#### Mandelbrot Set Basics

Introduction Fractal To Desktop renders the Mandelbrot Set onto your desktop. The Mandelbrot Set is defined by the equation Z^2+C. If you’ve seen the video on the homepage, this probably makes no sense. How can an equation this simple create shapes like the Mandelbrot Set? Complex Numbers Firstly, Z and C are not normal numbers, they’re complex numbers. Complex numbers follow the format a+bi where "a" is the real number ,"b" is the imaginary number, and "i" is the square root of negative 1. When showing the Mandelbrot Set, you’re seeing the complex plane where the x-axis contains the real number component and the y-axis contains the imaginary number component. What this means is that this function takes in two complex numbers and outputs a different complex number that can ...

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#### Perturbation Theory

Remember the program that I recommended in the last article called Xaos? If you downloaded that and zoomed in, you’ll notice that the screen gets very blurry. Here’s a picture of what I’m talking about: This problem has to do with the resolution of numbers inside a computer. Computers are either 32-bit or 64-bit and that corresponds to the size of the number the computer can use. The number of bits refer to the number of digits in base 2; a 64-bit number has ~15 digits in base 10 and 64 digits in base 2. As you zoom into the Mandelbrot Set, you need more precise numbers to show details. With a limit of 15 digits, you can’t show those details which causes the blurriness. We need to be able ...

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