Chapter 5: Two Dimensions and Beyond

Section 4: Substitution Systems and Fractals

[Substitution systems based on] other shapes

The systems on pages 187 and 188 are based on subdividing squares into smaller squares. But one can also set up substitution systems that are based on subdividing other geometrical figures, as shown below.

The second example involves two distinct shapes: a square and a GoldenRatio aspect ratio rectangle. Labelling each shape and orientation with a different color, the behavior of this system can be reproduced with equal-sized squares using the rule {3 {{1, 0}, {3, 2}}, 2 {{1}, {3}}, 1 {{3, 2}}, 0 {{3}}} starting from initial condition {{3}}.

Image Source Notebooks:

From Stephen Wolfram: A New Kind of Science [citation]