Some general information about emergence, especially as it relates to the natural world.

# Emergence

Emergence is, in the words of its Wikipedia entry, "the way complex systems and patterns arise out of a multiplicity of relatively simple interactions." Theories of emergence (and its cousin, self-organization) are some of more interesting things to come out of 20-21st century science and philosophy.

19th century perspectives on system and complexity were based on top-down thinking, in which a system could only exhibit interesting or complex behavior if it was designed that way from the start. Teleologies (when they could be spoken of at all) were reflections of theist, design-oriented thought.

But in the mid- to late 20th century a new vision of the natural world started to arise. In almost every domain examined it was observed that complex and ordered structures could arise from very simple interactions. In many cases the patterns were surprising; those observing them would not have guessed *a priori* that such complex structures could arise from such simple generative rules. In chemistry, biology, mathematics and computation, game theory, and social systems surprisingly complex behaviors were observed as the outcome of simple rules and interactions. Chaos theory, cellular automata, fractal mathematics, neural networks... many of the more interesting developments beginning in the 20th century (and extending into this one) have emergence as one of their foundational principles.

This project is an exploration of musical emergence. What follows are some examples of emergence taken from other domains.

# Examples of Emergence

## Chaos

Much has been written about *Chaos Theory* in the popular press over the past few decades, so many people have some knowledge of it. Mathematical chaos, or sensitivity to initial conditions, is often used to explain why many processes in the natural environment are impossible to predict and characterize with any accuracy. Complex systems are varied as the weather and the stock market are shown to be chaotic, unpredictable, and unknowable over long periods.

Chaos can arise from very simple system. If one imagines a universe of only two objects--say, a star and a planet--these objects will orbit each other in a well-behaved and predictable way. But if you add another object the system becomes unpredictable and non-linear. The real universe contains many more than three objects, so it is easy to see how the 18th century concept of a clockwork, ordered universe was laughably naive.

This plays well into the post-modernist perspective of an arbitrary, random, and uncaring universe. It supports our fears that the processes of life are fragile and unstable, and a small, inconsequential change in the present can have far-reaching (and usually dire) consequences in the future. And while the mathematics of Chaos Theory are true, such understandings miss an important point, perhaps *the* most important point.

Scientists and mathematicians often study chaos through *iterative functions*. In such functions a number is fed into an equation (often a very simple one), and the results of the calculation are fed back into the function. This process is repeated and the evolution of the system over time (the iterations) is examined. If the equations are of a specific form the system is chaotic, and arbitrarily small perturbations in the initial input give rise to arbitrary large changes in the system's evolution, given sufficient time. These systems are unpredictable, as all chaotic systems are.

But does that mean they are without form and pattern? Not at all.

If one examines the long-term evolution of the system patterns can become obvious. The value at any particular iteration in the future is unknowable, but when the system is examined as a whole obvious patterns emerge. The value of the function may not follow simple, predictable paths, but a path *is* followed, and there are points (the so-called "strange attractors") about which the functions orbit, albeit in a complex way. The illustration below is of the Lorenz system, and the large-scale structure is obvious. Some systems of this nature were discovered long ago, but the patterns were so surprising and unexpected that it took decades (and the advent of modern computing technology, which allows one to look at the systems over long times) to notice them.

(All images may be clicked to enlarge)

## Fractals

Fractals are another example of how complex patterns can arise unexpectedly from simple processes. By now most people are familiar with some of the more interesting properties of fractals, especially how they display detail and pattern at an infinity of scales. Given sufficient time and precision, a fractal equation can be examined at any arbitrarily small scale and still display detail. Fractals are by their nature self-similar and scale-invariant. What this means is that structures seen at any scale can be seen in similar ways at other scales.

Mathematically fractals are easy to define. A form is composed of smaller copies of itself (perhaps transformed in some way), which are in turn composed of even smaller copies. The process continues infinitely, and fractals occupy a dimensionality somewhere between the whole numbers. There is a Buddhist meditation that captures this idea nicely. The practitioner is asked to visualize the Buddha with light rays shining from his every pore. At the end of those rays sits another, smaller Buddha, from whose pores light rays also shine. This continues infinitely, with smaller and more numerous Buddhas.

Fractals can arise from iteration of some very simple equations. As with the study of chaotic systems, a number is fed into an equation, the results of which are fed back into the system. The process is repeated. For some initial values the cycle flies off to infinity (points which are considered out of the set), and for some it remains limited (the in-set points). The fractal arises at the border of the set of points which are "in the set."

The following is an illustration of one very small portion of the plane for one very simple equation. Strictly speaking, points in these sets are either in or out (black and white), but often--as in the following illustration--points are colored to characterize how quickly they start heading off towards infinity on the outside of the set. This helps to emphasis patterns which would otherwise be missed and also appeals to our sense of aesthetics.

Approximate fractals are seen widely in nature, from the wandering path of a river to the delicate leaf of a fern.

Fractals also arise unexpectedly in human culture. All too often in modern life, the organizational principles of modern society are based upon arbitrary intentions divorced from natural constraints. But in cultures that have to work within the constraints of nature (as opposed to strong-arming them into submission), fractal structures sometimes arise in things like the arrangement of living spaces. Tribal art sometimes repeats fractal patterns, possibly in part through working within technological constraints, and in part because we as humans find the patterns pleasing. For those who are interested in this, Ron Eglash (as Associate Professor at the Rensselaer Polytechnic Institute) has done extensive research and documentation into the fractal patterns that are found in some African indigenous design.

The above distinction between modern and indigenous cultures may sound somewhat arbitrary, but it illustrates an important point. Emergence takes place in the interaction of independent, local parts. But when a society--through force or strength or control--has the ability to ignore some of those parts, intention replaces emergence as a guiding principle. A tree's structure, for example, emerges out of the organism's adaptation towards its environment,and while that structure may not be the most efficient one could imagine, it is efficient and robust and balanced. Markedly contrasted to this are the so-called "mini-mansions" that dot our vanishing countryside. A surplus of energy, cheap building materials, and mechanized labor allow these to be built without regard to the environment, instead adopting as their guiding patterns the things which the builders and owners find of value, namely square footage and ostentatious displays of pseudo-wealth. (It is not surprising that many find the former beautiful and the latter an eye-sore.) The dwellings that have emerged from the interplay between man and environment tend to be exceedingly well designed, despite the fact that they haven't been "designed" at all. The mini-mansion, on the other hand is "designed," but it tends to provide space which is very un-livable, as many owners will regretfully admit after having finally obtained one.

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For more information on how emergence realates to music, see Music and Emergence.