```Newsgroups:     sci.math.research
From:           asimov@nas.nasa.gov (Daniel A. Asimov)
Subject:        Close-packing of 1-spheres in R^3
Organization:   University of Illinois at Urbana
Date:           Mon, 16 Mar 1992 19:03:08 GMT
```

```Let us define a hoop in R^3 to be any set in R^3 that is
isometric to the unit circle { (x,y,0) in R^3 | x^2 + y^2 = 1 }.

Denote the open ball of radius r about the origin by B(r).

Let X be a subset of R^3 which is a disjoint union of hoops.
Let us call such a set "hooped."  If X is also a measurable
set, we define the volume fraction vf(X) as follows:

vf(X) =  lim    ( vol(X intersect B(r)) / vol(B(r)) )
r -> oo

when this limit exists.  It is easy to see that vf(X) does not
depend on the location of the origin.

Define the packing fraction for hoops in R^3 to be the number

pf = sup( vf(X) )

where the sup is over all hooped measurable sets X in R^3 for
which vf exists.

QUESTION:	Does anyone know of any results on determining
the packing fraction for hoops in R^3 ?

(I have found that it is over 70 percent.)

Daniel Asimov
asimov@nas.nasa.gov
```