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Re: [Phys-l] a mathematical curiosity

J Denker wrote:

Similarly all the surface-area is near the equator. You can see this already
in D=3, on the earth, where the area of the tropical regions is
large compared to the area of the arctic regions ... “

I am befuddled. One could just as arbitrarily choose an equator running
through a rotational pole. Without the earth’s spin and/or its orientation
relative to the sun the choice of an equator is completely arbitrary.

From: John Denker
Sent: Monday, January 09, 2012 10:32 AM
To: Forum for Physics Educators
Subject: Re: [Phys-l] a mathematical curiosity
On 01/09/2012 07:43 AM, Carl Mungan wrote:

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!

The two appearances of D/2 rather than D always amaze me. The
power of pi in the numerator interacts with the gamma function in
the denominator, so that every time D increases by *two* the volume
picks up *one* more factor of pi. The even values of D show D/2
factors of pi ... but the odd values of D do not show any factor
of sqrt(pi). That's sneaky.

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.

As a related point, you can calculate the surface-to-volume ratio
for a hypersphere. It becomes large as D becomes large. In words,
we say that all the volume is near the surface. Similarly all
the surface-area is near the equator. You can see this already
in D=3, on the earth, where the area of the tropical regions is
large compared to the area of the arctic regions ... and in higher
dimensions the effect becomes even more pronounced.

This phenomenon is sufficiently important that it has been given
a name: sphere hardening:

As a consequence, this means that in high dimensions, two randomly
chosen vectors are very likely to be very nearly orthogonal.

This has important practical applications in cryptography and
communications. For example, consider the construction of an
error-correcting code. Sphere hardening means that randomly-chosen
codewords work almost as well as artfully-designed codewords.
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Bob Sciamanda
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