Self-Similarity

The simplest fractals are constructed by iteration. For example, start with a filled-in triangle and iterate this process:

For every filled-in triangle, connect the midpoints of the sides and remove the middle triangle. Iterating this process produces, in the limit, the Sierpinski Gasket.

Click the picture to repeat.

The gasket is self-similar. That is, it is made up of smaller copies of itself.

We can describe the gasket as made of three copies, each 1/2 as tall and 1/2 as wide as the original. But note a consequence of self-similarity: each of these copies is made of three still smaller copies, so we can say the gasket is made of nine copies each 1/4 by 1/4 of the original, or 27 copies each 1/8 by 1/8, or ... . Usually, we prefer the simplest description.