Fractals

            Science is constantly exploring new things with the help of math. Fractals are literally works of art. They are intricate pictures formed by repeating the same process infinitely. The exact definition is a geometric shape that possesses self-similarity and fractional dimension. They are found in nature and played with by the human mind. They can be seen at any scale and contribute to the idea of Britain’s coastline being infinite. With all the ins and outs of the land, the more detailed you get in your examination of it, the coastline around Britain ends up going on forever.  With much more to be learned, fractals are a fascinating part of science’s future.

            Fractals are geometric shapes that have two things:

                                    1) Self-similarity

                                    2) Fractional dimension

They provide a new way to describe, model, and analyze complex forms in nature. Examples include plants, weather, music, mountains, coastlines, and even the human brain. They also allow images to be kept in a fraction of the space normally needed. The reason for this is self-similarity. This means the object looks the same over all ranges of scale.

            Fractals also have dimension. In order to find the dimension, the length and width needs to be doubled. In the example below, doubling the bottom and sides to two triangles on each side produces three triangles. The chart shows how three fits into the pattern of the dimension of a line, square, and cube. Logarithms are needed to solve for the exponent, and the exponent is the dimension.

N=S^D where N= the number of miniature pieces in the final figure, S= the scaling factor, and D= the dimension. The following example shows this.
 

	4=4^1 pieces
	16=4^2 pieces
	64=4^3 pieces

Sierpinski’s triangle and Sierpinski’s carpet are two examples of fractals. They are both fairly easy to draw and show how self-similarity works.

            Sierpinski’s triangle was developed by Sierpinski in 1915 and was popular in Italian art. The triangle stems from Pascal’s triangle. By shading in all the odd numbers in Pascal’s triangle, the pattern for Sierpinski’s triangle takes shape. Sierpinski’s triangle is formed by connecting the midpoints of an equilateral triangle and then “cutting” out the middle triangle, leaving 3.

The process is repeated with each remaining triangle and can go on infinitely. Any part can be zoomed in on, and it will resemble the first triangle. This is self-similarity.
Sierpinski’s carpet basically follows the same steps. Only this time, a square is used. The original square is divided into nine smaller squares and the middle one is cut out. The process is repeated with the eight remaining squares and again, could go on infinitely. The dimension would be D=log8/log3 or 1.89.
 

Fractals are thought to have originated centuries ago. Certain types of fractals can be linked all the way back to geometry in indigenous African craftwork. German artist Albrecht Durer published The Painter’s Manual in 1525. Part of the book resembled Sierpinski’s carpet, but it was based on pentagons instead of squares.

The term “fractal” was coined by Benoit B. Mandelbrot while working as a research mathematician at I.B.M.’s lab in New York. He was experimenting with French mathamatician Gaston Julia’s theories. However, not all of Julia’s theories could be tested until thought caught up with technology. Today’s computers are capable of the millions of calculations that were needed to prove the Mandelbrot fractal. This fractal uses the formula z=z*z + c and is a set of coordinates whose representative numbers feed back on themselves when the equation is applied. After iterations, a number continues to infinity or winds up back at zero.

Today fractal geometry is only in its infancy. It is currently being used to describe shapes and structures with precise formulas. Plants and mountains show the use of fractals in nature.Self- similarity allows fractals to be viewed on any scale. Fractals also have fractional dimension. In the future, fractals may be used for architecture, medicine, and weather.
                                                                              Bibliography

Hall, Nina. Exploring Chaos: A Guide to the New Science of Disorder. New York: W.W. Norton & Co., 1991.

http://library.thinkquest.org/26242/full/fm/fm29.html

Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York: W.H. Freeman and Co., 1977.

http://www.math.umass.edu~mconners/fractal/fractal.html

http://mathworld.wolfram.com/sierpinskisieve.html

http://www.sunleitz.com/whatarefractals.html

http://en.wikipedia.org/wiki/Fractal

http://zeuscat.com/andrew/chaos/sierpinski.html