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Exploring Chaos and Fractals is an electronic textbook which includes full text, work sheets, sound, video and animation. Parts of the material have been placed on this Web server as an experiment in electronic publishing of hypertext based material. As it was already in hypertext form, conversion into html format was straight forward.
Exploring Chaos and Fractals covers the subject of chaos theory and its closely related field of fractal geometry. These new areas of research are causing a great deal of excitement world wide. Previously, many events were considered to be chaotic, unpredictable and random. The dripping of a tap, the weather, the formation of clouds, the fibrillation of the human heart, the turbulence of fluid flows or the movement of a simple pendulum under the influence of a number of magnets are a few examples. The slightest change in initial conditions, it was thought, caused results which appeared to follow no pattern. The advent of the calculating power of the computer has changed this belief. Patterns are being found, now that it is possible to have the machine do the immense number of calculations required to fully analyse the simplest behaviours. Yet the outcome is still unpredictable.
Chaos theory and fractal geometry and fractal geometry are cutting edge knowledge and yet are accessible to students. It is invaluable for students to see that mathematics, science, technology art and the nature of society are in a state of change.
Course material is presented with multiple entry points and multiple themes. For example, the user can start off with an image-based module where various views and/or animations or zooms of fractals can be examined, much like an electronic art gallery. Connections can then be made at any time to areas of interest. For example:
Exploring Chaos and Fractals has been designed to be used by individuals or groups in secondary or tertiary settings. It can be used within a single discipline or as a cross-curricular resource. All images and text modules can be printed.
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Out on the Web there are a variety of other resources on Chaos and Fractals.
The introduction to the course follows, or you can go straight to the overview with some examples of the content. To avoid excessive load on the net, at this stage no deep zooms or animations are available at present.
Do you need to be told about chaos, or is your desk a permanent example? As everyone knows, beneath what those intolerably neat and tidy people consider to be chaos, there is a form of order. The chaotic housekeeper can always find the item of their desire - as long as no-one tidies up!
Many systems which scientists have considered totally random, unpredictable and without form have now been found to be otherwise. There is form and pattern hidden within the CHAOS . It is a part of the natural form - a definitive ingredient of Nature itself.
The Oxford Concise Dictionary defines chaos as "Formless primordial matter; utter confusion." The day has come when there is a need for an update - Chaos Theory is changing the way scientists look at the weather, the way mathematicians plot equations and the way artists define Art. Population dynamics is one area which can be very sensitive to small changes in initial conditions. So can the weather. A butterfly flapping its wings in a South American jungle, it is said, can lead to a hurricane in China. This is the signature of Chaos Theory!
Think about the "new" disease, AIDS. Somewhere, sometime, maybe, a cell mutated. Think about the number of cells there are in the world. This was a pretty tiny event. BUT it happened to be a crucial mutation. Somewhere, somehow, the cell multiplied and spread. Or, maybe, the disease first moved into the human race - one infection. In terms of the population of the earth, a very small event.
Something determined the spread of this disease. From a tiny change in initial conditions a disease took a huge toll. There was a mechanism, a necessary set of conditions so the spread was not completely random. But it was unpredictable. Many other such mutations or infections must have occurred but had little or no effect. That tiny event somewhere led to large percentages of some populations to be struck down. It changed the nature of the whole human population. The butterfly of AIDS flapped its wings somewhere in Africa and cause a hurricane across the world.
As scientists studied these systems, a mathematics evolved which had already drawn interest from pure mathematicians. This mathematics involved ITERATION - taking the answer to the equation and feeding it back into the equation, over and over again. In watching the result of this process, some fascinating behaviours were observed. When the mathematicians and scientists got together, with the benefit of machines which could do their calculations within minutes, a new science was born.
In playing with these ideas, a new way of doing science grew. These computers could not only calculate they could communicate too. Information flew around the globe. You no longer had to be in the right place or talk to the right people. The equipment and information was available to masses of people all over the world. And their mathematics produced images which were stunningly beautiful, and, at times, awesomely like nature.
A new art form was born and a whole new set of questions arose about the nature of nature itself. These images, called FRACTALS were fun. They had an ever growing fan club who became obsessed with their generation. You can view Mandelbrot Images or use the to Fractal Generator plot fractals by clicking here.
Science, maths, computing, philosophy, art - where will Chaos end? Nowhere and never, we hope!
The Nature of Chaos
The natural world has always had a chaotic way about it. The mathematical world has always had amazing complexity resulting from simple equations. So why has Chaos theory just evolved as such a critical part of science, mathematics, art and the computing world?
A simple answer: computers.
The calculations involved are repetitive, boring and number in the millions. To produce the Mandelbrot Set on a single screen takes, it is estimated, about 6,000,000 calculations. No human would be stupid enough to endure the boredom. But a computer will. Computers are particularly good at mindless repetition. The computer is our telescope, our microscope and our art gallery. We cannot really explore Chaos without it, and we certainly can't produce fractals unaided.
But it is necessary to use the computer as an investigative tool. Most computer use is based on putting in data and instructing the computer on what output is required. Chaos Theory arose as scientists and mathematicians started to play. To put in numbers and watch as they careered around the plane, mostly the complex plane, in detailed patterns. They watched as the computer produced the numbers, and didn't just wait for the final result. And they tried different ways of plotting and exploring equations - mostly for the fun of it.
Playing with mathematics, science and computer programming produced images which looked like nature. Ferns and clouds and mountains and bacteria. They indicated why we couldn't predict the weather. They seemed to match the behaviour of the stock exchange and populations and chemical reactions all at the same time. Their investigations suggested answers to questions which had been asked for centuries - about the flow of fluids as they move from a smooth to a turbulent flow, about the formation of snowflakes, about the swing of a pendulum under the influence of magnets, about tides and heartbeats and cauliflower and rock formations and the behaviour of Hyperion.
