It was probably not the first thing that made me realize that math is beautiful, but it was something that amazed me the most and still does to this day: The fact that the Mandelbrot set is not only infinite - in a way that eg. the Koch snowflake is infinite - but that it is infinitely complex, the complexity never ends, you can zoom it forever and you will never find exactly the same patterns, the information that is contained in it is infinite and yet it is described by such a simple formula.
It made me wonder whether math was discovered or created, whether things like the Mandelbrot set existed independently from their discovery or not, whether the infinitely complex pictures existed if they were never seen etc.
I remember the sleepless nights in elementary school when I was writing programs to explore the Mandelbrot set, to find nice looking colors, to animate it - impossible to do live at that time so I had to learn how to script some animation program that I had, wait an hour to realize that I had the colors wrong, change one number, wait another hour, rinse and repeat.
I didn't know about complex numbers at that time. I only knew that I was looking at something most amazing in the world and just couldn't stop exploring. Fractals became my obsession and were probably one of the reasons why I started programming more seriously.
I was fascinated by the fact that I could zoom it so much that it was like finding some proton on the face of Earth and zooming it to the size of a planet, and then looking at that planet-sized proton with an electron microscope. I could print what I found and I knew that no one in the Universe has ever seen it before me and no one will ever be able to find it even after looking on my printout - the scale was so amazing.
I remember how I got scared when I eventually saw large pixels in my Mandelbrot set! Finally I realized that I hit the limits of the floating point number precision on my 386 but I knew that the complexity of the Mandelbrot set was there, somewhere, even if I couldn't see it with my computer at that time.
Those are some of my favorite pictures that I posted to Wikipedia:
You can download them from Wikipedia.
One of those pictures was magnified 248,034,982,258 times - probably the Cool Mandelbrot but I'm not sure because strangely all of them have the same description on Wikimedia Commons (something had to go wrong when they were copied from Wikipedia to Wikimedia Commons).
I would be honored if you'd like to use those pictures in your book. If you need higher resolution pictures or more information about them then I might be able to find something in some very very old backups.
Good luck with the book!