Fractal: A definition

A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension and may fall between the integers. Fractals are typically self-similar patterns, where self-similar means they are “the same from near as from far”. Fractals may be exactly the same at every scale,  or they may be nearly the same at different scales. The definition of fractal goes beyond self-similarity per se to exclude trivial self-similarity and include the idea of a detailed pattern repeating itself. (Wikipedia: http://en.wikipedia.org/wiki/Fractal)

Inspired by the study of weather patterns and Strange Attractors, I decided to use 2 axioms which at first glance are similar, but have very slight differences. Once repeated recursively, given a set of rules, these differences will explode into 2 very unique sets.
As an example:

Axiom 1

{A, B, A, B, C, A, B}

Axiom 2

{A, B, A, B, D, A, B}

Rules:

A = AB

B = ABC

C = BC

D = AC

Step 1

Axiom 1 {A, B, A, B, C, A, B} =

{A, B, A, B, C, A, B, A, B, C, B, C, A, B, A, B, C}

Axiom 2 {A, B, A, B, D, A, B} =

{A, B, A, B, C, A, B, A, B, C, A, C, A, B, A, B, C}

At this point, the differences have been abstracted further by the use of recursion. It is difficult to see, and even more difficult to hear the differences between the two.

Step 2

Axiom 1 {A, B, A, B, C, A, B, A, B, C, B, C, A, B, A, B, C} =

{A, B, A, B, C, A, B, A, B, C, B, C, A, B, A, B, C, A, B, A, B, C, B, C, A, B, C, B, C, A, B, A, B, C, A, B, A, B, C, B, C}

Axiom 2 {A, B, A, B, C, A, B, A, B, C, A, C, A, B, A, B, C} =

{A, B, A, B, C, A, B, A, B, C, B, C, A, B, A, B, C, A, B, A, B, C, B, C, A, B, B, C, A, B, A, B, C, A, B, A, B, C, B, C}

Now, just at the 2nd iteration, we can see the similarity grow farther apart:

{A, B, A, B, C, A, B, A, B, C, B, C, A, B, A, B, C, A, B, A, B, C, B, C, A, B, C, B, C, A, B, A, B, C, A, B, A, B, C, B, C}

{A, B, A, B, C, A, B, A, B, C, B, C, A, B, A, B, C, A, B, A, B, C, B, C, A, B, B, C, A, B, A, B, C, A, B, A, B, C, B, C}

This is a very simple example, using only 4 rules on 3 variables, but it gives the general idea behind this new piece I’m working on. I’m thinking the working title “Strange Attractors” is appropriate.

-S