Date: Mon, 9 Sep 1996 12:47:15 +0200
To: rusin@washington.math.niu.edu
From: eyal@hitech.technion.ac.il (Eyal Zussman)
Subject: Re: Unit sphere covering by equal circles (caps) ?

Dear Dr. Rusin:



The most popular (based on the sci.math newsgroup) approach for covering a
unit sphere by N equal circles (caps) is based on definition of circles
centers such that N electrons located in their centers will have minimum
potential energy on the unit sphere. Similar approach uses for even points
distribution on a unit sphere.
Of course, it is only one of intuitive approaches for the best covering and
possibly that it have some 'holes'. For example, possibly that some N's
influence all further and closest points will bring to different result
from one obtained by influence only closest points.

Consequently if that occurs, then probably we receive equal distribution of
N points on the unit sphere, but we don't receive centers for the best
covering circles. Similar problem occurs on the plane as well: on the plane
there are many equal point distributions, for example distribution of
square's vertexes, or equilateral triangle. Nevertheless, only distribution
of hexahedron (honeycomb) vertexes defines the location of circle centers
for the best covering of the plane.
If that problem really exist on the sphere as well, then solutions receives
by equilibrium of electrons (that have force  of 1/(R^2)) can be improved
by using force of 1/(R^M), where M>2; or/and by localization of the force
field, i.e., domain about each electron where can be discovered the
influence of its force field restricted by some maximal distance from the
electron.

I will be glad to receive yours opinion and comments for definition of
centers of N circles on unit sphere that covers it in the best way by other
law for force distribution field or / and by local force field (this
approach have not  well definition, because was not defined the maximal
radius of distribution, but assumes that force field of electrons have
influence only on closest electrons, when all electron equalized).

You can send me e-mail : zeev@HiTech.technion.ac.il
######################################################################
Ze'ev Fainberg, Faculty of Mechanical Engineering, Technion, Israel.

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Date: Mon, 9 Sep 96 14:51:42 CDT
From: rusin (Dave Rusin)
To: eyal@hitech.technion.ac.il
Subject: Re: Unit sphere covering by equal circles (caps) ?

Thank you for your email.

You are quite right; using an inverse-square-law force will leave  N
points well distributed around a sphere, but there is no guarantee that
this distribution is optimal in any of several other senses, including
the placement of non-overlapping disks. The latter requires maximizing
the minimum (over all pairs of distinct points) the distance between
pairs of points. You seem to be suggesting that the latter can be
attained by forcing the points to minimize a potential function 
associated with a force whose strength diminishes as  1/R^M  for
larger values of  M; in fact this is correct in the limit as  M -> oo.
(One can define the various potential functions  
	U_M = [Sum(1/dist(P_i, P_j)^M)]^(1/M)
and then one notes that  lim(U_M)  as  M-->oo  is
	1/min{dist(P_i, P_j)} * {number of pairs (P_i,P_j) being this close}.)

I must confess that I have not looked at much of the literature on this
subject, but it is likely a substantial amount of attention has been paid
to this issue. I assume you have found my web page
	http://www.math.niu.edu/~rusin/known-math/index/spheres.html
[URL updated 1999/01 -- djr]
on this topic; there are a couple of references cited there which may
be appropriate places to begin a more serious study.

dave