Introduction to fractal geometry: Definition, concept, and applications


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University of Northern Iowa
UNI ScholarWorks

Presidential Scholars Theses (1990 – 2006)

Honors Program

1992
Introduction to fractal geometry: Definition, concept, and applications
Mary Bond University of Northern Iowa

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INTRODUCTION TO FRACTAL GEOMETRY : DEFINITI0N7 CONCEPT7 AND APPLICATIONS
MAY 1992
BY MARY BOND UNIVERSITY OF NORTHERN IOWA CEDAR FALLS7 IOWA

•. . . clouds are not spheres, mountains are not cones~ coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line, . . •
Benoit B. Mandelbrot
For centuries, geometers have utilized Euclidean geometry to describe, measure, and study the world around them. The quote from Mandelbrot poses an interesting problem when one tries to apply Euclidean geometry to the natural world.
In 1975, Mandelbrot coined the term ·tractal. • He had been studying several individual ·mathematical monsters.·

At a young age, Mandelbrot had read famous papers by G.

Julia and P . Fatou concerning some examples of these

monsters.

When Mandelbrot was a student at

Polytechnique, Julia happened to be one of his teachers.

Mandelbrot became interested 1n iteration and, using

iteration and some mathematical monsters, took off in a

new dirction. Up until Mandelbrot's find, these monsters

were thought to be separate entities, with no correlation

between them. It was Mandelbrot who found the unity in

these constructions. In doing so, he introduced a new

language to the world of mathematics, providing a new

image for mathematics and revolutionizing the application

of non-Euclidean geometric constructs to science.

When Mandelbrot first coined the term ·fractal-, he

did not give a precise mathematical definition. It is truly

difficult to pin down an exact definition of the word

·fractal.· Mandelbrot would describe the shapes he was

studying (and calling fractals) as sharing the property of

being ·rough but self- similar.· He said that each shape

conforms to the loose sense of ·alike.· This quality is

probably the most definitive fractal characteristic. Looking at a particular fractal structure» the triadic
Coch curve~ will help to demonstrate this characteristic of self- similarity. The Coch curve is generated as follows:
Begin with a straight line. Let n stand for generation or time . As you can see» a straight line is the 0th generation.
To produce the 1st generation Coch curve» remove the middle third of the line segment and replace it with two segments that are each the same 1n length as the segment third which was just removed .

..
Repeating this process for each line segment 1n the n - th generation will produce the n+1-th generation. From the 1st generation, the 2nd generation is produced.
This procedure demonstrates another interesting phenomena. Notice that each stage of the Coch curve increases in total length in a ratio of 4/3. Therefore, the limit curve is of infinite length.
The Coch curve is a fine example of self-similarity. Each small segment is similar to the 1st generation. The 1st generation is underlying in all successive generations. A fractal is a shape which is made up of parts similar to the whole in some way.
A second definitive characteristic of fractals is the dimensional realm in which they exist. Fractals have dimensions in between 1 and 2, and in between 2 and 3.

In 1982, Mandelbrot finally gave a tentative definition of a fractal using the idea of dimension. A fractal is, by definition, a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension. Now, introducing an important theorem, if [. IS a curve and IS self- similar1 then the Hausdorff- Besicovitch dimension of [. is the same as its self-similarity dimension.
To find the self-similarity dimension of the K.och curve, begin by looking at its length function . The length of the n - th generation fractal is given by
The length of each of the small line segments IS
By noting that the generation number n may be written In the form
it becomes obvious that the length may be expressed as

follows:
This results In the equation
This is the self- similarity dimension of the :t.och curve. By the previously stated theorem, since the :t.och curve is selfsimilar, this dimension is equivalent to the HausdorffBesicovitch dimension. Now, each generation of the construction of the :t.och curve may be stretched to form a straight line, and therefore it IS concluded that the topological dimension of the Coch curve is 1. Since the Hausdorff-Besicovitch dimension ( ""-"' l.2618) strictly exceeds the topological dimension (1), this 1s a fractal curve according to Mandelbrot's definition. Furthermore, the fractal dimension equals the self-similarity dimension equals the Hausdorff- Besicovitch dimension, which IS approximately 1.2618.
Another important characteristic of fractals 1s their

I '

• •

artistic quality. The development of computer generated

graphics provided a rich soil from which fractals were

harvested. Computer graphics, adding visualization to

these mathematical monsters:, were a necessary factor for

growth in the field of fractal geometry. Interestingly

enough, when Mandelbrot was a Visiting Professor of

Mathematics at Harvard in 1979-80, Harvard's Science

Center had its first Vax computer (brand new), a Tektronix

cathode ray tube was used to view the pictures generated,

and a Versatec device was used to print hard copies of the

first computer generated fractal images. There has been

much improvement in computer generated fractals, from a

time when Mandelbrot said •. . no one knew how to set

properly [the computer devices].

to more recent

computer generated fractal forgeries, sometimes used in

the filming industry, of everything from plants to planet

rises. These computer generated fractal images are finding

their way into galleries:, and artists are even hand

painting huge portions of the fractal named the

·Mandelbrot set.· Much of fractal art resembles natural

phenomena such as mountain ranges, valleys, clouds, and the list goes on. Returning to the notion of Euclidean geometry being difficult to apply to nature, it seems as though fractal geometry provides a mathematical language which is perfect for application to the world around us. As a matter of fact, there are many items which we often see which have fractal characteristics such as clouds, ferns, and cauliflower (all have the quality of selfsimilarityt and lightning and dust (whose movement is of fractal nature).
Yet another characteristic of fractals IS their numerous applications. In metallurgy, which is the study of the properties of metals, a type of metal powder grain IS produced by the disintegration of a jet of liquid metal with subsequent cooling of the droplets as they fall down a tall cooling tower. This is called a shotting process. The structure of irregularly- structured aluminum shot is such that it can be described by a fractal dimension at coarse resolut ion. Also, it appears that chemical crusts on eroded surfaces look like fractals or have fractal characteristics,

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Introduction to fractal geometry: Definition, concept, and applications