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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 infinitesimal arc  (at an angle theta) is approximated by a horizontal line that is the approximate length of arc times the cosine of the angle, and the vertical line is the same arc times the sine of the angle.

Here's the rough visual:





Thus, each arc is being approximated by two lines, and we merely add all the approximations. This is where calculus and limits come in. For the one quadrant from 0 to 90 degrees, here is the result:

Multiply by four and you get  8*r (or 4*Diameter).

Voila!  

PI is not 4, because the approximate figure is never really the same as the circle, even in the limit of infinite number of approximations.

An interesting result arises from this: Most circles in digital representation (B&W) should have a brightness (or color density) of 4/pi - or about 27% brighter than a real circle of the same dimension in the real world.

Also, can anti-aliasing can be done in a more clever way to not only do edge smoothening, but also reducing the brightness so that the circle's relative brightness is same as physical reality - when the resolution of the picture is less than human eye's resolution?

Is this one of the reasons why Apple's move to Retina display - where the pixel resolution is better than  retinal resolution - makes the iPhone (and now iPad3) different?

More questions than answers.






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