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
|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
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.