Other than the ever-pervasive homework, my recent activities have included getting to the final world of Donkey Kong, playing racquetball, dining upon asian cuisine, kicking a giant yellow ball around and watching Beauty and the Beast. I was very impressed by how great that movie still is.

My goal this week will be to talk about fractal circuits. I choose to begin with a short discussion on fractals. A fractal is an object or image that is itself composed of that same object or image. This recursive definition will make more sense after this example:

This simple fractal is called the Sierpinski Triangle. Take a look in the picture at the diagram labeled #1; you will note that is contains three black triangles. Now imagine what would happen if you took each of those three triangles and replaced it with three more black triangles. Basically, you would be replacing each component of diagram #1 with the entirety of diagram #1. The result of this process is diagram #2. Carrying out this process on diagram #2 (replacing each of its nine black triangles with three more) would create diagram #3. This iterative process is how you create the next "level" of a fractal. Note that it is easy to say how many black triangles will be in any given level of the fractal; the first has 3 the second has 3*3=9, the third has 3*3*3=27, thus the nth level will have 3^n black triangles.

So basically the Sierpinski Triangle is a fractal because it is an arrangement of triangles that are themselves composed of triangles. The astute reader will also have noticed the other defining characteristic of fractals: they look really cool.

A tree composed of trees

A spiral composed of spirals

Fractals occur naturally too: this cauliflower looks like a tree composed of trees as well

I'll post the first iteration of one more fractal:

By replacing each of the red squiggles with the entire diagram, see if you can figure out (and perhaps sketch) the next few iterations of the fractal. I'll post what it looks like next time since this particular fractal will be important to my discussion.

After watching Beauty and the Beast I would be remiss to cite its antagonist: Gaston!

I'm in that peculiar position again when a perfect description of my villain already exists, and providing anything less than that would be a disservice. Everything you need to know about Gaston is in this video. Make sure you check out the lyrics. Whoops, almost forgot the other part.