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
