# Essay/Term paper: Fractal geometry

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The world of mathematics usually tends to be thought of as abstract.

Complex and imaginary numbers, real numbers, logarithms, functions, some

tangible and others imperceivable. But these abstract numbers, simply

symbols that conjure an image, a quantity, in our mind, and complex

equations, take on a new meaning with fractals - a concrete one.

Fractals go from being very simple equations on a piece of paper to

colorful, extraordinary images, and most of all, offer an explanation to

things. The importance of fractal geometry is that it provides an

answer, a comprehension, to nature, the world, and the universe.

Fractals occur in swirls of scum on the surface of moving water, the

jagged edges of mountains, ferns, tree trunks, and canyons. They can be

used to model the growth of cities, detail medical procedures and parts

of the human body, create amazing computer graphics, and compress

digital images. Fractals are about us, and our existence, and they are

present in every mathematical law that governs the universe. Thus,

fractal geometry can be applied to a diverse palette of subjects in

life, and science - the physical, the abstract, and the natural.

We were all astounded by the sudden revelation that the output of a

very simple, two-line generating formula does not have to be a dry and

cold abstraction. When the output was what is now called a fractal,

no one called it artificial... Fractals suddenly broadened the realm

in which understanding can be based on a plain physical basis.

(McGuire, Foreword by Benoit Mandelbrot)

A fractal is a geometric shape that is complex and detailed at every

level of magnification, as well as self-similar. Self-similarity is

something looking the same over all ranges of scale, meaning a small

portion of a fractal can be viewed as a microcosm of the larger fractal.

One of the simplest examples of a fractal is the snowflake. It is

constructed by taking an equilateral triangle, and after many iterations

of adding smaller triangles to increasingly smaller sizes, resulting in

a "snowflake" pattern, sometimes called the von Koch snowflake. The

theoretical result of multiple iterations is the creation of a finite

area with an infinite perimeter, meaning the dimension is

incomprehensible. Fractals, before that word was coined, were simply

considered above mathematical understanding, until experiments were done

in the 1970's by Benoit Mandelbrot, the "father of fractal geometry".

Mandelbrot developed a method that treated fractals as a part of

standard Euclidean geometry, with the dimension of a fractal being an

exponent.

Fractals pack an infinity into "a grain of sand". This infinity appears

when one tries to measure them. The resolution lies in regarding them

as falling between dimensions. The dimension of a fractal in general

is not a whole number, not an integer. So a fractal curve, a

one-dimensional

object in a plane which has two-dimensions, has a fractal dimension

that lies between 1 and 2. Likewise, a fractal surface has a dimension

between 2 and 3. The value depends on how the fractal is constructed.

The closer the dimension of a fractal is to its possible upper limit

which

is the dimension of the space in which it is embedded, the rougher, the

more filling of that space it is. (McGuire, p. 14)

Fractal Dimensions are an attempt to measure, or define the pattern, in

fractals. A zero-dimensional universe is one point. A one-dimensional

universe is a single line, extending infinitely. A two-dimensional

universe is a plane, a flat surface extending in all directions, and a

three-dimensional universe, such as ours, extends in all directions. All

of these dimensions are defined by a whole number. What, then, would a

2.5 or 3.2 dimensional universe look like? This is answered by fractal

geometry, the word fractal coming from the concept of fractional

dimensions. A fractal lying in a plane has a dimension between 1 and 2.

The closer the number is to 2, say 1.9, the more space it would fill.

Three-dimensional fractal mountains can be generated using a random

number sequence, and those with a dimension of 2.9 (very close to the

upper limit of 3) are incredibly jagged. Fractal mountains with a

dimension of 2.5 are less jagged, and a dimension of 2.2 presents a

model of about what is found in nature. The spread in spatial frequency

of a landscape is directly related to it's fractal dimension.

