Sorry if this is a stupid question, but I'm a physicist, not a mathematician, and fractals are pretty new to me.

Is there a simple relationship between the fractal dimension of a set and the fractal dimension of that set's boundary?

For non-fractals, the relationship is of course that the boundary dimension is one-less than the bulk dimension (e.g. the boundary of a 3d sphere is a 2d surface).

Any help (including the ways in which my thinking is totally wrong!) would be greatly appreciated.

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    $\begingroup$ Definitely does --not-- sound like a stupid question. But one question could help you understand your question better : how do you define the boundary of a fractal? In the case of a manifold, it's easy ; you define manifolds to be locally homeomorphic to an open subset of $H^n = \{ (x^1,\dots,x^n) \in \mathbb R^n \, | \, x^1 \ge 0 \}$, and those points whose local homeomorphism put them on the $x^1 = 0$ part of $H^n$ are called boundary points. The definition of boundary itself forces the dimension to drop by $1$. $\endgroup$ – Patrick Da Silva Oct 23 '13 at 15:00
  • $\begingroup$ @PatrickDaSilva, are there manifolds with fractional dimension? The mapping to $H^n$ seems impossible for fractals (my understanding of fractals is based mostly on examples like the Koch snowflake, etc.). In fact, a homeomorphism with $H^n$ seems like it immediately gives the set an integer dimension. $\endgroup$ – Zane Beckwith Oct 23 '13 at 15:08
  • $\begingroup$ The fractal dimension is typically not defined in the same way as the dimension of a manifold, because the dimension of a manifold is defined for manifolds (duh) and I don't think fractals are manifolds. So I also expect the boundary of a fractal to be defined in a different way. Do you have examples of fractals where you know what the boundary "is", but you don't exactly understand how it should be defined? In other words, are there examples of fractals where you know what you want to be the boundary, but don't know a general definition for what you're trying to describe? $\endgroup$ – Patrick Da Silva Oct 23 '13 at 15:54
  • $\begingroup$ I believe this is the standard definition of the fractal dimension : en.wikipedia.org/wiki/Hausdorff_dimension $\endgroup$ – Patrick Da Silva Oct 23 '13 at 15:56

It may depend on how you define "fractal", but typically a fractal is closed and nowhere dense, so the boundary is the set itself.

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  • $\begingroup$ My interest stems from computer simulations; the sets I'm talking about are essentially percolation clusters. Thus, my conception of fractal dimension is the way the number of sites in a cluster scales with some linear span of the cluster. So, the essence of my question is whether or not I can estimate the number of sites on the boundary of a typical cluster wth given linear span, if I know the fractal dimension of my clusters. $\endgroup$ – Zane Beckwith Oct 23 '13 at 15:30

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