Newsgroups:sci.math.researchFrom:asimov@nas.nasa.gov (Daniel A. Asimov)Subject:Close-packing of 1-spheres in R^3Organization:University of Illinois at UrbanaDate: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