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[Phys-l] a mathematical curiosity



I stumbled across the following mathematical curiosity:

One can calculate the volume of a unit sphere in D-dimensions (a "hypersphere"). For D=2 one gets pi (area of a unit circle), for D=3 one gets 4*pi/3, and so on.

It looks like the volume is slowly increasing with D. But that trend does NOT continue. The general formula for the volume is pi^(D/2)/gamma(D/2+1). One reaches a maximum volume at D=5. Thereafter the volume *decreases to zero* as D continues to increase. I find that surprising!

I am wondering what physical applications this result might affect. For example, volume of a hypersphere enters into one way of calculating the partition function of an ideal gas.
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Carl E Mungan, Assoc Prof of Physics 410-293-6680 (O) -3729 (F)
Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-1363
mailto:mungan@usna.edu http://usna.edu/Users/physics/mungan/