Just to make this a bit more interesting: one can associate to every side of the Newton polygon a line bundle on the modular curve. It is interesting to see in examples whether (after subtracting a multiple of the divisor of modular forms) the resulting point on the Jacobian is torsion.

@VD Does this only cover the case of $\tau= i$?

Indeed, the commutator becomes the loop around the puncture. Unfortunately, I don't know much about Fuchsian groups. It seems that the formulas should involve $\tau$, but how?

Right. I was also thinking that there is no way of getting anything like this from $\mathbb C$ or $\mathbb C\mathbb P^1$. The trouble is that this is far from explicit.

Thank you. While not directly related to the question, it was interesting to learn that, given the even terms, one can adjust the odd terms to ensure the involution property.

Thanks! It is a nice clean way of seeing the relation, and it may lead to a combinatorial description (the objects for $n+2$ being counted in terms of the objects for smaller $n$).

After thinking about it a bit more, there is a fairly easy formula for it. Specifically, if the angles are $\alpha,\beta,\gamma$, then up to a sign the ratio of the areas is the determinant of the matrix $((1,1,1),(\sin \alpha,\sin\beta,\sin \gamma)(\tan \alpha,\tan\beta,\tan\gamma))$ divided by $3(\sin \alpha+\sin\beta+\sin\gamma)(\tan\alpha+\tan\beta+\tan \gamma)$. In particular, in the case $\beta=2\alpha\to 0$ one sees that the ratio in fact goes to infinity.

Are there local maxima in $d=3$ that are different from the above configuration?

You may want to calculate what happens for a triangle with angles $\alpha,2\alpha,\pi-3\alpha$ as $\alpha\to 0$.

I think the same argument will work for any $b_1,b_2$. You will get the same result modulo $b_1-b_2$, but can never get a number less than $b_2$ unless you start with it.