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

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

Note carefully that as you get closer to 90 degrees,

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 (

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!

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

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.

**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:

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.