Mind F#ck?!
 

the perimeter of the circle is 3.14 which is pie, so the sum of the corners is 0.86 something, anyway, I don't get the whole picture and its point :p

the perimeter is 2*pi*r or d*pi d=1 so perimeter of circle=pi
the square parameter is 4, with the corners as such, it stays 4...you do corners till infinity it becomes like the perimeter of the circle so pi=4

I can explain it, but later.

I understand the third picture, he flipped the corners inwardly and thus the perimeter remained the same... but I am not sure how he go to the fourth part "removing more corners" if he removes 2 adjacent corners he might lose a part or a segment, that will cause the perimeter to decrease

It will never be a circle until you flatten out the corners, and when you do it'll be 3.14 :D

But if you do the corner thing till infinity times ( its similar to the method of integration) you will get a perimeter that is approximately equal to the perimeter of the circle.

chillax brothers, if pi is really 4 we wouldn't have cellphones, cars, went to the moon...
the only goal of this pic is to trollface Archimedes, and it did quite a good job.

True, you might think it will fully touch the circle, but it will not. It will touch the circle only at degrees of 0,90,180,270.