You can't assume that a graph 'inside' another will give a smaller length, in this case that a jagged path inside a square will have a lesser perimeter than the circle. Average them and you get an approximate value of 3.4142 for pi.Both methods will yield 4, not pi. The perimeter of the inside square is 2.8284. The perimeter of the outside square is 4. The proper method is to average the perimeter of the outside square and the inside square. The pi=4 result is obtained by choosing a bad method to solve the problem. To get a more accurate approximation, cut the corners of the square to make an octagon and make a similar octagon on the inside and average the perimeters (approximate value of 3.188). Average them and you get an approximate value of 3.4142 for pi. It was turned by hand, and could also be used to transfer water from a low. Archimedes' machine was a device with a revolving screw-shaped blade inside a cylinder. Then again, it is like a modified Zeno's Dichotomy Paradox.The pi=4 result is obtained by choosing a bad method to solve the problem. The numbers meet somewhere at infinity? I think it is the same reasoning used for π = 4: the square and the circle merge somewhere at infinity.
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