Some of the best applications of fractals in modern technology are

digital image compression and virtual reality rendering. First of all,

the beauty of fractals makes them a key element in computer graphics,

adding flare to simple text, and

texture to plain backgrounds. In 1987 a mathematician named Michael F.

Barnsley created a computer program called the Fractal Transform, which

detected fractal codes in real-world images, such as pictures which have

been scanned and converted into a digital format. This spawned fractal

image compression, which is used in a plethora of computer applications,

especially in the areas of video, virtual reality, and graphics. The

basic nature of fractals is what makes them so useful. If someone was

Rendering a virtual reality environment, each leaf on every tree and

every rock on every mountain would have to be stored. Instead, a simple

equation can be used to generate any level of detail needed. A complex

landscape can be stored in the form of a few equations in less than 1

kilobyte, 1/1440 of a 3.25" disk, as opposed to the same landscape being

stored as 2.5 megabytes of image data (almost 2 full 3.25" disks).

Fractal image compression is a major factor for making the "multimedia

revolution" of the 1990's take place.

Another use for fractals is in mapping the shapes of cities and their

growth.

Researchers have begun to examine the possibility of using mathematical

forms called fractals to capture the irregular shapes of developing

cities.

Such efforts may eventually lead to models that would enable urban

architects to improve the reliability of types of branched or

irregular

structures... ("The Shapes of Cities", p. 8)

The fractal mapping of cities comes from the concept of self-similarity.

The number of cities and towns, obviously a city being larger and a town

being smaller, can be linked. For a given area there are a few large

settlements, and many more smaller ones, such as towns and villages.

This could be represented in a pattern such as 1 city, to 2 smaller

cities, 4 smaller towns, 8 still smaller villages - a definite pattern,

based on common sense.

To develop fractal models that could be applied to urban development,

Batty and his collaborators turned to techniques first used in

statistical

physics to describe the agglomeration of randomly wandering particles

in two-dimensional clusters...'Our view about the shape and form of

cities is that their irregularity and messiness are simply a

superficial

manifestation of a deeper order'. ("Fractal Cities", p. 9)

Thus, fractals are used again to try to find a pattern in visible

chaos. Using a process called "correlated percolation", very accurate

representations of city

growth can be achieved. The best successes with the fractal city

researchers have been Berlin and London, where a very exact mathematical

relationship that included exponential equations was able to closely

model the actual city growth. The end theory is that central planning

has only a limited effect on cities - that people will continue to live

where they want to, as if drawn there naturally - fractally.

Man has struggled since the beginning of his existence to find the

meaning of life. Usually, he answered it with religion, and a "god".

Fractals are a sort of god of the universe, and prove that we do live in

a very mathematical world. My theory about "god" and existence has

always been that we have finite minds in an infinite universe - that the

answer is there, but we are simply not ever capable of comprehending it,

or creation, and a universe without an end. But, fractals, from their

definition of complex natural patterns to models of growth, seem to be

proving that we are in a finite, definable universe, and that is why

fractals are not about mathematics, but about us.

SOURCES

Magazine Articles:

"The Shapes of Citries: Mapping Out Fractal Models of Urban Growth",

Ivars Peterson,

Science News, January 6, 1996, p. 8-9

w

"Bordering on Infinity: Focusing on the Mandelbrot set's extraordinary

boundary", Ivars Peterson,

Science News, Novermber 23, 1991, p. 771

"From Surface Scum to fractal swirls", Ivars Peterson,

Science News, January 23, 1993, p. 53

"A better way to compress images", M.F. Barnsley and A.D. Sloan,

Byte, January 1988, p. 215-223.

Books:

McGuire, Michael. An Eye for Fractals.

Addison-Wesley Publishing Company, Reading, Mass., 1991.

World Wide Web Sites:

http://millbrook.lib.rmit.edu.au/fractals/exploring.html

http://www.min.ac.uk/%7Eccdva/

http://www.cis.oio-state.edu/hypertext/faq/uesenet/fractal-faq/faq.html

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