Why do fractals and Pi have a special relationship?

[Office_Shredder]

I agree, but just because not all of his logic is flawed doesn't mean \pi = 2

My point wasn't that he's correct, my point was that your criticism was incorrect; i.e. the act of assuming the existence of pi, and then having an argument that concludes pi=2, does not in and of itself invalidate the argument; rather you have to find a flaw in the argument itself to show the conclusion is incorrect.

 

[Archosaur]

My point wasn't that he's correct, my point was that your criticism was incorrect; i.e. the act of assuming the existence of pi, and then having an argument that concludes pi=2, does not in and of itself invalidate the argument; rather you have to find a flaw in the argument itself to show the conclusion is incorrect.

I think it does invalidate the argument.

If someone were to say, "The sky is blue, so... (some questionable leaps of logic) ...therefore the sky is green.", their argument would be invalid. You couldn't have proven that the sky is purple by first stating that it is green.

He didn't just "assume the existence" of \pi,
he defined \pi as \pi/2.
He concluded that \pi = 2.

That's not a valid argument.

 

[pallidin]

Pi is a consistent mathematical ratio that is simply non-controversial in that respect.

 

[Take_it_Easy]

well, there is uniform convergence of the succession of functions of circles to the function

constant = 0 in the interval (say) [0,1] now the elements of the succession have a graph that has constant length \pi, but the limit has a graph of length 1.

Well that's not a paradox, also more complicate things can happen when you consider uniform convergence.

think of the functions defined in [0,1]

f_n(x) = \sin(1/x)/n when x \in (0,1] and

f_n(0) = 0 they all have infinite length yet they converge uniformly to the costant 0 in [0,1]
that has finite (unit) lenght.

That is no paradox: just something not very intuitive, that happen!