Fractal Geometry

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