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?

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