must be done 11/19 8:00am EST
Must be done in eclipse, JAVA
If you can send me 2 variations of this problem I will give a $15-10 bonus!
The Mandelbrot set was first discovered by Pierre Fatou and Gaston Julia. Benoit Mandelbrot was among the first to use a computer to visualize the Mandelbrot set, so it was named in his honor. Perhaps the most eloquent description of the set was given by musician Jonathan Coulton, better known these days for his song “Still Alive”. In his song “Mandelbrot Set”, Coulton sings: “Pick a point called z in the complex plane, and let z1 be z2 + z, and z2 is z12 + z, and z3 is z22 + z, and so on. If the series of zs will always stay close to z and never trend away, that point is in the Mandelbrot set.” Your goal with this assignment is to essentially repeat Benoit Mandelbrot’s contribution to mathematics by writing a program to visualize the Mandelbrot set.
In case your math is a little rusty, a complex number is a number in the form a + bi, where i is the imaginary number, equal to the square root of -1, and a and b are real numbers. The number a is referred to as the real part, and bi is the imaginary part. Imaginary numbers can be plotted in a two dimensional plane in much the same way as ordered pairs are plotted in middle school algebra. The horizontal dimension of the complex plane is the real dimension (i.e. a), and the vertical dimension is the imaginary dimension (i.e. b). The magnitude of a complex number (sometimes called the modulus) is simply its distance from the origin, calculated using the Pythagorean Theorem.
Figuring out exactly which complex numbers are in the Mandelbrot set and which aren’t can get a little hairy. Some numbers are easily shown to be inside and some are easily shown to be outside, but other numbers are a little trickier to figure out, and this is where computers come in. To show that a given number is outside the Mandelbrot set, you can just calculate the series of zs and if the magnitude of any number in the sequence is bigger than 2, the number is not in the Mandelbrot set, and you can stop computing the sequence. For numbers that are in the Mandelbrot set, the sequence will go on forever without reaching a magnitude of 2, so you need to give up at some point and resign yourself to the fact that the number may be in the Mandelbrot set. The most spectacular images of the Mandelbrot set come not from plotting the numbers inside the set, but from plotting numbers that are just outside the set and assigning them colors based on how long it takes for the sequence to reach a magnitude greater than 2.
