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Showing posts from May, 2012

### Why PI is not 4, math is great, and other mysteries.

The other day, I found myself with an interesting problem of approximating a circle with the enclosing square which seems to prove pi = 4.

The paradox was forwarded by a most interesting puzzle collector, Surajit Basu, a friend and life long inspiration. See Sonata for Unaccompanied Tortoise for why!

Here is the offending paradox:

This is an example of how counterintuitive questions can be answered with a little calculus.

The key is to realize that no matter how closely we approximate the circle, the orthogonal lines of the approximation formed by inverting the square corners will never actually be tangential to the circle.

Note carefully that as you get closer to 90 degrees, the horizontal line is much longer than the vertical. Same goes with the approximation at 0 and 180 - the vertical line is much larger than the horizontal component.

If we take a quadrant of the circle - let's say the top left quadrant, moving counter clockwise from top to left -  we can imagine that each inf…