Fractal graphs by iterated substitution

Abstract

Space filling curves are well-known examples from analysis, which can be seen as examples of fractals produced by iterated substitution. Steps in the construction of three-dimensional space filling curves have also been popular subjects of mathematical sculpture, but suffer from potential structural problems due to the low levels of interconnectivity inherent in a curve. We introduce a generalization of the construction of space filling curves to constructions of fractal graphs by iterated substitution. These follow the same kind of substitution rules but allow vertex degrees to be greater than two. We also introduce the use of the Cheeger constant as a way to evaluate the structural strength of a sculpture based on a graph, from a purely graph theoretic point of view.

Keywords:

• space filling curves
• fractal
• iterated substitution
• graph theory
• Cheeger constant
• mathematical sculpture

AMS Subject Classification:

• 05C62

Acknowledgements

The author would like to thank Yla Tausczik for assistance in writing python programs to generate the graphs in Section 4.4, and the referees for many helpful suggestions. The 3d printed sculptures were produced by Shapeways.com.