Michael Sun -- Holmdel High School -- June 10, 1996
|
W elcome to my web project for my Calculus class in school. The topic that I am dealing with is Mandlebrot Fractals. Many people know of the the beautiful and artistic pictures that fractals can generate, yet they do not know or understand the mathematics behind them. In this web page, I will be explaining the basic concept of fractals, how they are generated mathematically and will deal specifically with the set of Mandlebrot fractals. |
|
|
|
A
fractal is a mathematical object with a rough or fragmented geometric shape. They are said to be "self similar and independent of scale" according to Ermel Stepp.
They can be subsdivided into parts, and each smaller part is a image of the original whole. Thus, zooming in on a fractal
would lead you to see the same image repeated over and over. This is why is is said to be "self similar" and "independent
of scale". Most fractals are generated from a mathematical equation where the results are iterated, that is the results from the equation are fed back into the equation,
and this process is continued until the number grows larger and reaches a certain boundary.
A "rigorous" mathematical definition of fractals was stated by Benoit Mandlebrot (famous for his Mandlebrot set, rediscoving fractals, and naming these mathematical objects) as "a set for which the Hausdorff Besicovich dimension strictly exceeds the topological dimension". Stepp cites that this definition is not totally satisfactory for it does exclude some sets that are considered fractals. There are many types of fractals--such as Sierpinski's triangle, the Kock snowflake, the Peano curve, the Mandlebrot Set, and the Lorenz attractor. Fractals have also been shown to describe real world objects that don't follow normal Euclidean geometry. Such examples are mountains, coastlines, and clouds. |
|
|
|
F
ractals are usually colorful, with colors assigned to values produced by the equation of the fractal.
An algorithm is used to assign these colors. Most fractals generated by computer for it is almost impossible to
create them by hand. In fact, it was the computer that led Benoit Mandlebrot to rediscover them. The first to discover fractals
was Gaston Julia but the inexistence of computer hindered the study of them. When the computer was available as a
resource for Mandlebrot, he began the first in depth study of them. The set he discovered and is famous for is the
Mandlebrot set. The black area of many fractals
is considered to be the Mandlebrot set. This fractal on the left is an example of a Mandlebrot fractal.
To see more examples of fractals, click here.
|
|
|
|
T
he Mandlebrot set of fractals is stated by Stepp as the set of all complex c such that iterating the function
z = (z-1)^2 + c does not go to infinity when starting with z=0. In other words, Mandlebrot fractals are just a graph of the equation
z = (z-1)^2 + c using complex numbers. The x-axis of the graph is the real line and the y-axis is
defined as the imaginary number line. Let us go through together an example of one iteration provided by Neal Kettler utilizing the Mandlebrot equation. z = (z-1)^2 + c In this equation the variable "z" is the next value to calculated. "z-1" is the current value and "c" is the value of the current pixel. We will then choose a starting pixel or point to begin the calculation, in this case Kettler chooses 1+i. To begin the iteration, we must have initially z = 0. Thus after one iteration, the equation looks as such.
z = (0)^2 + (1+i) Now, we begin a second iteration in which (z-1) = 1+i and c=1+i.
z = (1+i)^2 + (1+i) You repeat this process N times, trying to see if the value has reached infinity. If the value gets too large and is on course to reach infinity, then you exit the loop. If you finish iterating the loop N times, and the value has not gotten too large and is not going to infinity, then the point is most likely inside the Mandlebrot set. Now you are probably wondering how it is possible to tell whether the value is on its way to infinity. It has been proven and shown by Stepp that if |z| > 2, then the value will go to infinity and it is possible to exit the loop. Since z is a complex number, Kettler states that: |z| > 2 is the same as saying the (real part)^2 + (imaginary part) > 4 because of the Pythagorean theorem. Thus, given N iterations, if the value of z ever becomes greater then 2, then the value will go to infinity and become too large to calculate. At this point, the program generating the fractal will, according to Kettler, "bail out" of the loop. If the value is iterated through the loop N times and does not "bail out", then that point is determined to be part of the Mandlebrot set. These points are then assigned the color black. That is why the black regions of fractals are considered to be the Mandlebrot set. Points that do "bail out" are assigned a different color depending on the number of iterations needed for them to "bail out". Stepp also cites that the more iterations, N, performed the higher the detail of the fractal is created. However, it takes longer to generate fractals of high detail. It is through the different interations needed for values to "bail out" that the beautiful and plentiful color patterns we find in fractals are generated. Stepp explains that the reason that the initial z value must be equal to 0 is that zero is a critial point of the expression: (z-1)^2 + c This means the point in which the derivative of the expression is equal to zero:
d/dz [(z-1)^2 + c] = 0 If the equation of the function for the creating the fractal was different, Stepp states that a different starting value would be have to be found. The process of finding the starting value is the same as above--you must find the critical point with the use of derivatives. |
|
|
|
F R A C T A L L I N K S |
| B I B L I O G R A P H Y |
|
|
|
F R A C T A L P I C S |
|
