I'm preparing a presentation that would enable high-school level students to grasp that the (self-similarity) dimension of an object needs not be an integer. The first example we look at is the Sierpiński triangle and with some effort we learn that its dimension is $$s := \log(3)/\log(2) \approx 1.585.$$ After this I thought that it would be nice to mention what the actual Hausdorff $s$-measure of the triangle is, but all I found was measure estimates for a certain class of Sierpiński carpets and some estimates of the Sierpiński triangle.

I am literally shocked to learn that we apparently do not know the exact value of Hausdorff $s$-measure of the Sierpiński triangle! Especially since it's such a concrete and symmetric object. To comply to the idea of this site hosting questions instead of rants, I formulate my bafflement as follows:

Why is the $s$-measure of the Sierpiński triangle and other self-similar fractals so hard to calculate?

Are we missing a link to some complicated machinery or is the problem connected to some deep problem that one would expect to remain unsolved?

  • $\begingroup$ That's an interesting question! But I want to add that I would not know what to make of the fact that some specific set would have $0.7664$-dimenaional measure $1$. I suspect that the same is true for others and this may explain that some people would not put too much effort in such calculations. $\endgroup$ – Dirk Oct 21 '19 at 18:52
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    $\begingroup$ I don't know this subject at all, but is there any reason to expect the answer to be a closed-form number? If the answer is not a closed-form number, what would one mean by "knowing the exact value"? $\endgroup$ – Timothy Chow Oct 21 '19 at 20:46
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    $\begingroup$ @TimothyChow One could ask for a relatively fast algorithm for computing the digits of the number. Based on the reference above, it looks like it's not even known to one significant digit, so "closed-form" isn't really the issue.. $\endgroup$ – verret Oct 21 '19 at 20:55
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    $\begingroup$ @verret : Granted. I was reacting to the sentence, "I am literally shocked to learn that we apparently do not know the exact value." $\endgroup$ – Timothy Chow Oct 21 '19 at 21:53
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    $\begingroup$ Do you know exactly how many disks of radius 1 it would take to cover a disc of radius 1000000? I think this is the flavour of the question you’re asking. (Yes I know with Hausdorff dimension you’re allowed to have sets of different diameters, but morally...) $\endgroup$ – Anthony Quas Oct 22 '19 at 4:44

The latest I could find is

Móra, Péter, Estimate of the Hausdorff measure of the Sierpinski triangle, Fractals 17, No. 2, 137-148 (2009). ZBL1178.28007.

where the best values are given as $$ 0.77 \le \mathscr{H}^s(\Lambda) \le 0.81794 . $$ He also proves an upper estimate $\mathscr{H}^s(\Lambda) \le 0.819161232881177$ of which he says "everybody can check it easily".

He also explains why this is harder and more technical than the reasons found in the comments here; for instance, the obvious upper bound gives only $\mathscr{H}^s(\Lambda) \le 1$.

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    $\begingroup$ That's actually the paper behind the second links in the question. $\endgroup$ – Dirk Oct 24 '19 at 14:07
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    $\begingroup$ He says it is not so easy to check the best published upper bound because it was published in Chinese. $\endgroup$ – Matt F. Oct 24 '19 at 15:00
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    $\begingroup$ Paywall-free version of the papers: lower bound, upper bound. $\endgroup$ – LeechLattice Oct 25 '19 at 17:45

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