[Jprogramming] "n-volume" of an "n-sphere"

[Raul Miller]

Yes.

(That said, I've not tried coding up that simulation. If we simulate
points in a unit n-cube, discarding those outside the unit sphere,
this quickly becomes inefficient for higher dimensions. If we use some
other technique, though, we run into the problem of showing that the
distribution is valid. So I'm not confident that this would be an
illuminating approach.)

Thanks,

-- 
Raul


On Wed, Aug 16, 2017 at 3:47 AM, bill lam <bbill.lam at gmail.com> wrote:
> Is the distance from the origin
>  %:+/*: y
>
> can we use discrete points simulation to verify
> the number of points satisfying the inequality
>  R>:%:+/*: y
> is actually diminishing for large n?
>
> Вт, 15 авг 2017, Jimmy Gauvin написал(а):
>> The construction of the sphere implies it cannot be convex but you will
>> have to find a topologist to prove it to you.
>>
>> The sphere is the collection of points whose distance to the origin is
>> equal to the radius of the sphere.
>>
>> The ball or volume is comprised of the points whose distance to the origin
>> is equal or smaller than the radius of the sphere.
>>
>>
>> On Tue, Aug 15, 2017 at 10:41 PM, bill lam <bbill.lam at gmail.com> wrote:
>>
>> > Has the n-sphere become concave in higher dimension?
>> >
>> > Вт, 15 авг 2017, Jimmy Gauvin написал(а):
>> > > Funny how the n-Sphere volume dwindles for the higher dimensions.
>> > > Not quite intuitive but the factorial always "win" even with bigger
>> > radii.
>> > >
>> > > The hypercubes do not share this characteristic (V= edge ^ n)
>> > >
>> > >
>> > >
>> > > On Tue, Aug 15, 2017 at 3:33 PM, Ben Gorte - CITG <
>> > B.G.H.Gorte at tudelft.nl>
>> > > wrote:
>> > >
>> > > > A little surprise (to me) was
>> > > >    plot 1 sphvol i.30
>> > > > (for example)
>> > > >
>> > > > Can you predict it?
>> > > >
>> > > > greetings,
>> > > > Ben
>> > > > ________________________________________
>> > > > From: Programming [programming-bounces at forums.jsoftware.com] on
>> > behalf of
>> > > > Raul Miller [rauldmiller at gmail.com]
>> > > > 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.
>> > > >
>> > > > Thanks,
>> > > >
>> > > > --
>> > > > Raul
>> > > > ----------------------------------------------------------------------
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>
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