From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: min of some measure of N points in S^3 and S^4
Date: 3 Mar 1998 20:58:11 GMT

In article <6degm6$1ad4@r02n01.cac.psu.edu>, ale2 <ale2@NOSPAMpsu.edu> wrote:
>The recent post about minimizing some measure of N points on a sphere
>jolts me into wanting to know if anything interesting happens when the
>space in question is S^3 or S^4. Any basic papers or articles on this?

I don't know if this has been pursued.

Oh, sure, you can try to arrange  N  point on other sets too, but by
and large there isn't a fixed way to do this because of the same problem:
there's no single best way to describe what it means to be spaced
uniformly. As long as you have a metric space you can try something like 
maximizing the minimum distance between two of the  N  points, or the
sum of the squares of the reciprocals of the interpoint distances.
But is there any interesting space other than the round circle on which
the various notions could be expected to coincide for all  N?  I don't
know of any.

Once the minimizing criterion is selected, one could compute (locally) optimal
configurations for any  N  on any Riemannian manifold by following flows.
I don't know if there's anything exciting to expect.

>If point --> line;  S^2 --> S^3 and we try to find some minimum measure
>of N "straight" lines in S^3 do we get an interesting problem?

Um, there aren't any lines in S^3; do you mean circles? You could look for
arrangements of great circles in S^n by observing that each is the intersection
with  S^n  of a 2-dimensional plane in R^(n+1); thus the family of things
you're shifting around is the set of points in the Grassmannian manifold
of 2-planes in  R^(n+1)  (a fairly sphere-like topological space!). 

As above, one may ask about finding an optimal arrangement of points
in the Grassmannian (=great circles on the sphere) only after a choice
of optimality criteria has been made; but of course that has to follow
the choice of a metric on this manifold. I don't know if there's an
obvious choice of metric (in either the metric-space sense or in the
sense of Riemannian geometry). I've always worked in the topological category.

There's nothing to stop you from considering the other Grassmannians, of
course; if you can resolve the questions of metric and objective function,
you can ask about "well-spaced" collections of   N  great  k-spheres on S^n
(= collections of  N  (k+1)-dimensional subspaces of  R^(n+1)  ).

dave
==============================================================================

From: reidl@mast.queensu.ca (Les Reid)
Newsgroups: sci.math
Subject: Re: min of some measure of N points in S^3 and S^4
Date: 6 Mar 1998 13:26:53 GMT

Regarding arrangements of points on Grassmannians, see "Packing Lines, 
Planes, etc.: Packings in Grassmannian Spaces" by J.H. Conway, R.H. 
Hardin, and N.J.A. Sloane in Experimental Mathematics, v.5 (1996), #2, pp 
139-159. The bibliography also contains references to the problem of packing 
points in S^n.