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“First construct a line of finite length. Then, divide the line into thirds, and remove the middle third. This creates two smaller segments. One can then iterate on the two segments to make four. Actually, the operation can be iterated infinitely revealing an infinite number of sparnessness. Strangely, Mandelbrot had managed to apply a mathematical thought experiment to a natural occurrence.”

Another form of the chaos theory graphed, and perhaps the most famous, is the Mandelbrot set. Created by Mandelbrot in 1979, by the formulation of the simple equation: nn + 1 = n^2 + c. The way the number to be graphed is to be determined is as follows:

“Where c is the complex value that is being tested and n is the starting number (called the seed), which is initially 0. This fractal is graphed by the properties a particular a particular point has after being iterated a number of times by the equation. To compute the first iteration, we would take the seed (in this case 0), square it (it’s still 0), and add c (the value of point we picked). This new value is now designated as n. In the second iteration, we square n once again and add c.”

When this set is graphed, each of the five types of iteration appears in the pattern. What makes the pattern so beautiful is using a different color for the iteration. As this set is expanded, each small detail can be examined more closely, and each of those more closely, to an infinite amount of possibilities. These examinations and millions of possibilities based upon the starting point are the essence of the chaos theory.

Formally, the chaos theory is defined as the study of complex nonlinear dynamic systems. Complex implies just that, nonlinear implies recursion and higher mathematical algorithms, and dynamic implies non-constant and non-periodic. Thus the chaos theory is the study of forever changing complex systems based on mathematical concepts of recursion, whether in the form of a recursive process or a set of differential equations modeling a system. Many people hold misconceptions about what the chaos theory really is, and even looking at this definition some may still. “Jurassic Park” contained a direct explanation of the “Butterfly effect” with the following example: “Often called the Butterfly Effect, it is the notion that a butterfly stirring its wings in Peking can transform storm systems in New York.” Through this definition many people were led that chaos is the study of undefined variables in a random or disorderly system. This is hardly the case. The “chaos” in chaos theory is order – not simply order but the very essence of order. While it is true that the chaos theory says that minor fluctuations can cause huge fluctuations: “It is a theory describing the complex and unpredictable motion or dynamics of systems that are sensitive to their initial conditions,” one of the main concepts is that while impossible to exactly predict the state of a system, it is usually possible to model the overall behavior of the system.

The other basis for the chaos theory is that the exact patterns of a system are based upon the starting variables, and any slight and unaccounted for variable may easily change the course of the system. A good example of this phenomenon is the Lorenz Attractor, which represents the behavior of a gas at any time. If the initial conditions in the equation are changed by a tiny amount, great changes result. If someone plotting the Attractor were to change the initial conditions by the inverse of Avogadro’s number, checking the Attractor at a later time will yield numbers totally new and different from the original plot. This is because small numbers propagate themselves recursively until numbers are entirely dissimilar to the original system. Yet, with these new numbers, the plot of the two systems will still look incredibly close to identical. Even with these new number the plot of the system, and the overall behavior of both systems, remains very close. These shows that the chaos theory is not about the unpredictability of the end state, but instead is a representation of the predictability of the behaviors in the system, even the most unstable systems.

“Until recently, it was believed that if the dynamics of a system behaved unpredictably, it was due to random external influences.” Because of this scientists felt that if they could control the external influences of a system, the outcome could be predicted indefinitely. “It is now known that many systems can exhibit long-term unpredictability even in the absence of random influences. This is another reason that the chaos theory is so valuable it allows us to predict endings, even in the most complex and previously incalculable systems.

Now that you understand the chaos theory, you are probably wondering what good it is to society. First and foremost, it is still only a theory, but that doesn’t detract from its usefulness. The chaos theory is a great way of looking at events, which happen apart from the more traditional views, which have dominated science since Newtonian times. It allows scientists to look at events with a phase diagram, which rather than describing the exact position of a variable with respect to time,

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