I found this article on pi: http://blog.plover.com/math/pi.html and while I found it very interesting, it seemed unfinished. The basic point of the article is that pi is complex (for example e has a simple continued fraction representation: [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, ...], but pi does not), and the author claims that this complexity is due to the nonlinear nature of the euclidean distance metric. However, the author doesn't really have a conclusion, and I felt that I still had some questions that weren't satisfied:

1) How does such a complex constant (pi) arise from such a simple definition (a circle)?

a) Is it because the base-10 decimal representation is flawed, and there is another representation of numbers where pi is simple? If so, what is this representation?

b) Or is it because of some property of euclidean space, like the nonlinear nature of the distance metric. If so, where exactly does this property come into play in the definition of pi, and how does it create such complexity? It seems like a the simple square root of sum of squares metric shouldn't create such a bizarre constant (or if it did, that the constant would have something to do with the number 2).

Furthermore, if the answer is b, then are there any geometries or spaces that don't have this property, such that pi would be a simple constant?

I hope my questions aren't too vague! Thanks!

edit: By complex (I probably should have said complicated) I mean that, as pointed out in the article, whereas other irrational numbers like sqrt(2) or e have nice representations (in those two cases, they have nice continued fraction forms), pi does not have a nice continued fraction form. That's why I was wondering if there are any real number representations where pi does have a nice form, akin to e's representation in continued fraction form.

My main line of inquiry (which is the same line of inquiry of the linked incomplete article), is: how does such a simple definition of a circle: all points that are distance r away from a center, create such an incredibly complicated number?