LEWIS BOWEN, CHARLES HOLTON, CHARLES RADIN, AND LORENZO SADUN Abstract. Part of Hilbert’s eighteenth problem is to classify the symmetri...
LEWIS BOWEN, CHARLES HOLTON, CHARLES RADIN, AND LORENZO SADUN
Abstract.
Part of Hilbert’s eighteenth problem is to classify the symmetries of the densest packings of bodies in Euclidean and hy- perbolic spaces, for instance the densest packings of balls or sim- plices. We prove that when such a packing problem has a unique solution up to congruence then the solution must have cocompact symmetry group, and we prove that the densest packing of unit disks in the Euclidean plane is unique up to congruence. We also analyze some densest packings of polygons in the hyperbolic plane.
Part of Hilbert’s eighteenth problem is to classify the symmetries of the densest packings of bodies in Euclidean and hy- perbolic spaces, for instance the densest packings of balls or sim- plices. We prove that when such a packing problem has a unique solution up to congruence then the solution must have cocompact symmetry group, and we prove that the densest packing of unit disks in the Euclidean plane is unique up to congruence. We also analyze some densest packings of polygons in the hyperbolic plane.
I. Introduction
The objects of our study are the densest packings, particularly of balls and polyhedra, in a space of infinite volume; for a survey see the classic texts [Feje] and [Roge], and the review [FeKu]. Most interest has centered on densest packings in the Euclidean spaces En, notably when the dimension n is 2 or 3, but we will see that packing prob- lems in hyperbolic spaces Hn can clarify some issues for problems set in Euclidean spaces so we consider the more general problem in the n dimensional spaces Xn, where Xn will stand for either En or Hn. (It would be reasonable to generalize our considerations further, to sym- metric spaces, and even to include infinite graphs, but as we have no noteworthy results in that generality we felt it would be misleading to couch our considerations in that setting.) We will give results of two types. We prove (Theorem 2) that when a packing problem in Xn has a solution which is unique up to congruence then that solution must have symmetry group cocompact in the isometry group of Xn; and we prove (Theorem 1) that the densest packing of unit disks in E2 is unique up to congruence. In Section IV we analyze the symmetries of some densest packings of polygons in the hyperbolic plane. This will suggest a modified form of uniqueness for the solution of a densest packing problem. We now introduce some notation and basic features of density. We will be concerned with “packings” of “bodies” in Xn. By a body we mean a connected compact set in Xn with dense interior and boundary of volume 0. Assume given some finite collection B of bodies in Xn, for instance a single ball. By a packing of bodies we then mean a collection P of bodies, each congruent under the isometry group of Xn to some body in B, such that the interiors of bodies in P do not intersect. Denoting by Br(p) the closed ball in Xn of radius r and center p, we define the “density relative to Br(p)” of a packing P as:
(1) DBr(p)(P)≡Pβ∈P mXn[β ∩Br(p)] mXn[Br(p)] ,
where mXn is the usual measure onXn.
Then, assuming the limit exists, we define the “density” of P as:
where mXn is the usual measure onXn.
Then, assuming the limit exists, we define the “density” of P as:
(2) D(P)≡ lim r→∞Pβ∈P mXn[β ∩Br(p)] mXn[Br(p)]
It is not hard to construct packings P for which the limiting den- sity D(P) does not exist, for instance by the adroit choice of arbitrarily large empty regions so that the relative density oscillates with r instead of having a limit. (In hyperbolic space the limit could exist but depend on p, which we also consider unacceptable.) The possible nonexistence of the limit of (2) is an essential feature of analyzing density in spaces of infinite volume; density is inherently a global quantity, and funda- mentally requires a formula somewhat like (2) for its definition [Feje], [FeKu]. We discuss this further below. The most important examples for which we know the densest pack- ings are those for balls of fixed radius in En for n = 2 and 3. (For a recent survey of this problem in higher dimensions see [CoGS]). It will be useful in discussing these problems to make use of the notion of “Voronoi cell”, defined for each body β in a packing P as the closure of the set of those points in Xn closer to β than to any other body in P. A noteworthy feature of the n = 2 example is, then, that in the optimal packing (see Figure 1) the Voronoi cell of every disk (the smallest regular hexagon that could contain the disk) has the prop- erty that the fraction of the area of this cell taken up by the disk is strictly larger than for any other Voronoi cell in any packing by such disks. (Intuitively, the optimal configuration is simultaneously optimal in all local regions.) As for n = 3, it is generally felt that the dens- est lattice packing (i.e., the face centered cubic) achieves the optimum density among all possible packings, along with all the other packings made by layering hexagonally packed planar configurations, such as the hexagonal close packed structure; see [Roge]. There are claims in the literature by Hsiang [Hsia] and by Hales [Hale] for proofs of this, and there is hope that the problem will soon be generally accepted as solved.
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