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

[Don Kelly]

If one considers a point as infinitesimal -as usually considered-, then 
we have an infinite number of points at an infinitesimal distance from 
the origin and at a larger distance from the origin there are still an 
infinite number of points on the surface and  an infinite number of 
points enclosed . Isn't this getting into transfinite math?


What's the point?

Don Kelly


On 2017-08-15 8:23 PM, Jimmy Gauvin wrote:

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  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.go...@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-boun...@forums.jsoftware.com] on

behalf of

Raul Miller [rauldmil...@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
--
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

--
regards,

GPG key 1024D/4434BAB3 2008-08-24
gpg --keyserver subkeys.pgp.net --recv-keys 4434BAB3
gpg --keyserver subkeys.pgp.net --armor --export 4434BAB3
--
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

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

[R.E. Boss]

The central hypersphere, with radius <:%:d, where d is the dimension, touches 
the bounding hyperplanes of the hypercube with side 4 in dimension d=9, since 
they are at distance 2 from the origin.
In higher dimension that central hypersphere will stick outside the hypercube 
with side 4.
However, the centers of the contained hyperspheres are at distance 3 (%:9), so 
are outside the central hypersphere. And they remain outside that central 
sphere in all dimensions.
As you see, there is much room in high dimensions.


R.E. Boss


> -Original Message-
> From: Programming [mailto:programming-boun...@forums.jsoftware.com]
> On Behalf Of Raul Miller
> Sent: zaterdag 19 augustus 2017 06:46
> To: Programming forum 
> Subject: Re: [Jprogramming] "n-volume" of an "n-sphere"
> 
> You've got several kinds of hyper-circle/hyper-sphere's here:
> 
> Contained hyper-spheres, with radius 1
> 
> Enclosing hyper-spheres, with radius %:n
> 
> Difference hyper-spheres, with radius <:%:n
> 
> The reason the difference hyper-spheres have volume larger than the unit
> hyper-cube is that those hyper-spheres bulge out through the sides. But
> they still do not quite contain the cube at its corners.
> 
> Thanks,
> 
> --
> Raul
> 
> 
> 
> 
> On Sat, Aug 19, 2017 at 12:14 AM, Don Guinn  wrote:
> > I viewed the entire video of Hamming's and have noticed that there has
> been
> > no comment about the fascinating dilemma he presented at the end of the
> > video.
> >
> > He took a square with sides of 4 units, then placed 4 unit circles with
> > origins at
> >   4 2$1 1 1 _1 _1 _1 _1 1
> >  1  1
> >  1 _1
> > _1 _1
> > _1  1
> >
> > Then he asked what would be the radius of a circle at the origin just
> > touching the 4 circles. So, pick the circle with origin 1 1. It's origin
> > would be at the distance %:2 from the origin as the distance from the
> > origin is the square root of the sum of the squares of its coordinates.
> > Since the circle has a radius of 1 then the radius of the enclosed circle
> > must be 1 less than the distance of the unit circle from the origin.
> ><:%:+/*:1 1
> > 0.414214
> >
> > Now extend this to 3 dimensions. We have a cube with lengths 4 on each
> > side. Then put 8 unit spheres in the cube as before. Now the size of the
> > enclosed sphere is
> > <:%:+/*:1 1 1
> > 0.732051
> >
> > That sphere has a lot larger radius.
> >
> > So this can be generalized to hyper-cubes and hyper-spheres. Since we are
> > dealing with unit hyper-spheres with radius 1 the sum of the squares is
> > simply the number of dimensions of the hyper-cube and hyper-spheres. So
> for
> > the radius of the enclosed hyper-sphere for dimensions 1 2 and 3 are
> ><:%:1 2 3
> > 0 0.414214 0.732051
> >
> > Okay, it makes sense that the enclosed hyper-sphere for 1 dimension
> would
> > be zero as the 1 dimension hyper spheres would simply be 2 lines touching.
> >
> > The question is, what happens as we get to higher dimensions?
> ><:%:10
> > 2.16228
> >
> > Wow! the enclosed hyper-sphere is bigger than the enclosing hyper-
> spheres.
> >
> > How about 100 dimensions?
> ><:%:100
> > 9
> >
> > Wait! the radius of the enclosed hyper-sphere is larger than the size of
> > the hyper-cube. Is the enclosed hyper-sphere enclosed in the hyper-cube
> or
> > not?
> > --
> > 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