Introduction to Fractals and Chaos Theory
The picture above is an example of a Julia set. The picture is a graph of the results of entering a series of numbers into the VERY SIMPLE function f(x) = x^{2} + c. For this particular graph, the value of c was -0.74 + 0.11i.
There are several interesting things about this picture:
1. The image is not the work of a human artist but instead is the result of many repetitions of a simple mathematical calculation - and yet the image contains infinite complexity and a clear aesthetic quality.
2. It has similarity on many scales (zooming in closer reveals similar patterns.)
3. Mathematically defined, the boundary of the set has a fractional dimension.
Below is an animation which shows the result of zooming in and out on the Mandlebrot set.
The following are Java Applets that generate fractals:
The first Java Applet is in color and allows you to zoom in on regions of the
complex plane.
Click Here for a Mandelbrot Set generator.
Below are simple Java Applets that generate both Mandelbrot and Julia Sets - the code is included with each because they are short and simple enough to be understood with a minimum of programming knowledge. With these programs there is no capability for zooming-in and also they require that the "Java Plug-In" is installed on your computer. Many computers already have it installed, so try the programs first, you may not need to install the plug-in. If you don't see the fractals, then you don't have the plug-in. You can get it at this website:
http://www.java.sun.com/products/plugin).
Also, fractal animations and fractal music are both interesting and a lot of
fun...Click here to view some fractal animations (some also with sound) and here for
more about fractal music.
Definitons:
1. Nonlinear System - A system which has a rate of change which is not constant. Most models of real world systems are nonlinear. To have a nonconstant rate of change means to change at a changing rate. For example, the weather changes at different rates. Populations increase or decrease at different rates. Objects move at different rates. The stock market changes at a changing rate.
2. Chaos Theory - Study of nonlinear systems; Chaos theory studies pattern and organization within nonlinear systems. Nonlinear systems are typically characterized by unpredictability (weather, populations, stock market, etc.) Chaos theory is about discovering how simple predictable functions can create unpredictable results. Through the discoveries of chaos theory, we are able to understand how systems which were once thought to be completely chaotic actually have predictable patterns. Chaos theory originated in the 1960s. One of the early pioneers was a professor at M.I.T. named Edward Lorenz, who designed computer models of the weather.
3. Fractals - Geometric objects which have fractional dimension. Just as a line is one-dimensional, a table-top is two-dimensional and space is three-dimensional, a fractal has a fractional dimension. For example, a fractal could be 2.3-dimensional or 1.7-dimensional or .5-dimensional. There are many fractal objects in mathematics which have nothing to do with chaos theory and were around before the dawn of chaos theory. However, because of the technological advances of the modern computer, chaos theory has produced a whole new family of beautiful fractals.
4. Iteration - Repetition of a simple procedure such as a mathematical calculation.
Below are a few examples of fractal images generated entirely by mathematical formulas. The designs and the colors are not the work of an artist but rather the result of simple iterative functions.
These pictures are particulary interesting because the patterns and colors where not chosen by an artist. They are completely the result of a mathematical formula like the function f(x) above. Each point in the picture correponds to a complex number. A computer calculates the results of running those different complex numbers through a function. Values where the results tend to increase toward infinity are colored according to the rate at which they approach infinity. Values that tend toward zero are colored black.
A butterfly flaps its wings and the weather changes in China.
This is one of the catch phrases of chaos theory, called the butterfly effect. It refers to sensitive dependence on initial conditions. In nonlinear systems, making small changes in the initial input values will have dramatic effects on the final outcome of the system.
Theories abound as to real-life examples of this phenomenon:
1. The weather: small changes in weather effect larger patterns.
2. The stock market: slight fluctuations in one market can effect many others.
3. Biology: A small change in a virus in monkeys in Africa creates a "thunderstorm" of an effect on the human population around the world with the appearance of the AIDs virus.
4. Evolution: small changes in the chemistry of the early Earth gives rise to life.
5. Psychology: Thought patterns and consciousness altered by small changes in brain chemistry or small changes in physical environmental stimuli.
The butterfly effect occurs under two conditions:
1. The system is nonlinear.
2. Each state of the sytem is determined by the previous state. In other words, the output at each moment is repeatedly entered back into the system for another cycle through the mathematical functions that determine the system.
The result of the butterfly effect is unpredictability. Small differences in initial input can have dramatically different results after several cycles throught the system. In the fractals pictured above, points that are very close together can be different colors. The results can tend toward infinity at different rates or toward zero, even though the initial points are very close together.
Finally, one of the trademarks of these sorts of chaotic systems is self-similarity on different scales. If we were to change the boundaries in the pictures above, similar patterns would be found, no matter what scale we chose. Likewise, we may speculate about the fractal nature of nature itself. For example:
1. The weather has similar patterns on different scales. From the small whirlwinds that might blow across a parking lot before a storm, to the scale of a tornado and then to larger vortices such as the hurricane.
Click on the picture for a larger image...
2. The stock market in general follows a pattern reflected in the jagged up and down fluctuations of individual markets and in individual stocks.
3. There are similarities in biological systems, from large animals to single cells, consuming, elminating waste, and reproducing similar genetic code.
4. A tree has branches made of smaller branches, which are made up of smaller twigs, each smiliar to the larger pattern.
5. Many coastlines are jagged when viewed from a satellite and these same coastlines are made up of local bays and peninsulas when seen from an airplane. When walking along the water's edge even more jagged detail may become visible.
6. Clouds have a pattern when viewed on a global scale similar to a local scale.
7. Mountains have a shape which is repeated within itself creating the mountain as a whole.
8. Lightning bolts are made of patterns repeated to form the whole.
9. The universe itself might be viewed as one giant fractal. From the stars of a galaxy, each with their own solar system to the planets within that solar system, each orbited by moons. The matter that creates the universe is then made up of atoms about which orbit the electrons.
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