Are there any geometries/spaces where pi is a simple (or at least rational) constant? 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 complicated (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 complicatedness 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:

*

*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?
 A: First of all, for modern mathematicians $\pi$ is a particular number that can be defined in many ways, many of which have little to do with geometry.  In a different geometry the ratio of circumference to diameter of a circle might be
different, but it wouldn't be $\pi$.  
Another bit of terminology: don't say "complex" to a mathematician when you mean "complicated".  Complex numbers are something completely different.
For example, if you use the "$1$-norm" so the distance from $(x_1,y_1)$ to $(x_2,y_2)$ is $|x_1 - x_2| + |y_1 - y_2|$, then a unit "circle" has circumference $8$ and diameter $2$, so the ratio of circumference to diameter is $4$.  That's not $\pi$.
You could represent numbers in lots of ways.  For example, why not represent a number $x$ using the decimal representation of $x/\pi$?  Thus $\pi$ is represented by $1.0$ in this system.  It does not have many other virtues, though.
A: For your point 1(a), you could always write numbers using a non-rational base.  Trivially $\pi$ can be written as $10_{\pi}$ in base $\pi$, 
but you have now lost the ability to write four simply in the same base: it becomes about $10.220122\ldots_{\pi}$ and even has other representations such as $3.301102\ldots_{\pi}$.  Addition is not easy using this base.
