- Understand self-similarity
- Create a fractal shape
- Change the size of a shape using a formula and measure how complex a shape is and how it changes as it gets bigger or smaller
Fractals are mathematical sets, usually obtained through recursion, that exhibit interesting dimensional properties. We’ll explore what that sentence means throughout the rest of this section. For now, we can begin with the idea of self-similarity, a characteristic of most fractals.
self-similarity
A shape is self-similar when it looks essentially the same from a distance as it does closer up.
Self-similarity can often be found in nature. In the Romanesco broccoli pictured below[1], if we zoom in on part of the image, the piece remaining looks similar to the whole.
Likewise, in the fern frond below[2], one piece of the frond looks similar to the whole.
This self-similar behavior can be replicated through recursion: repeating a process over and over.
Suppose that we start with a filled-in triangle. We connect the midpoints of each side and remove the middle triangle. We then repeat this process.
If we repeat this process, the shape that emerges is called the Sierpinski gasket. Notice that it exhibits self-similarity—any piece of the gasket will look identical to the whole. In fact, we can say that the Sierpinski gasket contains three copies of itself, each half as tall and wide as the original. Of course, each of those copies also contains three copies of itself.
