> > The thing is that i can express Pi as an exact fractional ratio of the perimeter of a polygon, without using trigonometry just continued fractions. > > And that is both rational and a fact, you can wave your hands shouting that is not a circle as much you want.
It's not pi either. Pi is circumference/diameter *of a circle*. What you are doing with polygonal approximations to a circle will only yield approximations to pi.
-- Nam Nguyen in sci.logic in the thread 'Q on incompleteness proof' on 16/07/2013 at 02:16: "there can be such a group where informally it's impossible to know the truth value of the abelian expression Axy[x + y = y + x]".