Mandelbrot Set

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About Mandelbrot Set

I have always been fascinated by the existence of the Mandelbrot set. Humanity discovered (or invented?) something infinitely beautifully complex. In theory, you can zoom into the Mandelbrot set forever, it repeats its patterns infinitely and will never stop.

The Mandelbrot set is a set of complex numbers for which the function f(z)=z*z + c does not diverge when iterated from z = 0. What is c, what is a complex number, what does "not diverge" mean? A complex number is the combination of a real number and an imaginary number. One imaginary number is the square root of -1, which we denote as I, which does not exist, so we just imagine it. But imagining numbers can cause all kinds of weird shenanigans. Like the Mandelbrot set.

The canvas on which my Mandelbrot set is drawn is a two-dimensional plane. We can now call the x-axis the real part and the y-axis the imaginary part. For this set of complex numbers on the canvas, we can now use the magic function f(z)=z*z + c. Starting at z = 0, z = z*z + c, which is just c, because z = 0 and 0*0 is still 0. For the next iteration z = 1, f(f(z)) = c*c + c. c is just a complex number, so a + bi. c*c is then a*a + 2abi + b*b*i*i. We know that I*i = -1, so we can now write a*a - b*b + 2abi, which again, is just another complex number c. We can now compute these values over and over and check if they go to infinity or stay bounded. The Mandelbrot set is all of the numbers which do not go to infinity but stay bounded in absolute value.

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