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As can be seen in the David Morin's Classical Mechanics, there are some scaling strategies in order to calculate the moments of inertia of certain fractals, for example, the Cantor's set has a moment of inertia of $I_C = \displaystyle \frac{ml^2}{8}$, Sierpinski's Carpet has a moment of inertia of $I_S = \displaystyle \frac{3ml^2}{16}$ and Sierpinski's Triangle has a moment of inertia of $I_T=\displaystyle\frac{ml^2}{9}$ where $m$ is the total mass of the set and l it's longitude of its sides.

However this doesn't involve any integrals, that's why I'm asking this. Is there any way to calculate this moments of inertia by meanings of the usual integral? For Cantor for example:

\begin{equation} I = \int_C x^2 dm = \frac{m}{l} \int_C x^2 dx \end{equation}

Where $dm=\rho dx = \frac{m}{l}dx$)

At first sight, this integral has a value of $0$, as the Cantor set has Lebesgue measure zero. For physical "reasons" a finite number is expected, that's why I thought that maybe I need to consider some kind of "fractional integral" over the Hausdorff dimension of the Cantor's set. This dimension is $d=\displaystyle \frac{log 2}{log 3}$. So maybe should I try to compute this?

\begin{equation} \int x^2 \chi_C dm_d \end{equation}

With $dm_d$ the d-dimensional Hausdorff measure, and $\chi_C$ the characteristic function over the Cantor's set.

I ask a similar question before, and an answer was given by John Dawkins involving a probability approach:


I'll take $m=1$ and $l=1$. The natural "uniform" mass distribution on the standard Cantor set is the distribution of the random variable $$ X:=\sum_{n=1}^\infty {\xi_n\over 3^n}, $$ where $\xi_1,\xi_2,\ldots$ are i.i.d. with $\Bbb P[\xi_n=0]=\Bbb P[\xi_n=2]=1/2$. (This distribution is also the normalized Hausdorff measure of dimension $d=\log 2/\log 3$.) Clearly $\Bbb E[X] =1/2$, while the variance of $X$ (a.k.a. the moment of inertia about the vertical axis throough the mean of $X$) is $$ \eqalign{ \Bbb E[(X-1/2)^2] &=\Bbb E[(\sum_n{\xi_n-1\over 3^n})^2]\cr &=\Bbb E\sum_{m,n}{\xi_m-1\over 3^m}{\xi_n-1\over 3^n}\cr &=\sum_{m,n} 3^{-m-n}\Bbb E[(\xi_m-1)(\xi_n-1)]}. $$ The expectations in this double sum are $0$ or $1$ according as $m\not=1$ or $m=n$; therefore $$ \eqalign{ \Bbb E[(X-1/2)^2] &=\sum_{n=1}^\infty 3^{-2n} = {1\over 8}.\cr } $$


So now I ask if this can be used to obtain the moment's of inertia of the other two fractals above or the integral approach is required or something else. Of course it would be extremely helpful if you could show me any bibliography where this is solved or if you can find any new methods in order to solve this. Maybe some fractional calculus is required or something similar to it.

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    $\begingroup$ This is just a routine integration with respect to the natural measure on the fractal. Probably it's not in the literature because people have yet to see importance in the topic. $\endgroup$ Commented Apr 7, 2016 at 17:36
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    $\begingroup$ Does "routine" mean it's relatively easy to obtain? And do you know somewhere where information about how to compute this can be provided? Thanks. $\endgroup$
    – Rafa
    Commented Apr 7, 2016 at 18:47
  • $\begingroup$ @AnthonyQuas I can't quite agree with your comment here, but I really like the illustration of the piecewise isometry on your profile page! You're paper on the subject looks very interesting as well. $\endgroup$ Commented Apr 7, 2016 at 22:34
  • $\begingroup$ @MarkMcClure: Thanks! My favourite paper on the subject is the one with $3\log 2-\pi^2/8$ in the title. There are still many more unsolved than solved questions in the area. $\endgroup$ Commented Apr 7, 2016 at 23:27

2 Answers 2

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First, I don't think you want to integrate with respect to Hausdorff measure, whose exact value is quite difficult to compute. Rather, you want to integrate with respect to a uniform measure on $C$ or, more generally, over the attractor of whatever IFS you're dealing with. For a self similar set, this measure will also display a type of self-similarity. If the set $E$ is the invariant set of the IFS of similarities $\{T_i\}_{i=1}^m$ and the open set condition is satisfied and if $\{p_i\}_{i=1}^m$ are positive numbers adding to one, then there is a self-similar probability measure $\mu$ satisfying $$\mu(A) = \sum_{i=1}^m p_i \mu(T_i^{-1}(A)),$$ for $A\subset E$. There is a lovely technique for integration with respect to such measures elaborated by Bob Strichartz in his fabulous paper Evaluating integrals using self-similarity. In particular, that measure theoretic self-similarity implies a self-similar identity for the corresponding integral, namely: $$\int f(x) d\mu(x) = \sum_{i=1}^m p_i \int f(T_i(x)) \, d\mu(x).$$ This formula can be iterated to generate something like a Riemann sum to estimate integrals numerically. It can also be used to set up a recursion allowing us to compute the integral of any polynomial. To do so, we just need two more simple formulae.

