## Fractals

### Okay, but where are all the pretty pictures?

Fractals provide a much more visual insight into chaos theory. Like systems in chaos, the fractal undergoes iteration, except the iteration results in two conditions:

• self-similarity
• non-integral complexity

### Snowflakes?

Let us look at the Koch Snowflake above in order to understand these two concepts. Basically, we start off with a triangle, and then go to each side and divide it into thirds. For the middle third, we add two new sides that are the same length as the middle third. The result is the image above on the left. If we continue this process ad infinitum, we get the Koch Snowflake. A java example is provided here for a better understanding.

Self-similarity is the idea that smaller parts contain the whole design. For instance, if we went to one of the smaller sides, we would still see the sides break off in the same pattern, kind of like a miniature version of what is going on in the entire snowflake.

### Okay, but what is this complexity?

You have just been asked to count the number of people standing in a line of 20. It seems pretty simple: 20. Now we will apply this technique to counting the number of items in a 7x7 square table. Instead of counting every single item in the table, all we have to do is computer 7 squared. Now, what about counting the number of blocks in a Rubik's cube? We would use the same technique. One side contains three blocks, so now we cube the answer to get 27 blocks. Notice that for every dimension added, we raise the block to that power. We can generalize this in the equation:

N = r^D

In that equation, D represents the number of dimensions (also called complexity) with N representing the total count and r the scaling factor. Solving for D, we get that:

log N = D* log r
log N/log r = D

### Okay, so then wouldn't it be possible to get dimensions that aren't integers?

Exactly! That is the basic idea behind non-integral complexity. Let us look at the Koch Snowflake. We divided it into three parts, which would make 3 the scaling factor. The total number of resulting sides was 4. Solving for D, we get 1.26, a value that is not an integer, thus confirming that the Koch Snowflake is indeed a fractal.

### Could I see another one?

On the left is the Cantor set. If you will notice, it is self-similar... but is it complex? That should be easy enough. Okay, with an iteration, the three sections become 5, making that the value of N. This is done by dividing a line into three parts and removing the middle one, making the scaling factor r equal to 3. Calculating the dimension, we get 1.46, confirming that this, too, is a fractal.

Last updated 8/9/99