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other interlude

May 29, 2017

A cool zoom view of the classic mandelbrot set. If possible, watch w/ full quality.

take the series defined by  z[n] = z[n-1]^2 + c   , where the parameter c is a constant and z[0] = 0 . Also z and c are complex. (that really just means that under the hood there are 2 coupled series pretending to be one, as well as 2 constants, corresponding to the horizontal and vertical axes in all the pictures. The one corresponding to the vertical axis is the imaginary component, when we go back to talking about complex numbers. Imaginary just means when you square it we pretend it becomes negative in the process. Which is just an accounting rule that makes a bunch of formulas come out compactly and neatly. It can be written out using nothing but ordinary real numbers, which is what the poor gnomes in your computer actually have to work with.) There are as many such series as there are values of c. For some values of c, the series blows up to infinity, for other values of c , it stays bounded. What the pictures are is a plot of the boundary between the values that escape and the ones that do not – jazzed up with color. The color typically is mapped from the number of iterations that the computer program had to do for that pixel’s c value before it became clear that the series would escape for that particular c value. As we zoom in, the same shapes appear again and again…

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