
Re: Fractals
Posted:
May 4, 1995 11:08 AM


>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 selfsimilar 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 =====================================

