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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. 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 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.
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:
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. The URL for the answers webpage is now http://britton.disted.camosun.bc.ca/fractals_arcytech/answers.html. And at this point we are ready to move on to do some actual hands-on growth of some fractals with Java programs.
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