Fractals & Chaos Recap for 10/1

We had a brief discussion on Stephen Jay Gould’s essay “Size and Shape,” seeing some common threads with prior reading assignments. For one last follow-up, I recommend this NPR piece from 2007 – Size Matters: The Hidden Mathematics of Life (and/or this followup to the  Kurzgesagt video from last week).

From there, we moved on to revisit the problems with the current Hausdorff Dimension formula that we identified yesterday, specifically about what to do with fractals that are self-symmetric, but at inconsistent scales. We developed a new, Generalized Hausdorff Dimension formula:

Annotation 2019-10-01 084135
Generalized Hausdorff Dimension formula

Here, S_1, S_2, …, S_N are all the differing ratios each iterating piece form with the original whole (note that here, each S is a fraction, so a piece that is 1/2 the length of the whole has S = 1/2 instead of S = 2 as it was in the previous definition)

We then practiced this definition with a few more examples, found on this sheet.

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