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Circles, squares and 4-d graphs
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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