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Section 1.1 Images of complex dynamics

I remember the moment I found I could become a mathematician. It was the summer between my sophomore and junior years of college and I happened upon the August, 1985 edition of Scientific American. The cover featured one of the most captivating images I've ever seen with swirling colors radiating out of a convoluted central structure. I quickly turned to the article in the “Computer Recreations” column of that issue and found many more such images that, according to the article, lived in the complex plane.

To be clear, mathematics consists of much more than images. Even in that first, introductory article, I found several mathematical expressions, including the fairly benign looking \(z \to z^2+c\text{.}\) For me, mathematical intuition deals largely with the relationship between images and formulae. Yet, images alone can extremely captivating and don't need a lot of formal effort to appreciate. Thus, let's start by looking a few of the images that we'll learn about.