Chapter 8. Fractals


To my mind, there is something very “seventies” about fractal art. Even before James Gleick’s hugely popular book Chaos: Making a New Science (1987), which popularized the mathematics of chaos theory, fractal imagery, in particular the Mandelbrot set, was overly familiar from the kind of posters you might expect to see on the walls of hippie math grads.

We’ll never know why the renderers of early fractal art favored such lurid, psychedelic colors, but I assume it had a lot to do with the overconsumption of LSD. Anyway, I want this preconception of fractal art flushed from your mind as we explore self-similarity in this chapter. Fractal art can also be subtle and beautiful (see figures 8.1 and 8.2).

Figure 8.1. After experimenting with fractal structures, you’ll start to notice self-similar recursion everywhere. This illustration is from Ernst Haeckel’s 1904 book Kunstformen der Natur (Artforms of Nature)
Figure 8.2. Mandelbulb by Tom Beddard (2009), a 3D version of the Mandelbrot fractal. See Beddard’s site or for the math.

8.1. Infinite recursion

8.2. Coding self-similarity

8.3. Exponential growth

8.4. Case study: Sutcliffe Pentagons

8.5. Summary