The following question might contain more philosophical than actual calculus.
From $S^0$ the $0$-sphere, which was two dots on a line, to a $1$-sphere, which was a connected circle in a plane, $\pi$ emerged, and it was used thereafter for $n$-sphere recursion and all the nice function such as Fourier exponential, etc.
However, it suddenly bothered me a bit to see that the value of $\pi$ was unique, that upon rising from $S^0$ to $S^1$ it emerged, and that a single structure constant $\pi$ was used for whatever there was after. Couldn't there be a new $\pi_i$?
In some sense that it was true, since $\pi^2$ was evidently a completely different number than $\pi$ as the digit never repeat itself. But, still, why there wasn't additional new structure constant like $pi$ emerge from the transition of $S^1$ to $S^2$?
What's more interesting was the following observation. From $n=1$ to $n=2$, $\pi$ emerged, the continuous was where the physics changed, as nowadays it required complex domain. The single $\pi$ thus carried on until $n=3$. However, from $n=3$ to $n=4$, $\pi^2$ emerged, it's also where physics changed completely, as four vectors and spacetime conservation came in. Notice that the discussion to continue the current physics to $n=5$ was somewhat seriously discussed. Correspondingly, the new appearance of $\pi^3$ happened at $n=6$, the dimension where several notice of the string theory came into play(except the non-gravity ones). At first occurrence of $\pi^4$, i.e. $n=8$, lots of existing geometry understanding broke if recall it correctly.
It seemed as if, at each appearance of a new order of $\pi$, serious geometry and physical implication came into play, where, in fact, changed the understanding completely. I'd like to know if there's any thoughts regard to this relationship, or if it was just a coincidence.
Back to the original question, wasn't it so troubling that the $\pi$ was unique? Could there be new complementary constants appear in transition between $S^1$ to $S^2$, just like the appearance of prime number in integers, i.e. $\pi_1=\pi$, and what about $\pi_2$?