Some things I can do in 4D, a trick I learned as a child when first
confronted with the possibility.  I read Abbot et al and a couple of other
fascinating stories, then tried the graphing and model constructing.  It seemed
pretty clear after a few days of tinkering.
    The real helper was the N-dimensional packing algorithm for spheres.  It is
closely related to information encoding and how to specify the maximum number of
clearly distinct messages that you can put into a given number of bits (which
relates to the dimensions, n).
    Yes, there are two solutions for the cases that are most easy to see in 3D,
but there are infinitely many cases where there are an infinite number of
solutions (or no solutions) for either or both circles.

Cheers!

Chip Shults
My robotics, space and CGI web page - http://home.cfl.rr.com/aichip

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