What is Fractal Dimension?

In the realm of classical geometry, dimensions of objects are defined by
whole numbers. A point has zero dimension, a curve or line has one
dimension, triangles and circles have two dimensions, and spheres, cubes and
other solids have three dimensions. Fractals, however, have dimensions which
fall between whole numbers. This is because fractals will have a finite area
or volume but an infinite perimeter or surface size. For instance, the von
Koch snowflake (shown below) has an area which is precisely eight-fifths of
the the original triangle, but it is enclosed by an infinite perimeter.
[Image]

Consider the eastern coast of the United States. When viewed on a world
globe, it appears to be between 2 and 3 thousand miles long. If the same
coastline is viewed on a map of only the U.S., it would appear more like 4
or 5 thousand miles long with all the bays and capes. Navigational charts
pieced together could reveal 10 to 12 thousand miles and if a person were to
walk the length of it, he or she might well walk 15 thousand miles. Finally,
if an ant were to walk at one step from the shore, it would travel for more
than 30 thousand miles. This pattern would theoretically carry on infinitely
each time there is finer and finer viewing and consequently, the coastline
is infinitely long. The water does, however, close off a definite amount of
area. Because of this, shorelines have dimensions somewhere between 1 and 2.

How we come to the exact dimensional value of a fractal is simple. Consider
a straight line with a length of ten units. The line is made up of 10 one
unit long segments and therefore the a ratio is created equal to 10^1. For a
10 X 10 unit square, 100 one X one segments make up the figure. This
produces a ratio of 10^2. For a straight line, there is a dimension of one.
Likewise there is a dimension of two for the square. Volume is created with
three dimensions and can be seen in the same way. This shows how dimension
can be determined by a mathmatical ratio. For a two dimensional object such
as the square described above, the equation for the dimension was
log10^2/log10 = 2. The equation for finding a fractals dimension is:

d = log N/log (1/p)

p represents the fractional size after one iteration, and N is the
coefficient of new units after the fractal is raised to a power. d
represents the Hausdorff dimension of the fractal.

Example: When this equation is used on the Koch snowflake (shown above), the
curve is divided by thirds (p = 1/3), and after one iteration there are four
times as many segments as the previous curve (N = 4). Therefore the
Hausdorff dimension of the von Koch curve is log 4/log 3 = 1.2618. . . The
fractal's dimension shows up between 1 and 2. This can be used for all
regular fractals. Another example below (the Cantor set) shows how a fractal
can have dimension between 0 and 1.

[Image]

The first five iterations show how the fractal breaks into an infinite
number of points with no length. The equation for the fractal dimension of
this fractal is log 2/log 3 = .631 because each iteration divides the line
sets of two segments, each one third the size of the previous segments.

Other fractals can have dimensions between 2 and 3 which would be for a
jagged landscape or a sponge. Sierpinski pyramid is a pyramid which with
each iteration had smaller pyramids removed from it.

[Image]

The fractal dimension of the pyramid is surprisingly log 4/log 2 = 2. Behold
a 2-dimensional object.

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Created by Dave Mills,

millsd@lahs.losalamos.k12.nm