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Re: Fractals
Posted:
May 4, 1995 11:08 AM
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>I do not know what
>fractals are, or how the knowledhge of fractals is used in math
>professions.
>
>Can anyone help me with an answer?
>
>Thanks
>Eileen
>
Fractals are essentially self-similar objects: for example, a line of
telephone poles in perspective; cut off the first few poles and enlarge the
picture and it looks the same as it did before. A complicated coastine is
another example, as is, come to think of it (although most people don't
bother to say so) a straight line.
There are some middle grade activities using fractals. They have names
like Sierpinski's carpet. These activities probably help geometric
intuition, and sometimes can be hooked up to questions that are essentially
about infinite series but, because of their geometric nature, are
accessible to kids. (I'll give an example later.) The uses of fractals
are many, both within mathematics and outside it. They are used for
generating realistic computer animations, are good models for things like
blood vessels (big arteries branching into smaller arteries branching
into...) etc. They are a Very Big Thing in both pure and applied
mathematics these days. Although they had essentially been around for over
a hundred years -- those of us who've had math past calculus may remember
the continuous nowhere differentiable function, which is essentially a
fractal, or the Cantor set, another fractal -- but until we had powerful
desktop computers they were very hard to picture, which is why there's an
explosion of interest in them.
Fractals also can help kids think about infinite process (because that's
how you get a fractal).
Here's the fractal exercise I promised. It may be hard to follow without
graphics, but it's pretty standard and you can find it in any bunch of
middle grade or middle school activities on fractals. I think it's called
Sierpinski's triangle.
Take an equilateral triangle. Connect the midpoints of the sides and you
get an inner triangle (we'll call it the dual) and three outer triangles.
Throw out the dual. Now in each of the remaining triangles, construct the
dual and then throw these new duals out. You have nine triangles left. In
each of these construct the dual and then throw these new duals out.... You
get the picture, yes?
Easy question: how many small triangles do you have at each stage of the
process?
Harder but not impossible question: when you're done (i.e. after
infinitely many steps) what's the area of what's left?
Cheers.
====================================
Judy Roitman, Mathematics Department
Univ. of Kansas, Lawrence, KS 66049
roitman@math.ukans.edu
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