December 22, 2010

Tau vs. pi: hyperspheres

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 in there, deriving from its integral relationship with the circumference formula. By contrast, C=τr and A=τr2 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:

A=πr2 V=⁴/₃πr3
Mystery constant. But then I remembered the Tau Manifesto and thought, what would that make the volume formula?
A=τr2 V=⅔τr3
Hey! That's a pattern! And all of a sudden I'm curious what the hypervolume of a hypersphere is, and paging in my integral calculus and reading up on hypergeometry. It turns out that the pattern is a bit more complex than it seemed (of course), and at a first pass, the τ conversion doesn't help much (it cancels some 2s in the odd dimensions but seems to add complexity in the evens):

V2 (area) =πr2 =(1/2)τr2
V3 (volume) =(4/3)πr3 =(2/3)τr3
V4 =(1/2!)π2r4 =(1/2!∙4)τ2r4
V5 =(8/5∙3)π2r5 =(2/5∙3)τ2r5
V6 =(1/3!)π3r6 =(1/3!∙8)τ3r6
V7 =(16/7∙5∙3)π3r7 =(2/7∙5∙3)τ3r7
V8 =(1/4!)π4r8 =(1/4!∙16)τ4r8

But wait! What if we take that awkward extra power of 2 in the even-dimension formulas and distribute it over the factorial?

V2 (area) =(1/2)τr2
V3 (volume) =(2/3∙1)τr3
V4 =(1/4∙2)τ2r4
V5 =(2/5∙3∙1)τ2r5
V6 =(1/6∙4∙2)τ3r6
V7 =(2/7∙5∙3∙1)τ3r7
V8 =(1/8∙6∙4∙2)τ4r8

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 Keillor

Posted by blahedo at 10:26pm on 22 Dec 2010
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