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