A Meditation on the Mandelbrot Set
When Antonie van Leeuwenhoek first placed swabs of saliva under a microscope in the 1670s, he discovered a previously unknown world of microbes that lived all around us — on our bodies, in the water we drink, in the food we eat. Imagine the shock of discovering an entire realm of complexity and activity on a scale invisible to the human eye.
This is essentially what I felt when I was introduced to the Mandelbrot set, a mathematical object known for its self-similar intricacy and beauty. Videos on YouTube show hour-long zooms into deeper and deeper regions of the Mandelbrot set. From start to finish, these videos zoom in more than if they started with a view of the entire universe and zoomed in until only a Planck length was visible. Yet no matter how far you zoom in, there is no end in sight to the level of detail and intricacy contained in the fractal.
The Mandelbrot set is the set of all complex numbers that do not “blow up” under iteration of the complex-valued function f(z) = z²+c, starting at z=0. The function f(z)=z²+c, as simple as it is on its surface, is not at all simple when it comes to its behavior. In order to visualize this behavior for the images in this article, I wrote a computer program to do the computations for me (as is necessary for all but the most superficial glimpse at the Mandelbrot set’s workings).
