### Introduction

Here are some excerpts from the literature written by authorities on Chaos. They should help in the understanding of this great mathematical field that reaches out into the many branches of science.

- Music, Maths, Science and GIC (General Idle Contemplation)

he who has noble thoughts is never alone

Here are some excerpts from the literature written by authorities on Chaos. They should help in the understanding of this great mathematical field that reaches out into the many branches of science.

ISBN 0-7474-0413-5

First published 1987

z -> z^{2} + c

where z begins at zero and c is the complex number corresponding to the point being tested. So, take 0, multiply it by itself, and add the starting number; take the result - the starting number - multiply it by itself, and add the starting number. ... The program needs to watch the running total. If the total heads off to infinity, moving further and further from the center of the plane, the original point does not belong to the set and if the running total becomes greater than 2 or smaller than -2 in either its real or imaginary part, it is surely heading off to infinity. How many times depends on the amount of magnification. For the scales accessible to a personal computer, 100 or 200 is often plenty, and 1000 is safe. The program must repeat this process for each of thousands of points on a grid, with a scale that can be adjusted for greater magnification. And the program must display its result. Points in the set can be coloured black, other points white. Or for a more vividly appealing picture, the white points can be replaced by coloured gradations. If the iteration breaks off at ten repetitions, for example, a program might plot a red dot: for forty repetitions a yellow dot, and so on. The colours reveal the contours of the terrain just outside the set proper.

Barnsley quickly found that he could generate all the now classic fractals from Mandelbrot's book. Mandelbrot's technique had been an infinite succession of construction and refinement...By using the chaos game instead, Barnsley made pictures that began as fuzzy parodies and grew progressively sharper. No refinement process was necessary: just a single set of rules that somehow embodied the final shape.

Barnsley's central insight was this: Julia sets and other fractal shapes, though properly viewed as the outcome of a deterministic process, had a second, equally valid existence as the limit of a random process.

ISBN 0-14-029125-3

First published 1987

... The butterfly effect means that nearby points on the attractor tend to separate - but they stay on the attractor. Think of a ping pong ball in an ocean, with complicated currents at the surface. If you release the ball from below the surface, it floats upward. If you drop it from above, it falls downward. It is attracted to the surface... The ocean surface is the attractor, and the ball always ends up there; but the dynamics on the surface can be very complex and unpredictable.

wobble |
- | wobble |
- | wobble |
- | wobble |
... |

At a higher temperature, pairs of consecutive wobbles become slightly different:

wobble | - | wobble |
- | wobble | - | wobble |
- | wobble | - | wobble |
- | wobble |

The period doubles in length; you now have to wait for two wobbles before everything repeats...As the temperature rises, the wobbles group into fours...etc... This doubling of the period by the creation of ever finer differences between consecutive sequences of wobbles is called a period doubling cascade. Each successive step in the cascade occurs as the result of an ever-smaller rise in the temperature.

In mathematical models there is a particular critical temperature, and when this is reached, the period has doubled infinitely often, resulting in chaos. The period-doubling cascade is a route from order to chaos. It's important because it is one of the commonest such routes.

...Feigenbaum discovered that (taking our helium example) the amount by which the temperature must rise, in order to double the period, decreases geometrically as the period gets longer. Each such increment is roughly 1/4.669 times as long as the previous one.

...A decreasing geometric series is an easy way to cram infinitely many events into a finite space, and any such series must have some common ratio. But when Feigenbaum tried different mathematical equations with a period doubling cascade, he got the same ratio, 1/4669...a new, totally unexpected simplicity, emerging from some of the most complex behaviours known to mathematicians.

Emergent simplicities collapse chaos; they bring order to a system that appears to be wallowing hopelessly in a sea of random fluctuation.

...For example, your continued existence depends upon the atmosphere maintaining its normal range of pressures and its normal oxygen content. Because you are very big, on atomic scales, and your movements are very slow, your reaction to gas molecules is insensitive to tiny details. You "smooth out" the fluctuations by not noticing them, so you can build reliable structures upon the statistical features. Your lungs trawl the atmosphere for schools of oxygen molecules, and it doesn't matter how they get into the net as long as the total catch is its usual size. A very tiny, fast moving creature would have to hunt down individual gas molecules like a hunter-gatherer on the Savannahs, and would be terrible vulnerable to random shortages.

...Statistical regularities are certainly one important and widespread mechanism for emergence...However many emergent features do not come from statistics. There is nothing statistical about PI, the Feigenbaum number, the Mandelbroth set - or chlorophyl., DNA or homeotic genes, for that matter.

... Simplexity merely explores a fixed space of the possible. Complexity enlarges it. And both processes collapse the underlying chaos, producing stable features from a sea of complexity and randomness.

... Complicity arises when simple systems interact in a way that changes both and erases their dependence on initial conditions. The hallmark of complicity is the occurrence of the same feature or features in systems whose rules are either known to be very different, or are expected to be very different if only we could find out what they are. This carries an important consequence: Complicity is a convergent process; it homes in on the features regardless of fine detail in the rules...The prime example of complicity is evolution, for which the two systems of rules are the chemistry of DNA and the systematic ways in which organisms interact with their environment. Consider the animal feature "wings". Such a feature has evolved several times, starting from entirely different circumstances, in insects, pterodactyls, bats, birds...All 4 evolved wings because they solved the identical problem in their environment: to get off the ground. They built a bridge to "Flight island" and expanded the space of the possible.

... It is complicity, not quantum mechanics, that leads to consciousness. We can see why consciousness evolved: A good way to predators or find mates or generally manage your life is to decide between alternatives rather than blindly following predictable rules.