Pi Day!

Today is Pi Day.  So along with the Alamo and the Maine, remember the pi.

π r²

But as a friend once told me, "Pie are not square; pie are round.  Cornbread are square." 

If we superscribe a circle with squares of side r, it is clear that the area of the circle is less than four of these squares (left).  Four such squares is 4r².  If we inscribe the circle with squares of diagonal r, the area of the circle is clearly greater than four of these squares.  Thanks to Mr. Pythagoras, we know that if the diagonal is r, the side is SQRT(2X), so the area is r²/2.  Four of these is 2r².  Therefore, the area of a circle must lie between 2r² and 4r², and 3r² seems a reasonable guess.  If not for those curvey lines....  So it turns out to be "a little bit more than 3."  To wit:

3.1415926535897932384626433832795028841971693993751058209749445923078
164062862089986280348253421170679821480865132823066470938446095505822
317253594081284811174502841027019385211055596446229489549303819644288
109756659334461284756482337867831652712019091456485669234603486104543
266482133936072602491412737245870066063155881748815209209628292540917
15364367892590360011330530548820466521384146951941511609...

How's that for a pi in the face?  All hail, the mighty PI. 

However, there is seldom need for anything more than 3.14159; or even 3.1416, if we round.  Every really real circular object can be measured to a specific number of decimal places, and so the ratio C/d will always be a rational number, no matter how many decimals are in our instrument.  So what is this "irrational" pi anyway but a pure spirit, not found anywhere in the real world?  Hunh?  Credulous believers in PI even call it "irrational," but "rationalists" know there is no such thing in reality.  A fig for your pi.

There is apparently a movement afoot to replace π with 2π in radial formulae, calling it τ (tau).  There is a certain elegance in notation in trig if you do this.  Unfortunately, the area of the circle becomes τd²/8 (or τ r²/2 if you prefer), which is not so elegant.  The circumference of a circle is 2π r, which would be τ r.  So τ is the ratio of the circumference to the radius (C/r) while π is the ratio of the circumference to the diameter (C/r) or the ratio of the area to the squared radius (A/r²).  Notationally, tau may simplify linear formulae while pi makes more elegance dealing with square formulae.  Perhaps we should define a number for V/r³.