The mathematics of GraphGrow

GraphGrow in SVN now has code to calculate the Hausdorff dimension of the generated fractal. However, this calculation is only guaranteed to be valid if an open set condition holds. Moreover, there is no algorithm to check if the condition holds (because there is no algorithm to construct the feasible open sets required).

I guess I'll have to add a big fat warning sticker on the info pane of GraphGrow saying that the calculated dimension isn't necessarily valid, but it's frustrating that it seems impossible to check algorithmically if it's valid or not.

References

  • A Fractal Dimension Estimate For A Graph-Directed IFS Of Non-Similarities -- G A Edgar, Jeffrey Golds
  • Multifractal Decompositions Of Digraph Recursive Fractals -- G A Edgar, R Daniel Mauldin
  • On The Open Set Condition For Self-Similar Fractals -- Christoph Bandt, Nguyen Viet Hung, Hui Rao

Footnote: I'm not a professional mathematician, so my ramblings above might be completely wrong..