FractalsDate: 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 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 of my Math Doctor buddies can help you more. -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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