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Generating Fractal Equations

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Date: 7/1/96 at 22:26:3
From: Anonymous
Subject: Generating Fractal Equations

Is there any way to generate a fractal equation from a set of numbers?
For example, taking a million or so seemingly random numbers and
"reversing" the fractal process to provide a formula that will
generate the same series of numbers every time?
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Date: 7/7/96 at 0:43:6
From: Doctor Pete
Subject: Re: Generating Fractal Equations

In short, the answer is no.

Before I explain in further detail, let me set down some preliminary
notions; these you may or may not already be familiar with.

I am assuming what you refer to as "fractal process" is the iteration
of some mapping such as z |--> f(z).  That is, we start with an
initial value z0, and calculate the values f(z0), f(f(z0)),
f(f(f(z0))), ... and plot these iterates.  The set of iterates under
the mapping defined above is called the *forward orbit* (or for
simplicity, orbit) of z0 under the mapping z |--> f(z).  The *inverse
orbit* or *backward orbit* is the set of iterates of the inverse
mapping, z |--> f^-1(z), where f^-1(f(z))=z.  Note that such a mapping
may be multivalued, such as in the case where f(z)=z^2+c; another term
for the iterates of the inverse mapping of f is the *preimages* of z0
under f.

So this is basically how we generate sets of points (and pretty
pictures). Take some function f, and a point z0, and we plot the orbit
of z0 under f. There are other ways of plotting points, such as
numerical solution of differential equations, but in general the
concept is the same:  we use iteration to get the next point from the
previous.

Now for the explanation:

One of the things about iteration is that it has the property of
magnifying errors - that is, if two points z1 and z2 are initially
very close together, their difference will increase with every
iteration.  This is also the exact same property that gives fractals
their complex and unpredictable nature. So after, say, fifteen
iterations, the roundoff error created by the computer will be too
large to make the value of the point have any specific meaning; that
is, if you calculated 50 iterations, and then take 50 inverse
iterations, you will not get your original point back.  This is why
you cannot take a set of data and represent it by a mapping if the
data is fractal.

However, if you knew the order in which the points were plotted, you
have orbit information which *might* let you *approximate* the data
with some sort of mapping.  But this relies on understanding the
dynamics of the mapping beforehand, which is highly unlikely.

-Doctor Pete,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Equations, Graphs, Translations
High School Fractals

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