From:asimov@nas.nasa.gov (Daniel A. Asimov)Newsgroups:sci.math.researchSubject:Dense Sphere-Packings in Hyperbolic SpaceDate:11 Apr 1996 23:48:35 GMTOrganization:NAS - NASA Ames Research Center, Moffett Field, CA

(or maybe they should be called "ball-packings") Consider hyperbolic space H^n (the unique smooth simply-connected Riemannian n-manifold of constant sectional curvature = -1). For each radius r > 0, one can ask the "Kepler question" K(n,r): What is the (or a) densest packing of H^n by balls of radius r ??? However, there are certain problems of well-definedness that arise: Let Q(n,r) denote the set of all packings of H^n by closed balls of radius r (where a packing is a collection of such balls any two of which are are either disjoint or tangent). For any q in Q(n,r) we let U(q) denote the union of all the balls of q, and we let q(R; x) denote those points of U(q) that lie within a distance R of the point x of H^n. Given a packing q in Q(n,r), we can define the density d(q; x) as follows: d(q; x) = lim vol(q(R; x)) / vol(B(R)), R -> oo if this limit exists, where B(R) denotes a ball of radius R in H^n. ------------------------------------------------------------------------------- QUESTIONS: 1. Given a packing q in Q(n,r), if d(q; x) exists for one x in H^n, does it necessarily exist and have the same value for all x ??? Consider the packings q in Q(n,r) for which d(q; x) exists and is independent of x. (In this case we denote d(q; x) by d(q).) Let supden(n,r) denote the supremum of all such d(q). 2. Must there exist a packing q0 such that d(q0) = supden(n,r) ??? 3. Given n, is supden(n,r) a continuous function of r ??? Monotone ??? 3. Especially for n = 2 or n = 3, for which values of r is there an explicit packing q in Q(n,r) that is known to realize supden(n,r) ??? What about higher dimensions??? ------------------------------------------------------------------------------- Daniel Asimov Senior Research Scientist Mail Stop T27A-1 NASA Ames Research Center Moffett Field, CA 94035-1000 asimov@nas.nasa.gov (415) 604-4799 w (415) 604-3957 fax

From:kuperberg-greg@MATH.YALE.EDU (Greg Kuperberg)Newsgroups:sci.math.researchSubject:Re: Dense Sphere-Packings in Hyperbolic SpaceDate:12 Apr 1996 15:12:09 -0400Organization:Yale University Mathematics Dept., New Haven, CT 06520-2155

In article <4kk5oj$ntl@cnn.nas.nasa.gov> asimov@nas.nasa.gov (Daniel A. Asimov) writes: >1. Given a packing q in Q(n,r), if d(q; x) exists for one x in H^n, does > it necessarily exist and have the same value for all x ??? [d(q;x) here is defined as the limit of the proportion covered of round balls centered at x.] Certainly not, given that in the hyperbolic plane, a cone with an angle of 5 degrees contains a half-plane. >Let supden(n,r) denote the supremum of all such d(q). [assuming independence of x.] >3. Given n, is supden(n,r) a continuous function of r ??? Monotone ??? I doubt it, although I don't have a counterexample. >4. Especially for n = 2 or n = 3, for which values of r is there an > explicit packing q in Q(n,r) that is known to realize supden(n,r) ??? > What about higher dimensions??? There is a theorem due to Borocky that if you take a Delaunay simplex associated to a hyperbolic sphere packing, the proportion covered covered by the spheres is maximized when the simplex is regular. If this simplex tiles spaces, then I believe that you get an example of the kind you want. There is such a triangle with angles of 2*pi/n for every n>7 in the plane, and there is a regular simplex in four hyperbolic dimensions with angles of 2*pi/5. The ideal simplex in three dimensions has angles of pi/3, which gives you a horoball packing which is also optimal in the sense of Borocky; I don't know if horoballs are okay for you.