1. A. 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 iterate.
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.
"Big gaskets are made of little gaskets,
The bits into which we slice 'em.
And little gaskets are made of lesser gaskets
And so ad infinitum."
This implies fractals possess a scale ambiguity.

Return to Introduction to fractals.