That's easy: the diagonal point in the unit hyper-cube of dimension d, with coordinates (1 #~ d) has distance (%: d) from the origin, which grows above any R with increasing d.
R.E. Boss > -----Original Message----- > From: Programming [mailto:[email protected]] > On Behalf Of Jose Mario Quintana > Sent: donderdag 17 augustus 2017 00:27 > To: Programming forum <[email protected]> > Subject: Re: [Jprogramming] "n-volume" of an "n-sphere" > > 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, programming- > [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] > et>> > > > 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 > > > > > > > > > > ---------------------------------------------------------------------- > > For information about J forums see > http://www.jsoftware.com/forums.htm > > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
