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### Fractals

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Date: Wed, 9 Nov 1994 10:03:20 EST
From: Debbie Lloyd2
Subject: We need help Dr. Math

Dear Dr. Math,
We are doing a project on fractals. Is there any way that you could help
us understand what they are about? We have read all types of books and don't
understand what they are saying. Could you explain them in regular English for
us?
Thank You,
Brandon Cook
Aaron Bauldree
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From: Dr. Ken
Date: Wed, 9 Nov 1994 10:54:32 -0500 (EST)

Hello there, Aaron and Brandon!

Wir wollen einen functionieren finden, der den dinge hat, das es SO gross
ist, das man kein nummer in es stecken kann, um es umzubringen, und
daruber konnen wir zeigen...

Oh, wait.  You wanted the plain _English_ version.  Sorry.

Personally, I haven't worked a whole lot with fractals, but I have seen them
some, and I'll tell you what I know, and try to give you a couple more
sources to look at.  One thing I do know, however, is that the mathematics
involved is QUITE sophisticated (it deals a lot with complex numbers and
such), and it would take a long time for a high school student to gain a
very good understanding of it.

1) The word "fractal" is a contraction of "fraction dimensional."  You know
that a line is one-dimensional, a plane is two-dimensional, and the space
around us is three-dimensional.  Well, you can also think of any curve (like
a bendy, windy piece of string) as a one-dimensional object, and any surface
(like a sheet that you bend around) as a two-dimensional object.  Well, a
fractal (like the snowflake curve; have you seen that yet?  Try to find it
in a book on fractals) has a dimension that's between these values.  I'm not
sure what the exact dimension of the snowflake curve is, but I'm pretty sure
it's between 1 and 2.

2) How long is the coast of Britain?  The answer is that there's no right
answer.  See, you could never measure all the little nooks and crannies on
the coast, every atoll and bay, and every pointe, so you have to decrease
your resolution when you're trying to measure it.  If you truly did measure
EVERY little crannie and nook, you'd come up with the answer that the coast
of Britain has an infinite(!) length.  We say that it is a fractal.  It has
dimension between 1 and 2.

3) I can direct you to the book "The Joy of Mathematics" by Theoni Pappas
and its sequel "More Joy of Mathematics", which explain things in relatively
simple terms.  If the mathematics behind fractals proves too challenging
(Theoni doesn't put much of the math in the book, she just kind of describes
what's going on), there are all kinds of other neat things in the books, and
it might help you figure out a different topic.  If you are pretty set on
doing fractals and you still want to know more, write back, and maybe one

-Ken "Dr." Math

X-Sender: steve@mathforum.org
Date: Wed, 9 Nov 1994 11:38:14 -0500

To add to what Ken wrote:

Imagine a triangle with each side one foot long.  Attach a new triangle on
the middle third of each side.  Make the new triangles one third the size
but the same shape.  The whole shape now looks like a six-pointed star.
Now take each of the twelve resulting sides and repeat the process.  Keep
repeating forever.  This is called the Koch curve, and looks a little like a
snowflake.

This curve has some interesting aspects.  It never crosses itself, but the
length of it increases as you keep repeating the process (it increases by
four thirds, can you see why?).  If you repeat the process forever then the
length becomes infinite.  As you add the smaller and smaller triangles,
space is being filled, but you could draw a line around the original
triangle and the curve would never go outside of it.  So the curve is
infinitely long and it fills space, but it only fills a finite amount of
space.

Normally we think of a line as having only one dimension and not filling
space.  But this one has more than that yet less than a plane, which has
two dimensions.  So how many dimensions does it have?  Somewhere
between 1 and 2.  It's a fractal, with fractional dimension.

What use is this?  Aside from the beauty of fractals, it gives a way of
measuring the degree of roughness or irregularity of an object.  So in
terms of our discussion of the Koch snowflake, it reflects how well it
fills space.

-- steve ("chief of this great staff")
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Associated Topics:
High School Fractals

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