[This proof is actually held by vihart, and shared by me, while the extra comes from Spanish Wikipedia, which inspired me to look for more digits of the not periodical sequence, which I found in http://mathworld.wolfram.com/PiContinuedFraction.html.]
Get a square, and draw a circle that touches all sides of the square
Let's say that the square's sides all measure 1.
Now modify the sides like this
so while the perimeter is still 4, now the circle is touched in 8 areas.
Modify it again and again, until you get...
A circle !
But is this REALLY a circle?
A real circle has 0 sides, not ∞ like on this infinitely wrinkled circle.
So it is not a real circle, but an infinitely close imitation.
But what is the perimeter (remember this is an infinitely wrinkled circle with no curves, but ∞ sides)?
We started with 4, and by only modifying the sides, we didn't change the perimeter, only to get a final 4 as the perimeter!
So pi isn't four as anyone would have thought with this proof about making an imitation of a circle at first sight.
Pi as a continued fraction?
To get pi to an accuracy of 11 digits, try "3+(1/(7+(1/(15+(1/(1+(1/(292+(1/(1+(1/(1+(1/(1+(1/(2+(1/3)))))))))))))))))", and continue the pattern with more digits I found here!
This is a great approximation, but can't you simply say 3.14159265358979.../1?
Not so exiting, but if you aren't worried about all numbers in the equation being integers, more understandable.
-The Roaring Thunder