Re: Packing different sized spheres

In article <[EMAIL PROTECTED]>,
Ian Wanless  <[EMAIL PROTECTED]> wrote:

>Is it possible to achieve a density arbitrarily close to 1 by using
>spheres of arbitrary sizes?  This is such a natural question that it
>must have been pondered before. Can someone enlighten me?

Yes, if you can use an arbitrary finite set of sizes you can get
arbitrarily close to 1.  Start with a lattice packing of spheres of
one size r_1, obtaining the density d corresponding to that lattice.
Now consider packing the "holes" remaining with spheres of size 
r_2 << r1, again using a lattice packing (omitting the spheres that would
intersect the first set of spheres).  If r_2 is small enough, the effect
of the boundary is negligible, so for given epsilon you can obtain density
more than (1-epsilon) d in these "holes", or d + (1-d)(1-epsilon)d.
Repeat the process.  After using n sizes, the density will be more than
1 - (1-d)(1-(1-epsilon)d)^(n-1), which goes to 1 as n -> infinity.

Robert Israel                                [EMAIL PROTECTED]
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2