Enzymind Labs

The most common example of a 3x3 rule system is the Serpinski carpet. The 3-D generalization is called a Menger sponge. The 1-D system, of which the Serpinski carpet is a generalization, is the Cantor set; this system was an important example in the history of topology and the theory of infinite sets.

The basic 1-D rule (for the Cantor set) is:

Start with a;

a -> [ a b a ], b -> [ b b b ]

Paul Bourke has a nice page of examples of related gaskets and sponges , including a physical device (an antenna) based on the Serpinski gasket.

How to make Serpinski cookies, using stretching of dough to accomplish scaling between itterations. In principle, any simple recursive system cookies can be realized with this technique.

There are other 3x3 rule recurrent systems that lead to other fractal shapes. Here are a few:

Systems with the same symmetry as the Serpinski carpet

2 symbol 3x3 rule system #1

3 symbol 3x3 rule system # 1