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Virtual
Ecosystems -- Fractal Dimension
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"Effective
dimension concerns the relationship between mathematical
sets and natural objects."
Benoit Mandelbrot
The Fractal Geometry Of Nature |
You are in an
airplane taking a movie of a shark swimming on the
surface of the ocean. You are trying to decide whether
the shark is hunting for food or just resting. You would
expect that the predator would have a highly convoluted
and space filling search trajectory as he searches for
food. If he has no information about the location of his
prey, this kind of track increases his chances of
finding his prey. If he's not hunting, his track is
probably not as irregular.
In the dimension
page , we noted that dimension is a constant
characteristic number that can quantify the complexity
of a geometric object. If this is true, dimension could
quantify the shark's track. If we had a dimension for a
resting shark's track, we could use that dimension as a
basis for comparison as we attempt to define a shark's
behavior. In the applet
, you will have the opportunity to define different
shark tracks and determine their complexity. But, you're
wise to read ahead to get a feeling for how fractal
dimension is calculated.
There are many shapes and patterns which do not conform
to the integer based idea of dimension (i.e. D = 1, 2,
or 3) described in the dimension
page . There are patterns that lie in a
two-dimensional plane, but if they are linearly scaled
by a factor S, the area does not increase by S squared
(D = 2), but by some non- integer amount. In other
words, in our formula N = SD, D would be some
value between one and two (i.e. 1.367). These geometries
are called fractals and fractals describe a large
percentage of all patterns in nature! A non-integer
value for dimension is called a fractal dimension. And,
the fractal dimension of a geometric pattern that
represents an ecological phenomenon is that
characteristic constant number that permits us to
quantify the pattern.
Through the example of a shark's swimming track, this
page describes the concept of fractal dimension. We will
describe one of several ways to compute fractal
dimension. We will also compare the different methods
and describe when each method should be used. The
computational details for the various methods can be
studied in the additional resources listed below.
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The shark's
swimming track describes a jagged, irregular
line over the surface of the water. And, since
your observations are recorded at some
predetermined time interval, it is highly likely
that the track is composed of a number of small
segments that you didn't observe.
The question at hand is, what is the dimension
of the shark's track??
Is it a one-dimensional line? That's impossible
because the track moves within the
two-dimensional plane of the ocean surface. |
But, the track isn't a two dimensional plane either. The
track would only become a plane if it covered every
molecule of the water surface in the plane's area of
observation. That would be much like a child randomly
and completely coloring a piece of paper with crayon
lines. So, the track's dimension is somewhere in between
a straight line and a plane because it is neither a
straight line nor is it completely space filling.
Let's first examine the two possible extremes of a
shark's track. If the shark were to swim only in a
perfectly straight line or curve, we could say that his
track is one-dimensional. If the shark were to swim in a
totally random fashion, describing lines that eventually
"fill" the ocean's surface, we would say that
his random swimming is two-dimensional because it covers
an entire two-dimensional plane.
But, the shark is controlling his movements. In effect,
he is constraining his track so that his track is
neither one-dimensional or two-dimensional. We would say
that his track is fractal. And his track would have a
fractal dimension of somewhere between that of a
straight line ( i.e. 1 ) and a plane ( i.e. 2 ).
Fractal dimension is a measure of how space filling a
pattern is. Any pattern in the two-dimensional space of
a plane must have a fractal dimension of between 0 and
2. A point has fractal dimension of zero, a line has a
fractal dimension of one, and a pattern which completely
and randomly fills all the space in a plane has a
fractal dimension of two. The coastline of Britian is an
irregular line on a plane. It has a fractal dimension of
about 1.31. The Australian coast has a fractal dimension
of 1.13 and the South African coast (one of the
smoothest coastlines) has a fractal dimension of 1.02.
In addition to two-dimensional space, we can talk about
the fractal dimensions of objects or patterns in
three-dimensional space. Here, a pattern could have a
fractal dimension up to 3.0 -- the topological dimension
of three-dimensional space. Natural landscapes and
topography are fractal. A landscape that portrays
latitude, longitude, and altitude is an example of a
fractal with a dimension somewhere between 2.0 and 3.0.
In fact, one could speak of very high fractal dimensions
if we are dealing with multi-dimentional data sets. In
our discussion, we will keep things simple by dealing
with two-dimensional space.
The higher the fractal dimension of an object up to its
limit (two for a limiting plane) , the more complex it
is. At a fractal dimension of two, a pattern on a plane
is totally random. At a low fractal dimension (say
1.00003), the pattern is not very complex.
So, our shark has a highly convoluted and space filling
search trajectory as he searches for food. The fractal
dimension of his track would be quite high. On the other
hand, a prey wants to minimize his chances of getting
caught. Therefore his movement or track would be more
linear and less irregular. With a movement that is close
to linear, his chances of getting caught are much less.
The fractal dimension of a prey would be expected to be
quite low.
So, how do we calculate fractal dimension? The objective
is to find D in the formula N = SD. Our
logarithmic form of the formula (D = log N/log S)
provides the means.
Fractal dimension is based a measure of how an object's
mass, volume, length, area, or some other property
changes over varying scales of measurement. There are
several methods that can be used to calculate fractal
dimension. The additional resources (below) describe
five methods for computing a fractal dimension. They are
the similarity dimension, the caliper (or compass)
method, the box-counting (or grid) method, the
pixel-dilation method, and the mass-radius method. Each
method has it's strengths depending on which kind of
fractal pattern is being analyzed.
This page describes only the box-counting method because
it lends itself well to non-exact, statistically
self-similar fractal patterns that are generated from
empirical data and photographs. The box-counting
dimension can measure pictures that are not exactly
self-similar. And most real-life patterns are
statistically self-similar but not exactly self-similar.
This is probably the most common scenario for marine
ecologists. Furthermore, the box-counting dimension is
very easy to program. But, there are limitations to the
method which are described in the resources noted below.
Since there are numerous methods to compute fractal
dimension, it is important to know which method suits
the project. The various methods and their utility are
described in some of the additional resources.
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To calculate
the box-counting dimension, we place our image
on a grid. In the illustration to the left, we
are using an idealized Koch fractal to mimic our
shark track. s can be the total number of grid
cells. Count the number of grid blocks that the
pattern touches (114). Label this number n(s).
Now, resize the grid. In the example, the grid
square size is halved (the number of cells is
doubled). Again, count the number of grid blocks
that the pattern touches (273). The process can
be performed over a range of box sizes. Plot the
counted values found on a graph where the x-axis
is the log(s) and the y-axis is the log(N(s)).
Draw in the line of best fit and find the slope.
The box-counting dimension is equal to the slope
of that line. Or, follow the example to the left
and calculate D directly. On my calculator, I
actually got 1.259867. If you perform the same
exercise for a figure that is a line and another
figure that is a square, you will get the
expected dimensions of 1.0 and 2.0 respectively.
So, with a method for computing fractal
dimension, we are able to quantify the
complexity of ecosystem patterns. |
Lacunarity is a counterpart to the fractal dimension
that describes the texture of a fractal. It has to do
with the size distribution of the holes. Roughly
speaking, if a fractal has large gaps or holes, it has
high lacunarity; on the other hand, if a fractal is
almost translationally invariant, it has low lacunarity.
Different fractals can be constructed that have the same
dimension but that look widely different because they
have different lacunarity. There are applications of
lacunarity in image processing, ecology, medicine, and
other fields. I provide two lacunarity references below
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