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:[email protected]] > On Behalf Of Raul Miller > Sent: zaterdag 19 augustus 2017 06:46 > To: Programming forum <[email protected]> > 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 <[email protected]> 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
