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Virtual
Ecosystems -- Self Similarity
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"The most
useful fractals involve chance ... both their
regularities and their irregularities are
statistical."
- Benoit B. Mandelbrot. |
Applet Designed After Ideas
By Eckhard
Roessel |
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The two basic characteristics of fractal objects are self
similarity and fractional dimension. This page examines the idea
of self similarity.
In geometry, for objects to be similar they must have the same
shape. While their size may differ, they must be proportionally
the same. The key word here is proportional.
A self similar object is exactly, approximately, or
statistically similar to a part of itself. When we look very
closely at patterns that are created with Euclidean geometry,
the shapes look more and more like straight lines, but that when
you look at a fractal with greater magnification (scale) you see
more and more detail.
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A
geometric figure has exact self similarity if it
contains a repeating pattern of smaller and smaller
parts that are the same shape as the larger figure and
differ only in size. The square shown at the left is
progressively broken down into smaller squares that are
the exact shapes of their larger brothers. In other
words, each larger figure is composed of four smaller
versions of itself each scaled down to 1/4th its size.
If we were to perform n levels of construction, we would
end up with a total of 4n squares. It is said
that self similar objects are scale invariant because
they look the same no matter what the magnification, or
scale. All fractal objects are self similar but not all
self similar objects are fractal. The self similar
square is not a fractal because it has an integer
dimension (2) and not a fractal dimension (e.g.
2.05798). |
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The figure to the left is a Sierpinski triangle. We
generated this figure using the "chaos game"
method in the Fractal
Construction page. As we will demonstrate in the Fractal
Dimension page, the Sierpinski triangle is a fractal
because it has a fractal dimension. This figure is an
example of a fractal object that is exactly self
similar. At each magnification, the same pattern of
triangles will appear. In fact, at any magnification, it
is difficult to tell whether or not the figure has been
magnified because the components look exactly the same
at every level of scale. |
One of the most studied fractal figures is the famous Mandelbrot
set. While the Mandelbrot set has little to do with Marine
Ecology, it is an easily demonstrated example of a self similar
fractal. The applet at the top of this page portrays the
Mandelbrot set. With the left mouse button down, drag your mouse
caret over the figure to create a zoom rectangle. The best
places are the "feathery" edges where a black area and
a blue area meet. When you lift the left mouse button, the
figure will regenerate to show you the zoomed area. You should
now see feathery swirls. Do another zoom operation on one of the
feathery swirls. You will see more feathery swirls at a higher
magnification. Continue this process to generate higher and
higher magnifications. You can return to the original Mandelbrot
set image by clicking on the figure without dragging.
The Mandelbrot set is an example of self similarity, or scaling
invariance, in fractal figures. If you carefully examine each
magnification, you will see that the objects may not be exactly
self similar. And, they may be rotated to another orientation.
This is an example of approximate self similarity. Things appear
to be self similar, but there are slight physical differences.
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Many
things in nature look the same way no matter how you
magnify them. You can find self similarity in trees,
root systems, mountains, clouds, rivers, coastlines, the
swimming track of sharks, and human lungs. However, self
similarity in nature is rarely exact in two important
ways. First, self similarity is limited in range of
scale. Lung tissue will look self similar between two
ranges of magnification (scale). But, beyond that range,
the tissue takes on another pattern that is self similar
in a different way. A second issue is that natural
things are rarely physically exact even within limits of
scale. There is a qualitative aspect that leads one to
believe that the object is self similar. But close
examination reveals random physical differences.
Nonetheless, the object's statistical parameters (e.g.
mean and variance ) are self similar. This kind of self
similarity is the one most common in nature. It is
called statistical self similarity or statistical
scaling invariance. |
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