☞ Happy Accidents

Glitches, the Mathematics of Protozoa, and Computational Delight

A biomorph, using the method of Clifford Pickover.

In the late 1980’s, the mathematician and writer Clifford Pickover was playing around with fractals. Pickover is incredibly prolific and known for his explorations of strange attractors, fractals, and many other topics in mathematics. Playing around with Julia sets, he accidentally programmed them into his computer incorrectly, and instead of generating the expected forms, he ended up generating a wide variety of protozoa-like shapes. Essentially, he found a mathematical algorithm for single-celled organisms [pdf]:

Of course, this is not really true. These shapes simply have the appearance of microscopic protozoa floating on a microscope slide. But there are two intriguing takeaways from this:

The first is that biological forms can arise from quite simple mathematics. The basic functions that describe different forms are pretty basic. This is a delight and a hallmark of the world of fractals and chaos: the organic, despite its seeming complexity, can sometimes be described using mathematics.

But the other takeaway is related to the nature of the discovery: Pickover discovered these shapes by accident. He made a programming error. When it comes to the computational exploration of these kinds of spaces, accident and chance can actually be a powerful means for exploration. Because often, when you are working with a high-dimensional system, you sometimes need some randomness or mistake to perturb the system into something interesting.

A similar kind of result even appears in Mandelbrot’s seminal The Fractal Geometry of Nature. In the text, Mandelbrot includes an image that is described as “The Computer ‘Bug’ as Artist.” It’s a wonky and glitchy artistic image. As he writes, the image “can be credited in part to faulty computer programming…The change that had been wrought by a single tiny bug in a critical place had gone well beyond anything we had expected.”

A couple years ago, I even created a computational means for exploring these kinds of glitches more deliberately with a computer program I called Happy Accidents. It randomly mutated certain types of Processing code in order to expand the space of creative possibilities. (Programming enthusiast alert: because the version of Processing I used is based on Clojure, I was able to easily do the kind of mutations found in genetic programming). You could easily pour in preexisting code, which would be perturbed, and you would get to discover new visualizations.

In the end, just as mathematics can be a source of biology, a raw material of biology—mutation—can be a source of novel mathematics.


A few things worth checking out:

  • An upcoming book from Stripe Press called Scientific Freedom: The Elixir of Civilization. This book makes the argument that we need more basic and transformative research. We need to foster the weird stuff that is less incremental, but potentially far more change-making. In other words, research that is low-risk but high-reward. It explores how science operates (or often, fails) and what can be done to allow for more fundamental and important discoveries. It includes thoughts on scientific funding agencies, scientific research institutions, and much more. Very thought-provoking.

  • Message Not Understood: This is a Kickstarter campaign for a documentary being produced by my friend Daniel Krasner. The film is about the prehistory of the personal computing revolution. The video on the Kickstarter page is fascinating.

  • Novel Technosignatures: A paper about ways of detecting alien civilizations. Lots of fodder for science fiction stories in this one.

Until next month.