TITLE: How can we not know the $s$-measure of the Sierpiński triangle?
QUESTION [24 upvotes]: 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?

REPLY [5 votes]: 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 $\mathscr{H}^s(\Lambda) \le 1$.