First, the integral of a constant is exactly that constant: $$\int c d\mu(x) = c$$ We're using here the fact that $\mu$ is a probability measure. If it had a different total mass $m$, we'd simply pick up $m\times c$ instead.

Second, the integral is linear $$\int (af(x) + bg(x)) \, d\mu(x) = a\int f(x) \, d\mu(x) + b \int g(x) \, d\mu(x).$$

As an example, let's focus on the Cantor set and try to compute your moment of inertia. In this case, the self-similar identity takes the form $$\int f(x) \, d\mu(x) = \frac{1}{2}\int f\left(\frac{1}{3}x\right)\,d\mu(x) + \frac{1}{2}\int f\left(\frac{1}{3}x+\frac{2}{3}\right) \, d\mu(x).$$ Thus, $$ \begin{align} \int x \, d\mu(x) &= \frac{1}{2}\int \left(\frac{1}{3}x\right)\,d\mu(x) + \frac{1}{2}\int \left(\frac{1}{3}x+\frac{2}{3}\right) \, d\mu(x) \\ &= \frac{1}{6} \int x \, d\mu(x) + \frac{1}{6} \int x \, d\mu(x) + \frac{1}{3} \int d\mu(x) \\ &= \frac{1}{3} \int x \, d\mu(x) + \frac{1}{3}. \end{align} $$ Solving for $\int x \, d\mu(x)$, we find $$\int x \, d\mu(x) = \frac{1}{2}.$$ We can go through the process again to compute $\int x^2 \, d\mu(x)$. $$ \begin{align} \int x^2 \, d\mu(x) &= \frac{1}{2}\int \left(\frac{1}{3}x\right)^2\,d\mu(x) + \frac{1}{2}\int \left(\frac{1}{3}x+\frac{2}{3}\right)^2 \, d\mu(x) \\ &= \frac{1}{18} \int x^2 \, d\mu(x) + \frac{1}{2} \int \left(\frac{1}{9}x^2 + \frac{4}{9}x + \frac{4}{9}\right) \, d\mu(x) \\ &= \frac{1}{9} \int x^2 \, d\mu(x) + \frac{2}{9} \int x \, d\mu(x) + \frac{2}{9} \int \, d\mu(x). \end{align} $$ Again, we know the lower order integrals so we can solve for the integral of $x^2$ to get $$\int x^2 \, d\mu(x) = \frac{3}{8}.$$ Finally, it's easy to expand your integral to get $$\int (x-1/2)^2 \, d\mu(x) = \int(x^2-x+1/4)\, d\mu(x) = 1/8$$ in agreement with your probabilistic approach.

Again, this can be applied to any weighted IFS of similarities as long as OSC is satisfied. For an on $\mathbb R^2$, of course, the computations are more involved. I do have Mathematica code that automates the procedure, however, and could verify your other computations, if you like.

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    $\begingroup$ For the middle-thirds Cantor set, the Hausdorff measure is the uniform measure (from Hausdorff's original paper). But you are correct: cases where the exact Hausdorff measure is known are very rare. $\endgroup$ Commented Apr 7, 2016 at 21:15
  • $\begingroup$ Thank you so so much, with this I've been able to successfully compute the expected physical value for the Sierpinski's Carpet. You've been really helpful. $\endgroup$
    – Rafa
    Commented Apr 7, 2016 at 21:52
  • $\begingroup$ @GeraldEdgar Yes, I seem to remember a question about that on my general examination - as well as a companion question about the Packing measure. $\endgroup$ Commented Apr 7, 2016 at 22:26
  • $\begingroup$ @Rafa - Glad to hear it! $\endgroup$ Commented Apr 7, 2016 at 22:26
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I used these techniques for some self-similar fractals (Cantor set, Koch snowflake, Menger sponge) to get exact (but mostly nonrigorous) formulas for the decay of the electrostatic potential with distance from the fractal, and the decay (or otherwise) of the Fourier transform. This was during my PhD in theoretical physics - please excuse the mathematical immaturity...

  • Potential theory and analytic properties of a Cantor set C. P. Dettmann and N. E. Frankel, J. Phys. A.: Math. Theor. 26,1009-1022 (1993)

  • Structure factor of deterministic fractals with rotations C. P. Dettmann and N. E. Frankel, Fractals 1, 253-261 (1993)

  • Potential theory and analytic properties of self-similar fractal and multifractal distributions. C. P. Dettmann and N. E. Frankel, J. Stat. Phys. 72, 241-275 (1993)

  • Structure factor of Cantor sets C. P. Dettmann, N. E. Frankel and T. Taucher, Phys. Rev. E 49, 3171-3178 (1994)

There are many earlier and later works on the subject; see for example "Fourier analysis and Hausdorff dimension", P. Mattila, (Cambridge University Press, 2015) and references therein.

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