Is Pi more than just a ratio for circles? Is there a easy way to understand the universality of Pi?
In: 1018
This is extremely tricky to answer as ELI5. The answers you get with “circles are kinda everywhere,” are good, but I think they leave out a significant amount of the mystery.
The ones I saw trying to explain using Euler were not wrong, but quickly left ELI5 territory as well.
I think it’s worth acknowledging that the fact that Pi shows up quite as often as it does *is* surprising. Those of us in mathematics or related fields have gotten so used to the idea that we just take it as a given. That should not take anything away from the strange ability for Pi to show up in the absolutely weirdest places.
For instance, if we were to take the sum of the following numbers: 1/1 + 1/4 + 1/9 + 1/16…, we get the answer: π^(2)/6. I believe that any sane person looking at this should be wondering: where the hell did pi come from? If you are interested in this, you can look up the Basel Problem and find any number of wonderful rabbit holes to go down.
But that is beside the point for the moment. It is just plain *weird* that pi shows up there. Sure, once you start tearing the problem apart, you can see where it eventually creeps in, but I still think it is magical.
So what is the answer? I’m not sure there is one; at least, not one better than the answers saying that circles are everywhere.
It is legitimately strange that i ends up creating a relationship between π and e. It’s there. We can work it out. But it is not like we created i to do this. It just sort of happened. And I sometimes like to sit back and simply marvel at the fact that it did this.
My apologies for not really offering an answer as such. But I really want to emphasize just how wonderful and mysterious that π does end up everywhere, even in spots that would not seem to have anything to do with circles at all. It sometimes makes me wonder if we have mistakenly linked π to circles, simply because this was the first place we ran across it.
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