"It turns out that an eerie type of chaos can lurk just behind a facade of order - and yet, deep inside the chaos lurks an even eerier type of order"

-- Douglas Hostadter

There are some significant changes in the eBook that are not in this draft. You may purchase the eBook here

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Complexity theory is the study of how complicated patterns can result from simple behaviors of individuals within a system. Chaos is the study of how simple patterns can be generated from complicated underlying behavior. Chaos theory is really about finding the underlying patterns in apparently random data. It is unfortunate that science has chosen the word "chaos" to describe this form of order because the word "chaos" is at odds with common parlance, which suggests complete disorder. Nonetheless, science defines "chaos" as a form of order that lacks predictability.

Chaos theory helps us to understand patterns in nature. It has been used to model biological systems, which are some of the most chaotic systems imaginable. Chaotic patterns show up everywhere around the world, including cloud patterns, the currents of the ocean, the flow of blood through fractal blood vessels, the branches of trees, astronomy, epidemiology, and the effects of air turbulence.

Chaos theory states that, under certain conditions, ordered, regular patterns can be seen to arise out of seemingly random, erratic and turbulent processes. Chaos theory does not emphasize the inherent disorder and unpredictability of a system. Instead, chaos theory emphasizes the order inherent in the system and the universal behavior of similar systems.

Computer graphics makes it possible to study how these patterns appear and disappear with changes in the system parameters. Many patterns, such as the vortex of a tornado, stock market trends, and crowds of people can now be subjected to computer modeling.

Chaos can be simulated with simple computer graphics and a process called cellular automata (CA). With CAs, a fixed rule of pattern development is applied to a series of totally random initial conditions. With CAs, it possible to simulate how simple behaviors are generated from complex rules.

Picture a fine grid of squares where each square can be painted either black or white. In the first line of squares, randomly select which squares are to be painted black. The resulting random pattern generated in this first line represents the initial condition or state. Then develop a set of rules that determine the state (black or white) of each square in the next line of squares based on the colors of the three nearest neighbors in the previous line. From these rules, iteratively develop the black or white states of each successive line of squares. This process of evolving a pattern from complex (random) initial conditions on the first line of cells and a simple set of pattern development rules results in an endless array of pattern structures.

For example, the following three CAs are products of three different rule sets. The CA represented by rule #1 contains a very regular, repeating pattern. In fact, no matter how many times we randomly change the initial conditions (the pattern of black squares on the first line), the fixed nearest neighbor rules will produce different patterns that are all periodic.

The second CA represented by rule #30 contains a very irregular pattern. With careful examination, one intuits some kind of order. This CA is a picture of chaos. It is not random. It simply appears to be disordered but with some kind of implied order.

The third CA represented by rule #110 is a mixture of the first two CAs. Local structures are complex - some being regular and some being chaotic. Rule #110 portrays complexity -- a mixture of order and chaos. This type of CA represents many patterns in nature.

All three of these CAs represent systems where there was evolution of order (or patterns) despite the fact that the initial conditions were random. While the initial conditions (the randomly colored cells on the first line) defined the details of a pattern, the type of pattern was defined by the nature of the rule set and not the initial conditions.

Chaos models uses fixed rule sets for pattern development. However, the same pattern development rule set applied to two very close but different sets of initial conditions will result in two final patterns which differ from each other.

This is shown for the two CAs produced using the same rule set. The outputs are clearly different even though they are both periodic types because the initial conditions have changed.

This "sensitivity to initial conditions" that is present in certain systems has come to be called "dynamical instability", or simply "chaos". Because long-term predictions made for chaotic systems are no more accurate that random chance, we can get only short-term predictions with any degree of accuracy. Even the smallest imaginable discrepancy between two sets of initial conditions would always result in a huge discrepancy at later or earlier times, the hallmark of a chaotic system. It is now accepted that weather forecasts, which are chaotic systems, can be accurate only in the short-term, and that long-term forecasts, even made with the most sophisticated computer methods imaginable, will always be no better than guesses. Thus the presence of chaotic systems in nature seems to place a limit on our ability to apply deterministic physical laws to predict motions or patterns with any degree of certainty.

An interesting finding of chaos theory is that, even though the systems and their associated patterns may be quite different, there is a commonality in the rules which permit one to classify rule sets into higher level categories. In the case of the relatively simple CAs shown above, Stephen Wolfram discovered that the each of the 256 possible rule sets can evolve a system from a completely disordered state into one of four categories:

- Category 1: Initial random patterns terminate quickly. Any randomness in the initial pattern disappears.
- Category 2: Initial random patterns evolve quickly into stable or oscillating structures which do not terminate in any number of time steps.
- Category 3: Initial random patterns evolve in a chaotic manner. Any stable structures that appear are quickly destroyed by the surrounding noise. Local changes to the initial pattern tend to spread indefinitely.
- Category 4: Initial random patterns evolve into long lived complex structures. These patterns can be a combination of categories #2 and #3. Many patterns in nature are similar to category #4 structures.

The two characteristics of chaos, then, are:

- Chaotic patterns use a fixed and definable set of rules for pattern formation.
- Chaotic patterns are unpredictable because any small change in initial conditions could result in huge changes in resulting behavior.

One of the foremost contributors to chaos theory was Benoit Mandelbrot. In 1975 Mandelbrot pioneered a new geometry that helped visualize natural patterns. Among other things, fractal geometry helped describe the actions of chaos on a computer screen. Mandelbrot showed that many of the irregular shapes that make up the natural world are not random. Instead, these patterns have simple organizing principles.

The following five minute movie offers an easily understood explanation of the ideas associated with chaos theory.