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*From*: "Bob Sciamanda" <treborsci@verizon.net>*Date*: Mon, 09 Jan 2012 12:25:22 -0500

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:

http://www.google.com/search?q=%22sphere+hardening%22+dimension

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.

_______________________________________________

Forum for Physics Educators

Phys-l@carnot.physics.buffalo.edu

https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l

Bob Sciamanda

Physics, Edinboro Univ of PA (Em)

treborsci@verizon.net

http://mysite.verizon.net/res12merh/

**Follow-Ups**:**Re: [Phys-l] a mathematical curiosity***From:*brian whatcott <betwys1@sbcglobal.net>

**Re: [Phys-l] a mathematical curiosity***From:*John Denker <jsd@av8n.com>

**References**:**[Phys-l] a mathematical curiosity***From:*Carl Mungan <mungan@usna.edu>

**Re: [Phys-l] a mathematical curiosity***From:*John Denker <jsd@av8n.com>

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