A bit of Fractal Math
Lets do a bit more math now. Remember the Plusses fractal from a few pages back? One can do a lot of math with this simple fractal. Lets start by finding a simple sequence; that of the number of line ends for each iteration. If you look at the first figure, it is easy to see that there are 4 line ends in the 0th iteration. We now have to figure out how many line ends are added with each iteration. Lets take any one line end to see how it gets transformed to go to the next iteration. Can you see what happens? Remember that the exact same thing happens to each line end.
Fractal dimensions. Some of the interesting aspects of fractals has to do with looking at how some of their dimensions evolve as they grow from iteration to iteration. We'll finish this section with two problems that deal with fractal dimensions. Both these problems are not that easy, that is why I have the following offer. If you solve either of them, send me your name and as much information as you want and I'll be very happy to publish it right here. If you attempt to solve it or are simply curious to see the solution, let me know and I'll send you the answer.
Problem 1 -- Area and Length of the Plusses Fractal. This problem has to do with calculating two dimensions of the Plusses fractal, the area of the rhombus as well as the total length of all the lines within it. Here are a few hints:
- Hint 1: Assume that the lengths of the initial lines are 1 inch long. This means that you start with an area of:
b x h = 1 x 0.5 = 0.5 in2 and a total line length of 1 + 1 = 2 - Hint 2: At every step, the size of the line of the new +'s is half of the one before. So for iteration 1 they would be 1/2 in. Additionally, the new +'s are added exactly at the line ends. That means that, for example, the base of the rhombus at iteration one becomes 1 + 1/4 + 1/4 = 1.5 in.
- Hint 3: You could use a program like Excel to help you figure out how the different dimensions evolve.
- Calculate how the area of the rhombus increases as the fractal grows
- What happens to the area? Does it keep growing at every iteration or does it stop growing?
- Now calculate the total length of all the lines inside the fractal at each iteration
- What happened to the total line length? Does it keep growing at every iteration or does it stop growing?
- Check what happens to the rate of growth of both the area and the total length.
- What do you think about the results? Did you guess that that would happen? Aren't the results interesting?
Problem 2 -- Total number of +'s on the Plusses Fractal. The next problem that one could work on is to figure out how fast the number of +'s grows. You start with 1, then 5, etc. Finding a formula to predict the total number of +'s for each iteration is not that simple. That is why I'm including a few hints here:
- Hint 1: Remember that the process consists of adding a + sign at each line end, that is why the answers to the questions right above will come very handy.
- Hint 2: You may want to start by finding out how many +'s are added at each iteration and then totaling them. Perhaps write a table like this:
| Iteration | Number of line ends | Number of +'s to add for this iteration | Total number of +'s for this iteration |
| 0 | 4 | 1 | 1 |
- Hint 3: Take the last column and write it, not just as the total sum, but write it as an addition of terms. Say something like: 1 + 5 + 15 + 30 + etc. You can write one such equation for every line of the table
- Hint 4: Work with the previous equations to find a sequence within a sequence
- Hint 5: Get the final equation to predict the total number of +'s for each iteration.
And at this point we are ready to move on to do some actual hands-on growth of some fractals with Java programs.
Last Updated: Sunday, 06-Apr-2003 19:20:33 GMT
 Arcytech | Java Home Page |  Provide Feedback |