The Mandelbrot Set



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The Mandelbrot Set

By Thomas Raymond, Brian Hopper, and Michelle Limoge

October 12, 2005

MA 111

Benoit Mandelbrot was born on November 20, 1924 in Poland. His family moved to France in 1936. ( [1], page 122). Mandelbrot attended the Lycée Rolin in Paris up to the start of World War II, when his family moved to Tulle in central France. This was a difficult time for Mandelbrot because he feared for his life on many occasions because of the impeding war. Because of these harsh conditions he faced while being educated, Mandelbrot did not receive a traditional education. After his success in later years with the Mandelbrot Set, he attributed most of this success too his unconventional education. It caused him to think in ways that a “normal” educated person would because he was not taught how to learn in a standardized way ([1], page 1). In 1945, he read Gaston Julia’s 199 page masterpiece, but he was not too impressed by the paper. He could not relate to the style of mathematics within the paper, therefore he did not agree with it. Mandelbrot explored mathematics in his own way, but he eventually was lead back to Julia’s work in 1977. By using computer technology, he was able to show that Julia’s work was a source of some of the most interesting aesthetically pleasing fractals known today ([1], page 122-123).



The Mandelbrot set is a picture in the complex “c” plane of the fate, which is whether the function goes off to infinity or has a stable cycle, of the orbit, which is the path of the function under iteration, of 0 under the iteration of the function x^2 + c. A c-value is in the Mandelbrot set if the orbit of 0 under the iteration of the function x^2 + c and the value C does not go off to infinity. If the orbit of 0 does go to infinity, then that c-value is not in the set. For example, take a complex number “c.” Next, put 0, the orbit, into the function x^2 + c and one will simply get c from the calculation 0^2 + c. This can be done again by putting c back in for x in the original equation because it was the output from the first time through the equation, and this will yield c^2 + c. You can continue the equation to acquire (c^2 + c) ^2 + c, when the previous answer, c^2 + c is put back into the equation. This process can be repeated by replacing the previous value into the equation. Therefore, a list of complex numbers is generated. If these complex numbers get larger and larger, and consequently farther away from the origin, the c value would not be in the Mandelbrot Set. If it does not get larger and larger and farther away from the origin, then it is considered inside of the Mandelbrot Set. Some examples of numbers in the Mandelbrot Set are: 0, -1, i, and -2. Below is an example of four iterations of the c value -1.

Ex: -1—the answer fluctuates from 0 to -1 and keeps repeating this pattern.
x0 = 0^2 + 0 = 0

x1 = 0^2 – 1 = -1

x2 = -1^2 -1 = 0

x3 = 0^2 – 1 = -1

x4 = -1^2 – 1 = 0…

All of these examples are within the Mandelbrot Set because they never go off to infinity. They keep a pattern and never fluctuate too far from the origin. To see a picture of the Mandelbrot Set, see figure 1.



Figure 1

As can be seen above, this is an actual picture of the Mandelbrot Set. All the values that are within the black space are inside of the Mandelbrot Set. If a c-value is within the color spaces, they will go off to infinity, but those colors that are closer to the black space will have more iterations within the Mandelbrot Set than those that are farther away. Values that are green and purple have iterations in the set for sometime before going off to infinity, where the yellow, orange, and red values of c go off to infinity quicker. The main black region of the Mandelbrot Set is called the main cardioid. The many bulbs which are attached to the main cardioid are called, primary bulbs. The primary bulbs do not include the “sub-bulbs” or “antenna-like structures” that are attached to it.

The Mandelbrot Set and the Julia Set are very similar but slightly different. In a filled Julia set, first, you are given a complex value “c” value. In the filled Julia set for x^2 + c, it is a collection of all seeds, all the possible values of x for that particular value of c, whose orbit does not escape to infinity under the iteration x^2 + c. Therefore, there is a different filled Julia set for each value of c. In figure 2, there is an example of a Julia Set. All of the white dots with their corresponding letters represent a seed. Each seed except D is in the Julia set because it is outside of the of the black/green/purple region. Each seed has a different orbit, whether it moves around the Julia set or whether it remains stationary. Those that remain stationary consist of an orbit with one position.

The Julia Set can take on two different shapes. The first shape is formed by c values that lie within the Mandelbrot Set. This produces a filled Julia set that is connected, or in other words is a single piece. It can take on many shapes but all of them are connected. The second shape of the Julia Set is produced by c values that lie outside of the Mandelbrot Set. This produces a filled Julia Set that is shattered, which means that it breaks into infinitely many pieces. A Julia Set begins to “shatter” when the c value crosses the boundary of the Mandelbrot Set.

Figure 2


If the c value of a Julia Set lies within the main cardioid of the Mandelbrot Set all of the orbits are attracted to a fixed point. As the c value crosses into one of the primary bulbs of the Mandelbrot Set there are certain points at which the Julia Set “pinches” together at certain points. Also a new “attracting cycle” is born, which means that there may be multiple points at which the orbits are attracted to.

When c lies within a primary bulb the orbit is attracted to a cycle of a given period. The period value is the number of iterations that it takes for the orbit to begin repeating itself. The period can be determined in multiple ways. When looking at the image of the Mandelbrot Set you find the primary bulb of which you would like to know the period. The main “antenna” has a number of “spokes.” If you count the number of “spokes”, including the “spoke” that attaches the “antenna” to the primary bulb, it will give you the period number. You can also find the period number by looking at the Julia Set. By counting the number of black regions that meet at a “junction point,” which is the point where the main black region meets the side regions, you can get the period number. The main black region is included in this count. The period number is the same for any c value that lies inside that primary bulb.



Bibliography
Chaos and Fractals
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Mandelbrot.html



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