Making fractalsDate: Sat, 8 Jul 1995 15:26:28 -0400 From: Anonymous Subject: How do you make a fractal ? Question: How do you make a fractal? -- Gustavo Moreira Date: Mon, 10 Jul 1995 09:34:05 -0400 (EDT) From: Heather Mateyak Subject: Re: How do you make a fractal ? Hi Gustavo, Fractals are part of the mathematical field called dynamical systems, or, sometimes, chaos theory. Any object that does not have an integer dimension (1D, 2D, etc.) is called a fractal. Some fractals are very geometrical in shape, and these are the easiest to understand without much difficult math to go along with them. In one of these types of figures, the von Koch curve, you start with a triangle, and then add a triangle onto each side with sidelength 1/3 as long as the original size of the triangle's size. The length of each side of the triangle increases by 4/3 with each new addition of a triangle onto each side of the previous form. If you are familiar with exponential growth, you will see that if something increases by 4/3 each time, 4/3 x 4/3 X 4/3 ...., or (4/3) to the nth power, this number goes off to infinity as n gets infinitely large. To find the area enclosed by this curve, you can look at the area of the shape at each step. Take the length of a triangle side to be 1. Figure the area of the triangle (you need to use some trigonometry to find the height, and then use 1/2 x base x height). Now add the triangle to each side of the previous form, and calculate the area. If you keep doing this for a while, you may notice a pattern. If you want to try this, you will need to know something about geometric series. It's a really interesting problem, though. There are many other kinds of fractals, simpler than the von Koch curve. Take a line segment of length 1. In the next step, remove the middle third of the line. In the next step, remove the middle thirds of the remaining segments. If you do this forever, you have a set that is called the Cantor Ternary Set. This is another example of a fractal. This has dimension between 0 and 1. One way that mathematicians measure the dimension of highly geometric sets (you will see what I mean by this; those that are formed by removing pieces, but pieces that look exactly like the original whole) is a measure called the Capacity Dimension. The precise definition is a little more complicated, but I will give you a definition that is easy to handle (that is, if you are familiar with logarithms). Dc = (ln m)/(ln 1/r) Now, the m stands for the number of copies of the original figure that exist in a shrunken size at the next step. For example, with the Cantor Ternary Set, when we go from the first step to the second step, we end up with 2 copies of the original line. At step 3, we end up with 2 copies of the picture that is in step 2. So, m=2. Now, r stands for the amount of 'shrinkage' of the sidelength from one step to the next. For the Cantor Ternary Set, we see that the length of each part in step 2 is 1/3 as long as the original line. So, r=1/3. So, the capacity dimension of the Cantor Ternary Set is Dc = (ln 2)/(ln 3), or I think it's about .63 or somewhere around there. I highly suggest drawing the pictures as you go along, looking at the set from one step to the next, seeing how the set develops. There are several other sets of which you can measure the capacity dimension. For example, there is a set called the Sierpinski Gasket. To form this set, start with an equilateral triangle (filled in). For step 2, remove the middle triangle (imagine the triangle being split into four sections). For the next step, remove the middle triangles or the three triangles that remain. If you repeat this process infinitely, you will have the Sierpinski Gasket. Can you calculate the capacity dimension of this set? There is another figure called the Sierpinski Carpet. If you start with a filled-in square, split up the square into 9 squares. For the first step, remove the middle square. Then, for the next step, remove the middle square of the remaining squares. If you keep doing this, you'll have the Sierpinski Carpet. An example of a fractal that is between 2 and 3 dimensions is the Menger Sponge. Start with a Cube that is split into 27 equal smaller cubes. For the first step, remove the middle cube, and the cube that is in the middle of each face of the cube. Repeat this action on the remaining cubes for the next step. One thing to notice about all of the highly geometric fractals is that if you keep focusing in on a part of the set, say of the Menger Sponge, and then blow up the size of that small part, it will look exactly the same as the set as a whole. Some objects in nature exhibit a fractal form, like a fern, for example. Each part of the fern resembles the fern as a whole. Or a tree, to a certain extent. Each branch has a part with no branches (like the trunk), and then it has more branches with leaves, like the tree as a whole. There are many examples of fractals in nature. Just look around you. I have mentioned only these 'highly geometric' fractals. There are other sets that we encounter in dynamical systems that have non-integer dimension, though there are several that cannot be easily measured (or accurately measured) using the capacity dimension. There is another measure of dimension called the Lyapunov Dimension, which is used to measure the dimension of sets that are less regularly shaped than the ones I have mentioned. The definition of the Lyapunov Dimension is quite complex, so if you want to study in detail, you will have to look it up in a textbook. The textbook I used for my class was 'Encounters With Chaos' by Denny Gulick. Anyway, I'll throw out a name of a set that one uses the Lyapunov Dimension to measure the dimension of -- the Henon Attractor. Knowledge of linear algebra is necessary for an understanding of this object, so if you want to look it up, be my guest. The most famous fractals are the Julia Sets and the Mandelbrot Set. There are lots of places on-line that have pictures of these sets. If you've ever seen the pictures of these fractals, you know they are quite beautiful. However, as Ken mentioned, to understand these fractals, you need a thorough understanding of complex numbers and in addition, a thorough understanding of dynamical systems. The Julia Sets and the Mandelbrot Set exist in the complex plane. Mathematicians can clearly define these sets, and many of the properties they exhibit, but there may be more to discover. These things have been studied only since the latter part of this century. So whoever says math is dead, having been established by stodgy old-timers from a long time ago, is just plain wrong! Anyway, if you want me to further explain anything of what I said, feel free to write back. If you want to learn even more, I could try to tell you more, or at least tell you where to start. One good place to look is: http://mathforum.org/library/topics/fractals/ Hope you like this, Heather Mateyak (occasionally moonlighting as Dr.Math) |
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