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Subsubsection 1.1.1.1 The Mandelbrot set

The Mandelbrot set is one of the most iconic images in all of complex dynamics. As we will learn, it arises in the study of the family of functions

\begin{equation*} f_c(z)=z^2+c. \end{equation*}

Note that the variables are all complex numbers and the Mandelbrot set itself lives in the complex plane.

Figure 1.1.1 The Mandelbrot set

Figure 1.1.2 The region displayed on Scientific American's cover

Figure 1.1.3 A zoom into the cover image

Figure 1.1.4 A much deeper zoom into the cover image

The preceding images were all generated with the following web app, where you can easily make many more such images:

https://marksmath.org/visualization/mandelzoom/.

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