2008-12-23T11:20:00.000-08:00

“Big whorls have little whorls,

Which feed on their velocity;

And little whorls have lesser whorls,

And so on to viscosity.”

-Lewis Fry Richardson

automata
Performance for two pianos and automechanical organ, live in Łódź, Poland

automata
Performance for recorder, flute, 2 saxes, bass clarinet, 2 pianos, 2 electric organs, live in Amsterdam

On self-similarity and emergence

Self-similarity can be found just about everywhere. Structures that exhibit this property possess details, on large and small scales, that mirror the entire form (and vice versa). Consider your body. From the largest arteries to the tiniest capillaries, the self-similar branching design of your circulatory system makes it possible for oxygen-carrying blood to reach every part of your body, on all crucial scales. The self-similar branching network of the neurons, clusters of neurons, and clusters of clusters of neurons that make up your brain and nervous system also make it possible for electrochemical signals to be transmitted efficiently throughout the assembly that is you. Consider the surface of the ocean, made up of tiers of waves upon waves upon waves, each smaller wave nested within a larger wave. Consider a mountain range, with peaks and valleys within peaks and valleys on scales ranging from kilometers to micrometers. Consider trees, galaxies, and lightning bolts. Using the satellite imagery software google earth, one can zoom in and out to any height above the earth’s surface and see similar structures on all scales.
Self-similarity also occurs in the art world, integrated into ornamental design, the drawings of M.C. Escher, and the paintings of Jackson Pollock. In music, the canons and fugues of J.S. Bach exhibit a temporal self-similarity that is pleasing to the ear. In mathematics and geometry, beautiful abstract structures have been discovered in expressions of the Mandelbrot set, logarithmic spirals, and other fractals.

The concept of emergence has parallels with self-similarity. Throughout all scales of nature, order can emerge from the interactions of a multitude of tiny chaotic particles, just as new, complex behavior can emerge from simple rules. Consider an ant colony. A single ant behaves in a simple way, but the self-organizing dynamics of a colony behaves as an intelligent and complex superorganism – information is processed by the collective and not by individuals. Consider your own consciousness as you read this. A single neuron in your brain behaves according to relatively simple rules, but the complex network of billions of these cells somehow gives rise to intelligence (we are not even close to understanding this most mysterious phenomenon). In contrast, scientists have recently discovered that simple computer programs governed by simple rules can produce infinite complexity. Cellular automata is one notable model that gives rise to a wide variety of complex patterns and behavior, an analysis of which may illuminate some of the deepest mysteries in nature.
How did the universe get so complex from simpler beginnings?
How does order emerge from such complexity?

Automata explores these ideas in music. Its structure is self-similar: the whole resembles the details, identically, on smaller scales. If one were to speed up the tempo to any order of magnitude on a computer, the structure would sound the same. This is similar to zooming out of a forest or desert in google earth – one would see larger structures but details would be lost. While the visual score mimics the behavior of cellular automata programs, the music itself is constructed using simple rules that give rise to complexity over time. As the piece could theoretically go on forever, complexity would emerge and grow as time passes. Improvisation is incorporated into the structure so that complexity and freedom inevitably bloom out of an initial mathematical regularity.

click score to enlarge

jacobadler.blogspot.com