A Mandelbrot image is produced by zooming into the border of the Mandelbrot set, an amazing mathematical construction exhibiting infinite variety and self-similarity.
Each point in the Mandelbrot set is produced by feeding the results of a complex number function back into itself many times (recursion) and observing how the function behaves. At each step (aside from the first) the previous output of the function is squared, and then added to complex number constant which corresponds to the original location of that point in the image. If the function stays bounded (i.e., does not become extremely large), that point is considered to be inside the set.
The interesting fractal patterns happens at the border edge of the Mandelbrot set. However, most of the rich color in Mandelbrot images is actually applied to the area outside the set, based on how quickly those functions escape to infinity.
The basic Mandelbrot equation can be adjusted to produce a other interesting sets and images. For example, a simple adjustment is to cube the previous value, instead of squaring it.
Mandelbrot fractals are interesting on a philosophical or theological level. The Mandelbrot set (and variations) are constructed in the sense of being the output of functions we created. Nevertheless, the infinite variety and beauty of the fractal is not really designed by us, but are somehow inherent in the mathematics themselves. So, studying the Mandelbrot set is an unending experience of exploration and discovery.
By analogy the Mandelbrot set answers the theological question: “What would we do in Heaven after millions of years had passed? Wouldn’t we get bored?” We would, indeed, if Heaven was nothing more than gold streets and pearly gates. Actually, though, God will be in Heaven. If God is a being of not only perfect order, but also an infinite being in knowledge and ability, then it would be possible for us to spend forever studying God — exploring and experiencing all that He is.
O the depth of the riches both of the wisdom and knowledge of God! how unsearchable are his judgments, and his ways past finding out! — Romans 1:33 (KJV)
[There are] things which eye has not seen, and ear not heard, and which have not come into man’s heart, which God has prepared for them that love him. — 1 Corinthians 2:9 (Darby)