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Virtual
Ecosystems -- Fractal Construction
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| A
Koch Snowflake Iterating Through Six Generations |
A
Serpinski Triangle Generated With The Chaos Game Method |
A
Barnsley Fern Generated With The IFS Method |
The best way to understand the
way something works is to build it. With that philosophy
in mind, this page describes how simple fractal objects
are constructed . The construction process will lay the
groundwork for understanding self-similarity
and fractal
dimension .
Euclidean shapes are described by simple algebraic
formulas. r2=x2+y2
completely describes a circle of radius r. For a given
radius, one calculates the point at x for each of a
number of given y points
But, the crude snowflake pattern shown above doesn't
have a nice formula to describe itself. Instead, as a
natural object, it requires a continuously iterated
algorithm to be constructed. At each iteration, each
line segment of the triangle is replaced by four new
segments each 1/3rd the length of the parent line. The
process for the generating Koch snowflake is shown
below.
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 | Start with an equal sided triangle. A side length
equal to 1 will make things easier. This triangle is
often called the initiator.
| In the middle third of each side of the triangle,
build a smaller equilateral triangle with its sides
equal to 1/3 of the length of its parent. Then erase
the base side of the smaller triangle. The result
for each of the sides of the original triangle
should be should be four line segments, each 1/3rd
the length of the original line -- a total of 12
line segments. This process is often called the
generator because it is repeated in all subsequent
steps.
| For each of the 12 line segments, use the same
process. The result should be 48 line segments 1/3rd
the of the previous line segments or 1/9th the size
of the original triangle's line segment.
| Repeat the process to the desired number of
iterations. |
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Some interesting things happen as the Koch snowflake
is constructed:
| The iteration of a simple rule can produce complex
shapes.
| No matter how many iterations are performed, the
object always stays within the bounds of the
original square without ever intersecting itself.
| Although the algorithm for generating the Koch
curve is concise, simple to describe, and easily
computed, there is no algebraic formula that
specifies the points of the curve.
| The perimeter length of the snowflake object grows
toward infinity as the number of iterations
increase. At each iteration, the length of the curve
increases by a factor of 4/3rd. The panel #1 size is
3. The panel #2 iteration is 12 x 1/3 or 4. The
object in panel #3 has a perimeter size of 48 x 1/9
or 5.333. And so forth. The growth in peripheral
length of the Koch snowflake is non-linear and
exponential. This characteristic relates directly to
the concept of fractal
dimension . Look at the status bar on your
browser. The Koch snowflake applet is calculating
and displaying the perimeter size as each iteration
is processed.
| If you were to examine the Koch snowflake using a
magifying glass (or a computer zoom operation), you
would find that the smallest piece is identical in
shape to the largest triangle. Unlike Euclidean
shapes, fractal shapes have detail at all levels of
magnification. Each small portion of the snowflake,
when magnified, reproduces the larger portion
exactly. This characteristic is called self-similarity
. It is said that fractal objects have no scale
because they look the same at all magnifications.
| Because the Koch snowflake is composed only of
lines and sharp corners, there is no way to fit a
unique geometric tangent at any point. Since fitting
tangents is a central feature of differential
calculus, calculus cannot be used as a tool to
describe or analyze the Koch snowflake and other
fractal objects. |
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Another method for generating fractals is the
"chaos" game. This method uses random numbers
combined with a set of rules. The rules are:
| Step 1: Choose an equilateral triangle in the
plane.
| Step 2: Label the vertices A,B, and C.
| Step 3: Randomly pick the initial point inside the
triangle.
| Step 4: Randomly choose one of the three vertices.
| Step 5: Move to a point 1/2 the way between that
vertice and where you are now. Plot that point.
| Step 6: Go to step 4. |
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After many thousands of points, the Serpinski
triangle shown at the top of this page will emerge.
Notice that the pattern structure inside Serpinski
triangle looks a lot like the Wolfram
CA Rule 22 -- mollusk shells. The characteristics of
the "chaos game" approach are:
| We use a deterministic rule algorithm.
| Unlike the Koch curve, we emply a
non-deterministic, random process.
| When we plot the points generated by this random
process, we always get the same result -- a figure
of great regularity.
| Our starting point is unimportant.
| No matter what our magnification, we see a
Sierpinski triangle. -- i.e. there is self
similarity |
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Yet, a third method for generating fractal forms is
the Iterated Function System (IFS) approach. IFS
fractals use a rule set that is supplied in a table. The
table contains coefficients for rotation, translation,
and scaling at different levels of detail. The computer
graphics industry uses rotation, translation, and
scaling to render dynamic visualizations of all kinds.
With these coefficients, we can use a random number
iterative process (much like the "Chaos Game"
with the Sierpinski triangle) to randomly pick the
coefficient set to generate a dot in the Barlsey fern
shown at the top of the page. The IFS method is used
extensively in image compression and storage. Instead of
storing huge amounts of pixel information, the IFS
method only requires that the coefficient table be
stored. This significantly reduces the size of the data
file that needs to be transmitted or stored.
We have just demonstrated three fractal methods for
generating objects that can better appproximate natural
objects than can Euclidean geometry. But, as natural as
it appears, there are no irregularities in the fern. It
is too perfect. The fractals that have been generated at
this page are called exact fractals because their
precision is exact -- a characteristic rarely found in
nature.
On the other hand, cauliflower, broccolli, and our
own lungs are all fractal objects with irregularities
between adjacent points. The extent of physical
self-similarity is limited. Nonetheless, they are all
fractal attractors. Natural objects differ from exact
fractals in two distinct ways:
| There is always a lower and upper cutoff of the
scale on which a natural fractal structure can be
observed. With cauliflower you could see self
similarity within certain limits. The stalks were
one group of self similar objects and the flowers
are another group of self similar objects.
| The self similarity within a group is statistical
instead of being physically exact. With statistical
self similarity, an analysis of mean values,
variances, etc. will give the same statistical
properties at each rescaling (magnification) of the
object. In other words, it is the statistical
properties (not the physical dimensions.) that are
self-similar. The mean and variance (not the
physical form) stay the same at each level of
magnification. Since there are always fluctuations
(variances) in natural processes, they never lead to
physical structures with an exact level of symmetry
or self similarity. |
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We will look more at natural objects and processes to
a greater extent after examining self-similarity
and fractal
dimension .
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