The statement, "Suppose that R is fixed. Then the volume of an n-ball of
radius R approaches zero as n tends to infinity." is in the link that Raul
mentioned in his second post in this thread together with a couple of
proofs.
A related statement (assuming I am not mistaken) is: Given a fixed R
(regardless of how large) there is a dimension n such that the unit n-cube
centered at the origin is not contained by the n-ball of radius R centered
at the origin.
How come? The following J sentence gives a clue,
( (+/&.:*:)@:(0.5$~ ])("0) i.222 ) i. (0 1 2 3 4 5 6 7)
0 4 16 36 64 100 144 196
OEIS A016742
On Wed, Aug 16, 2017 at 10:12 AM, Murray Eisenberg <[email protected]>
wrote:
> Why get decimal approximations when you can get the exact values?
>
> With Mathematica, for example, one finds:
>
> Table[RegionMeasure[Ball[n]], {n, 1, 10}]
> {2, Pi, (4 Pi)/3, Pi^2/2, (8 Pi^2)/15, Pi^3/6, (16 Pi^3)/105, Pi^4/24, (32
> Pi^4)/945, Pi^5/120}
>
> (Actually, the output from Mathematica gives an actual Greek letter
> version of “Pi” and displays the fractions as actual fractions.)
>
> Of course, an exact formula for the n-dimensional measure of the unit
> n-ball is known:
>
> Pi^(n/2)
> V(n) = ————----------
> Gamma(1+n/2)
>
> (If the ball has radius r rather than 1, the volume is multiplied by a
> factor of r^n, as one might expect.)
> In even dimension n = 2 k, this reduces to:
>
> Pi^k
> V(2 k) = ——-,
> k!
>
> and this makes readily apparent that the n-volumes readily decrease for k
> >= 3 as k continues to grow.
>
> (The formula is more complicated for odd n.)
>
>
> > On 16 Aug2017, at 8:00 AM, [email protected]
> wrote:
> >
> > Tue, 15 Aug 2017 19:33:09 +0000
> > From: Ben Gorte - CITG <[email protected] <mailto:
> [email protected]>>
> > To: "[email protected] <mailto:[email protected]>" <
> [email protected] <mailto:[email protected]>>
> > Subject: Re: [Jprogramming] "n-volume" of an "n-sphere"
> > Message-ID:
> > <[email protected]
> <mailto:[email protected]>>
> > Content-Type: text/plain; charset="us-ascii"
> >
> > A little surprise (to me) was
> > plot 1 sphvol i.30
> > (for example)
> >
> > Can you predict it?
> >
> > greetings,
> > Ben
> > ________________________________________
> > From: Programming [[email protected] <mailto:
> [email protected]>] on behalf of Raul Miller [
> [email protected] <mailto:[email protected]>]
> > Sent: Tuesday, August 15, 2017 19:55
> > To: Programming forum
> > Subject: [Jprogramming] "n-volume" of an "n-sphere"
> >
> > sphvol=: (1p1&^%!)@-:@] * ^
> > 1 sphvol 3
> > 4.18879
> > 1 sphvol i.7
> > 1 2 3.14159 4.18879 4.9348 5.26379 5.16771
> >
> > Left argument is the radius of the "n-sphere".
> >
> > Right argument is the number of dimensions.
> >
> > I put "n-volume" in quotes, because if the dimension is 2 (for
> > example), the "n-volume" is what we call the area of the circle. (And
> > if the dimension is 1 that "n-volume" is the length of a line
> > segment).
> >
> > Anyways, I stumbled across this and thought it might be interesting
> > for someone else.
>
> ——
> Murray Eisenberg [email protected]
> Mathematics & Statistics Dept.
> Lederle Graduate Research Tower phone 240 246-7240 (H)
> University of Massachusetts
> 710 North Pleasant Street
> Amherst, MA 01003-9305
>
>
>
>
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