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low cost private java pages The Mandelbrot Set The Mandelbrot set is a classic application of complex arithmetic. It is the example of a fractal set. Here and in the future we are going to try to separate the mathematical definition of our data from its display on the screen. In fact we won't even add the screen display till the second iteration of the program.Our data structure will be a two dimensional array, each element of which represents a fixed point in the complex plane.
low cost private java pages The array is defined by the value of the lower left point, the gap between points and the number of pixels in the x and y directions. Thus given that element (0,0) of the array matches the complex point (x0,y0), each point is separated by a value gap, we know that element i,j of the array represents the point (x0 + i*gap, y0 + j*gap) in the complex plane. Since we know this by position of the array element alone we dont' need to store this value in the array.What will we store in each element of this array? We'll calculate a number to go there in the following fashion. Let z = 0 + 0i and let c be the position of that array element in complex space.
low cost private java pages calculate z = z*z + c and iterate as follows up to 200 times:for (i=0; i < 200; i++) { z = z.Times(z) + c; }The Mandelbrot set is composed of those elements which, no matter how long we do this, never approach infinity. Since infinity can take a mighty long time to approach, it's fortunate that a fairly elementary theorem in complex variable theory guarantees that any number whose magnitude ever exceeds two in this iterative scheme will become as large as one might wish. (i.e.
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