Essay/Term paper: Fractal geometry

Essay, term paper, research paper:  Science

Fractal Geometry


"Fractal Geometry is not just a chapter of mathematics, but one that helps
Everyman to see the same old world differently". - Benoit Mandelbrot

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

"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