- Level: AS and A Level
- Subject: Maths
- Word count: 3016
Fractals. In order to create a fractal, you will need to be acquainted with complex numbers. Complex numbers on a graph are characterized by the coordinates of (x,y)
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Introduction
Ruzeb Chowdhury June 09, 2012
Fractals
Usually when one speaks of fractals to an audience, the audience refers to a piece of art that consists of repeating shapes and self-similar patterning (“same as near as far”). However, they do not realize that fractals were made and recognized due to a mathematical concept that was developed by many people such as Gottfried Leibniz, Georg Cantor, Waclaw Sierpiński, Gaston Julia, and Benoit Mandelbrot. Mandelbrot was the one to coin the term “fractal”, but the other mathematicians paved the way for Mandelbrot’s findings. Also, some people may think that there is just one way to make a fractal, which is by using math. This is not totally true, as there ways to make the same fractal in a different approach, such as by using the L-system.
A fractal created with the Julia set under constant -.74434 - .10772i.
The mathematics behind fractals, such as the Koch Curve, is rather simple, yet it becomes a complicated, yet ingenious method when referring to the Mandelbrot Set. In order to create a fractal, you will need to be acquainted with complex numbers. Complex numbers on a graph are characterized by the coordinates of (x,y) in which the x-coordinate is any rational number, whereas the y-coordinate contains an imaginary number, denoted by “i”. An imaginary number is the square root of a negative number. An example of a complex number can be: (5 + 4i).
Middle
z0 = 0
z1 = 0 + 2i
Enough of whole numbers, the Mandelbrot Set requires the use of fractions for the values of the real and imaginary parts, or else we’d only be talking about a few points. For example, the iteration of .7 + 6i is not considered a point in the Mandelbrot Set only after three iterations:
z1 = 0.7 + 0.6i
z2 = 0.83 + 1.44i
z3 = -0.6847 + 2.9904i
z3 = 3.06778523531
Usually, smaller fractions for the complex and imaginary parts tend to stay in the Mandelbrot Set, even after fifty iterations! For example, when iterating .2 + .3i , it stays inside the circle of radius 2:
z1 = 0.2 +0.3i
z2 = 0.15 + 0.42i
z3 = 0.0461 + 0.426i
z4 = 0.02064921 + 0.3392772i
z5 = 0.0853173714338 + 0.314011612302i
z50 = 0.079204221383 + 0.356467487952i
z50 = 0.365160757272
In order to create the Mandelbrot Fractal, we just ask the computer to color the pixels black if the value of c is bounded; else we color the pixel white if it goes to infinity. Different colors were determined by the quickness of the equation to go until infinity, thus the presence of blue in the image below.
Fractal created by the Mandelbrot Set
French mathematician Gaston Julia had published her work at the age of 25, yet it was not recognized as an input to the mathematical world until after about sixty years, when computers were able to visualize Julia’s creation.
Conclusion
The dimension of a square is two, and is proven below:
dimension = log (self similar pieces)log (magnification factor)
= logN2log N
= 2 log Nlog N = 2
Similarly, to find the dimension of a fractal, we use the same formula. The Sierpińksi Triangle consists of three self-similar pieces under the magnification of two.
Therefore:
dimension=log3log2≅ 1.585
The fractal dimension of the Sierpińksi triangle is approximately 1.58, less than a square yet more than a line. Fractal dimension is the measure of how "complicated" a self-similar figure is and how large it is. The Sierpińksi triangle is more complicated than the Koch Curve (1.26) and the Cantor Dust (.63), thus it is larger than them.
Bibliography
“Creating Fractals: The Mathematics.” Adam Lerer. Angelfire. n.d. Web. 25 May. 2012.
“Escape Criterion.” Professor Michael Frame. classes.yale.edu. Fall 2000. Web. 2 June.
2012.
“Fractal Dimension.” Professor Robert L. Devaney. Boston University. 2 April. 1995.
Web. 4 June. 2012
“Fractals and Fractal Geometry.” Thinkquest.org. Oracle Thinkquest. n.d. Web. 23 May.
2012.
“Introduction to the Mandelbrot Set.” David Dewey. cs.washington.edu. n.d. Web. 22
May. 2012.
“Julia Set Generator.” www.easyfractalgenerator.com. n.d. Web. 5 June. 2012.
“Julia Sets.” Professor Michael Frame. classes.yale.edu. Fall 2000. Web. 2 June. 2012.
“L-system.” Wikipedia.org. n.d. Web. 30 May. 2012.
“Mandelbrot Calculator.” Gary Rubinstein. MATHE 6500C. 2009. Web. 26 May. 2012.
Mandelbrot, Benoit. The Fractal Geometry of Nature. U.S.A: Macmillan, 1983. Print.
"The Math of Fractals." Coolmath.com. Coolmath.com, Inc., n.d. Web. 22 May.
2012.
“The Mandelbrot Set.” Professor Robert L. Devaney.Boston University. n.d. Web. 29
May. 2012
“Fractals and the Fractal Dimension.” Vanderbilt.edu. n.d. Web. 9 June. 2012
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