Fractal Theory

 

There are a number of fractal types. I will only briefly discuss a few here which can be created using the Fractal Explorer program. I realize most are not interested in the mathematics, but I encourage you read the information below to get an idea of the beauty of this theory.

The Mandlebrot Set

Click to enlarge The Mandlebrot set contains points generated by an equation which define a type of fractal aptly called a Mandlebrot fractal. (Click on the image to the left ). Due to the iterative (repeating) nature of the equation, fractal images possess the unique quality that as you zoom into the image, they retain their complexity and display a similar image with slight variation in form and color. This quality allows you to magnify the fractal and continue to see the pieces of the image replicated indefinitely, making fractals and the underlying theory so interesting for mathematicians, digital artists and viewers alike. It you would like to see this effect for yourself, click the link here courtesy of The Math Forum @ Mathforum.org and you will be taken to their site where you can zoom into a Mandlebrot fractal.

To understand the basic geometry of fractals, one must first understand a few underlying mathematical concepts. In Euclidian geometry, we think of dimensions consisting of real numbers. For example, a line is "1-dimensional", a surface is "2-dimensional", and an object with depth is "3-dimensional". Mandlebrot's vision defined a new dimension consisting of one set of numbers in the Complex plane which, when plotted, form a unique fractal image.

In the Complex plane,numbers are defined in the form a + bi, where a and b are real numbers, and i is defined as the square root of "-1". You can visualize this by drawing a graph where the x-axis is the real number line with real values a and the y-axis is the imaginary number line with imaginary values bi, where b is the y-axis coordinate value. Mandlebrot was able to create a fractal dimension using a subset of numbers in the complex plane defined by the iterative (repeating) function:

Z(n+1)=Z(n)^2 + C [Z and C are complex numbers, with |C| <= 2, and C = square root of (a^2+b^2).]

This translates to " the new value of Z is defined by multiplying Z by itself and adding the value C". Note: the symbol ^ means raised to the power and |C| means the absolute value of C. The set of points Z resulting from repeating this calculation a number of times form the Mandlebrot set. By plotting these points, a fractal image is formed. By limiting the value of C within the range "-2" and "+2", the resulting Z values do not become infinite in value and can be plotted on a graph. In the Mandlebrot set, all these numbers Z define the set and the resulting fractal image shown above.

Julia Sets

Mandlebrot's work was inspired by the mathematicians Gaston Julia and Pierre Fatou, who did their work in the early 20th century. Julia and Fatou used the same equation as Mandlebrot to generate Julia sets using the same limiting rule that |C| <= 2.

Click to enlargeClick to enlargeFor each complex number C, there is a unique Julia set which defines an associated fractal image. Since there are an infinite number of points C, there are an infinite number of Julia sets. With Julia sets, some of the images when plotted display a single connected image and some display separated pieces in the image. I have created two examples here to show the distinction. The image to the left is made up of disconnected areas. The image to the right has one continuous connected area. Click on them to see more detail in each image.

An interesting aspect of fractal theory is that The Mandlebrot set is the union of all connected Julia sets. As you zoom into a Mablebrot fractal, you can see different images of connected Julia sets.

The different colors for both Mandlebrot and Julia set images are determined by the fractal generator program. In each iteration of the equation, the program's algorithm varies the color for the points generated, thus enhancing the image we see.

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