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?
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