A beautifully told story with colorful characters out of epic tradition, a tight and complex plot, and solid pacing. -- Booklist, starred review of On the Razor's Edge

Great writing, vivid scenarios, and thoughtful commentary ... the stories will linger after the last page is turned. -- Publisher's Weekly, on Captive Dreams

Monday, March 14, 2011

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:
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. 

No comments:

Post a Comment