Firstly, you do realise that I'm not the author, I just submitted it, don't you?
Secondly, there are some curious things about pi. It does turn up in places that apparently have nothing to do with circles. It's a bit like e in that regard.
Next, we don't have to assume it's "magical," but some of the properties are noteworthy. The fact that 355/113 is such a good approximation, and yet it doesn't appear to turn up naturally in any of the proven convergences is a bit odd. Why does it not turn up? The only place it does turn up is when you write down the ad hoc continued fraction to express the value you already know. That seems unnatural, and immediately leads to a desire for further investigation.
And finally, mathematicians have a feel for things that are "natural," and that's what they end up exploring. Often it leads nowhere interesting, but sometimes it leads to unexpected connections, and occasionally to equally unexpected applications. But in all, some questions just feel right for exploration, and some properties of pi fall into that category. You never really know exactly what will advance math - we only have intuition to guide us in deciding what is an "interesting question."
Secondly, there are some curious things about pi. It does turn up in places that apparently have nothing to do with circles. It's a bit like e in that regard.
Next, we don't have to assume it's "magical," but some of the properties are noteworthy. The fact that 355/113 is such a good approximation, and yet it doesn't appear to turn up naturally in any of the proven convergences is a bit odd. Why does it not turn up? The only place it does turn up is when you write down the ad hoc continued fraction to express the value you already know. That seems unnatural, and immediately leads to a desire for further investigation.
And finally, mathematicians have a feel for things that are "natural," and that's what they end up exploring. Often it leads nowhere interesting, but sometimes it leads to unexpected connections, and occasionally to equally unexpected applications. But in all, some questions just feel right for exploration, and some properties of pi fall into that category. You never really know exactly what will advance math - we only have intuition to guide us in deciding what is an "interesting question."