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Name: yendor
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995


Question:
Draw a 2-d cartesian plane. Then draw four circles of coordinates 1,2 four points = (1, 1) (1, -1) (-1, 1) (-1, -1). Now draw in a fourth, so that it just barely touches the other four; i.e. take the radius as approaching zero with the center at the origin then increase the radius until it barely touches the other four. Draw a square that encompasses the four original circles. The question in 2-d is does this square encompass the 5th square circle, (not square). It is obvious that it does, and obvious if you expand this to 3-D using 8 spheres and another sphere with aa cube. My real question is whether this continues for higher dimensions it is not so obvious that it continues for 4-D, but it does. For 5-D+ I am not so sure. Could someone help me and tell me why it does continue if in fact it does, or why it does not?






Replies:
Hey, I am glad you finally got it a little clearer. I think everybody was kind of confused earlier, though maybe because they just did not understand why such a problem could be interesting. Anyway, I think I have got the idea now, so let us see how it goes.

In N-dimensions, we have 2^N hyperspheres located at all the points (+-1,+-1, . . .,+-1) in N-dimensional space. We also have a hyper- cube enclosing them, centered on zero, with all side lengths equal to 4 (each unit hypersphere has maximum coordinate value 2 or minimum coordinate value -2 in any dimension, so it must be enclosed in that hypercube. Is there a smaller one?) The sphere centered on the origin that just touches those unit spheres centered on (+-1 . . . +-1) must touch them at their nearest points, on the line joining the origin to the sphere centers. The distance to the sphere centers from the origin is sqrt(N), so the distance to the nearest point must be sqrt(N) - 1. Thus the radius of the inner sphere is sqrt(N) - 1. The question is, can sqrt(N) - 1 ever be greater than 2? And the answer, of course, is YES - for N > 9, with the inner sphere touching the enclosing cube in the 9-dimensional case. So, I hope I have solved this properly and not forgotten some peculiarity of higher dimensions. This problem in itself points out how strange things become as dimensions increase. Actually, there is a very interesting book by Conway and others on high-dimensional lattices.

asmith



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