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The importance of beings fractal
Prof A.J. Roberts
Department of Mathematics and Computing
University of Adelaide
23 July 1996
"Good science is the ability to look at things in a new way and
achieve an understanding that you didn't have before ... It is opening
windows on the world ... you perceive a little tiny glimpse of the way
the Universe hangs together, which is a wonderful feeling." Hans
Kornberg
Fractal geometry, largely inspired by Benoit Mandelbrot during the
sixties and seventies, is one of the great advances in mathematics
for two thousand years. Given the rich and diverse power of
developments in mathematics and its applications, this is a
remarkable claim. Often presented as being just a part of modern
chaos theory, fractals are momentous in their own right. Euclid's
geometry describes the world around us in terms of points, lines and
planes---for two thousand years these have formed the somewhat
limited repertoire of basic geometric objects with which to describe
the universe. Fractals immeasurably enhance this world-view by
providing a description of much around us that is rough and
fragmented---of objects that have structure on many sizes. Examples
include: coastlines, rivers, plant distributions, architecture, wind
gusts, music, and the cardiovascular system.
\section{Some fractal models}
Before discussing in detail the common feature of the previously
mentioned examples, I present a few examples of fractals and the type
of physical applications that they have.
\subsection{Noise and natural events}
Have you ever noticed that there are some days where nothing goes
right? times when you just cannot get a decent telephone
connection? years when drought follows drought? long periods when
gusts of wind come thick and fast? That events often occur in bursts
is a well documented aspect of the world. The Cantor set is a model
for such bursty phenomena. Construct the Cantor set in the following
manner:
start with a bar of some length; then remove its middle third to
leave two separate thirds; then remove the middle thirds of these to
leave four separate ninths; then remove the middle thirds of these to
obtain eight separate twenty-sevenths; and so on. Eventually we just
obtain a scattered dust of points. However, this dust is specially
distributed into pairs of points, of pairs of pairs of points, and so
on. If the original bar represented a time interval, then the dust
represents times when events occur and the striking feature is that
there are long quiescent periods separating the short bursts of
activity that a clump of the points represents.
\subsection{Coastlines and rivers}
The line of a coast or the path of a river is tortuous. On a
small-scale map of Australia or any other country the coastline has
lots of wriggles. Upon examining a larger-scale map the wriggles
will be resolved clearly into gulfs and peninsulas. However, many
smaller scale wriggles will still be seen. These can only be
resolved by looking at an even larger scale map, whereupon they will
be seen to be inlets and spits. But once again there will be
wriggles in the coast. Similarly for rivers---they exhibit bends and
meanders upon many scales of length. The Koch curve models these
phenomena.
Starting with an equilateral triangle, we replace the middle third of
each side by two segments of the same length (as if we pasted an
equilateral triangle of one-third the size onto each side); this
forms the second picture above showing large-scale peninsulas and
bays. Repeating this process of extracting the middle third of each
straight side and replacing it with two segments of the same length,
the next stage of the construction gives the third picture; it shows
smaller inlets and spits. Continually repeating this process leads
to a very wriggly line that is the Koch curve. It is perhaps too
convoluted for a coastline, but on the other hand, it looks far more
realistic than a routine curve!
\subsection{Turbulence}
Most mathematicians, physicists and engineers would give their right
arm to understand what is going on in this picture. It shows
something of the highly complex motion that is turbulence in a fluid
as expressed by the following ditty by L.F. Richardson:
Big whorls have little whorls,
Which feed on their velocity;
And little whorls have lesser whorls,
and so on to viscosity.
When air or water moves, a smooth flow quickly breaks up into
swirling eddies. These eddies are of a wide range of sizes and, as
on a windy day, there are often quiescent periods separating the
various wind gusts. The structure of turbulence may be epitomised by
a Sierpinski sponge which is formed from building blocks in much the
same way as the "Eiffel tower." Form a small unit by putting 20
blocks face to face along the edges and corners of a 3x3x3 cube,
leaving the middle of the six faces and the middle of the cube
vacant. Make a bigger unit by connecting 20 of these units together
in the same 3x3x3 pattern. And so on to as large a scale as possible.
\section{Scaling and dimensionality}
The common theme in these examples is not just that they have detail
on many lengths, but also that the structure at any scale is much the
same at any other scale---the coastline around a continent looks just
like any small part of the coastline. If we take a magnifying glass
or microscope to any of these examples then, no matter what the
magnification, the geometric detail that we see is the same. This
property of looking similar at all scales is termed self-similarity:
exact in the artificial examples, and statistical or random in
practical applications. In order to tease out the self-similar
characteristics of such objects we need to explore the fractal over
many lengths and sizes.
The coarsest characteristic of fractals is their dimensionality.
While we normally expect a dimension to be an integer, a natural
number such as 1, 2 or 3, fractals are best described by means of a
dimension which is fractional, such as 1.2 or 0.69. This dimension
is obtained by blurring the fractal at some size, counting the number
of blobs in this blurred picture, and then seeing how the count
varies with the size of the blurring. Another explanation of this
process is to count how few "clumps" of a certain size the object can
be broken into, and then see how this count varies with the size of
the clump.
\subsection{Points, lines and planes}
Lets become familiar with the argument via some well known geometric
examples: points, lines, and planes. Consider a line of some length
L as shown below. The line
could be curved, but for simplicity we take it to be straight. To
"blur" the object at a size d I mean that we try to cover as much of
the object as possible by discs of diameter d. In this picture I
have used N=9 discs to cover the line segment. If the discs were
half the diameter, then we would have to use twice as many of them to
cover the line; if the discs were one-third the diameter then we
would have to use three times as many to cover the line; and so on
for other sizes. Typically, the number of discs needed to cover a
line of length L is N=L/d. The important aspect of this relation is
that the number of discs is inversely proportional to the first power
of the size (diameter) of the discs: .
Contrast this with what happens when we cover an area A of the plane
with discs of some diameter d: N=34 in the above example. Typically,
the number of discs needed to cover an area A is inversely
proportional to the second power of the size of the discs: ; the
number would be close to the area, A, divided by the area of each
disc, , to be if it were not for the wastage around the perimeter
of each disc.
See that the exponent of this relation between N, the number of
discs, and the size of the discs, as measured by the diameter d, is
precisely the dimensionality of the object: a line is
one-dimensional; an area of the plane is two-dimensional. This
relation between the exponent and the dimensionality is true in
general. For another example, consider a small number, n, of points
distributed in space---for all d smaller than the minimum separation
between the points the number of discs needed is precisely the same
as the number of points in the set. Thus , and the 0 exponent
matches the zero-dimensionality of a point or a finite number of
points.
For the geometric objects introduced earlier, the relation between N
and d involves a fractional exponent D. It is only reasonable for us
to say that the dimensionality of such an object is the fraction D.
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