Fractal Geometry Essay, Research Paper

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