This new theory was infiltrating a vast range of intellectual domains. And then they started plotting the fractals. Some mimicked nature. These caused a stir. Some were stunningly beautiful. These created a debate: Was this Art? And some were just fascinating. Mandelbrot and Julia became the jargon of a new group of enthusiasts.
Chaotic systems are not random. They may appear to be. They have some simple defining features:
1. Chaotic systems are deterministic. This means they have some determining equation ruling their behaviour.
2. Chaotic systems are very sensitive to the initial conditions. A very slight change in the starting point can lead to enormously different outcomes. This makes the system fairly unpredictable.
3. Chaotic systems appear to be disorderly, even random. But they are not. Beneath the seemingly random behaviour is a sense of order and pattern. Truly random systems are not chaotic. The orderly systems predicted by classical physics are the exceptions. In this real world of our, chaos rules!
The link between chaos and fractals is strong. Fractal geometry is the geometry which describes the chaotic systems we find in nature.
Fractals are a language, a way to describe a geometry. Euclidean geometry is a description of lines, circles, ellipses and so on. Fractal geometry is described in algorithms - a set of instructions on how to create the fractal. Computers translate the instructions into the magnificent patterns we see as fractal images.
They are images of the process of a mathematical exploration of the space in which they are plotted. Let's take the computer screen as representing a space. Each point on the screen is tested in some way. Usually an equation is iterated with this point as its starting value. That means a result is calculated using the equation, and this value is fed back into the equation leading to a further result being calculated. This process is repeated over and over and over. As a result of this calculation, the point on the screen at which we started is plotted in a particular colour. Then the computer repeats the process for the next point on the screen. And so on over the whole screen until the coloured image is fully produced. This is one method - there are others, you will meet in this course.
This image, however, is far more than a flat picture in magnificent colours. A fractal is infinitely complex. That is, if you zoomspider1 in on any part of the fractal you will always find more detail. Each stage tends to have the same form as the original. So the fractal lacks scale. A small portion of the fractal is just as detailed as the original.
The amazing thing about fractals is that the formulae used to generate them are often extremely simple. A simple formula can lead to complex images. These images are sensitive to the initial conditions. Sound a bit like Chaos theory?
A piece of cauliflower has a fractal form. Each segment of the cauliflower is similar to the whole. And the segments then break down into littler, similar segments. Cauliflowers are limited by the real world. Mathematicians, computers and imaginations are not. Hence mathematically generated fractals go on forever, deep within themselves.
Benoit Mandelbrot, in 1979, was playing with a rather simple little equation:
(z is mapped as the square of z plus c) where z and c are complex numbers, and c is a constant. The computer screen became the complex plane. Each point on the plane was tested into this equation. If the iteration went out of control, the point on the screen was plotted white. If it stayed within some bounds, for ever, then the point was considered to be inside the set and plotted black. The Mandelbrot set was born.
>Out of this seemingly simple equation and simple steps performed on it arose what some claim to be the most complex mathematical object ever discovered. Many mathematicians are still exploring it, finding more patterns, twirls, filaments, mini-Mandelbrots, dragons and spirals as they plunge deeper and deeper into its depths.
Colour was added, indicating how quickly the iteration went out of control, and programmers made their computers cycle the colours.
So the image needs a computer to exist. It needed the computer to be able to perform the vast number of calculations, to produce the colours at rapid speeds and to allow us to choose any part to explore further.
Mandelbrot coined the word "fractal" to describe his new object and those like it. He argued that the edge of the set was more than a line (of dimension 1) and less than an area (of dimension 2). He claimed it had a dimension somewhere between the two. A fractional dimension.
Since then fractal enthusiasts have appeared anywhere the technology existed to allow them to play with this new discovery. One group, the Stone Soup Group, became so enamoured of these creatures they collaborated, via the computer networks, never meeting, to produce a piece of software to generate an infinite variety of these images. They allowed the user to colour cycle the fractals, to change the colours, to zoom in on any part and to recalculate the work of others lodged on the network bulletin boards. The program, Fractint, is unrivalled and free. the authors do not permit it to be sold. More of them later. But what can stir people up so much to spend countless hours working on something for no financial gain? Fractals!
Chaos theory is riddled with strange patterns which underlie seemingly random and unpredictable behaviour. Although we don't know which weather pattern will occur ten days down the track, there are weather patterns which are possible, and likely, and others which are not. The likely patterns - snow in Antarctica, heat in India - are called ATTRACTORS. These patterns ATTRACT the system into their state. In examining the process of the iteration of many non-linear equations it is found certain patterns occur and often lead to some kind of bounded behaviour. These are the attractors for the equation.
The concept of an attractor is crucial to the understanding of chaos theory. Some of the attractors discovered are most surprising in form. The attractors which arise for chaotic systems have been called Strange Attractors - because they are!
This is a course about chaos and fractals, so you wouldn't expect some nice, linear way of learning about it to evolve, would you? It hasn't!
This course is a "choose your own adventure". You can start anywhere and pop all over the place. Some topics require a sequence, such as the mathematics and programming streams. Others can be used as individual sessions.
The themes which emerged as students worked with the authors to develop this course were:
For each topic within each theme there is a Work Sheet and, in most cases, an associated Reference Sheet. The Work Sheet suggests a way of exploring the topic. The sheet will note at the top as prerequisite if there are any sheets which need to be done first to fully appreciate the concepts covered by this sheet.
The Reference Sheet gives an idea of what you will find out and detailed references to that particular aspect. There may also be suggestions for further work in the aspect under consideration.
These are numbered logically. For example the first sheet in the Mathematics theme is named as WSMA1 for the Work Sheet. An OVERVIEW of the sheets is given.