The images in the website banner and on some of the pages in the "Math Learner's Guide" are representations of regions of the Mandelbrot set. The Mandelbrot set is one of the most famous and beautiful sets that arise in the study of fractals, which are useful in modeling many natural phenomena, including trees, coastlines, and snowflakes. Check out the linked pages to zoom deep into intriguing images. Part of what amazes me is that something as intricate and beautiful as the Mandelbrot set is generated by this simple complex function:
$$f(z)=z^2+c$$
"This simple complex function" may sound oxymoronic, but "complex" here doesn't mean complicated. It refers to the set of complex numbers, the domain of the function.