"they say" fractals exist in greater dimensions then the two-dimensions we make them in. Maybe like, two and a half dimensions almost, there the going between dimension 2 and 3 has to be difficult for the fractal to maintain,, what do you think?
The embedding space of a Cantor dust may be only one dimension. A Cantor dust constructed by beginning with a line, dividing into three equal parts, removing the middle, and at each successive generation doing the same with the lines that are left has a Hausdorff dimensionality of Log(2)/Log(3), or roughly 0.631. The boundary of the Mandelbrot object is embedded in two dimensions, but has been proven to have a fractal dimensionality of 2:
Shishikura, Mitsuhiro. "The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets." Annals of Mathematics (1998): 225-267.
http://arxiv.org/abs/math/9201282... although the area of the boundary is believed to be zero.
As for maintaining a constant fractal dimensionality, yes, of course, the fractal structure of natural objects necessarily breaks down at some point. Things are after all composed of atoms and subatomic particles. However, given the appropriate dynamics, a fractal structure is actually easier for a biological system to maintain than other structures given its scale independence. You can see that given the simple set of rules for constructing a snowflake curve or Cantor dust. Thus scale independence means less information needs to be encoded. But at the same time, the fractal structure of one part or another of a biological system will usually break down after three or four levels. A different dynamics takes over.
October 02, 2006, 09:19:59 PM
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