I'd read a while back about Tau Day and the idea that τ=6.28... is a better mathematical constant than π=3.14..., for a variety of reasons. (Go read the Tau Manifesto and learn several of them if you haven't already.)
One of them was the idea that far from being a strength of π, the area formula A=πr2 is actually a weakness, because it camouflages the fact that there should naturally be a ½ in there, deriving from its integral relationship with the circumference formula. By contrast, C=τr and A=½τr2 display on their face the same relationship as, say, that between velocity and distance (under constant acceleration) or spring force and potential energy.
So anyway. I was thinking about the volumes of spheres, and I recalled that the formula was V=⁴/₃πr2; of course I knew that because I'd memorised it many years ago, not that it had any reason behind it:
But wait! What if we take that awkward extra power of 2 in the even-dimension formulas and distribute it over the factorial?
Check it out! Even if we don't have a deep understanding of what a double factorial is or how to compute the Γ function, we can clearly see the recurrence relation among the various dimensions, and the relationship between the even-numbered dimensions and the odd-numbered dimensions, and that they're much more closely related than might first appear from reading the Wikipedia article on n-spheres that I linked above.
So, chalk up one more success for the τists!
"When I go to get a new driver's license... or deal with the city inspector... or walk into a post office... I find public employees to be cheerful and competent and highly professional, and when I go for blood draws at Quest Diagnostics, a national for-profit chain of medical labs, I find myself in tiny, dingy offices run by low-wage immigrant health workers who speak incomprehensible English and are rud to customers and take forever to do a routine procedure." --Garrison KeillorPosted by blahedo at 10:26pm on 22 Dec 2010