This is the written version of a talk given by the author at the János Bolyai Conference on Hyperbolic Geometry in Budapest in 2002. In the first part of the paper, the author takes the reader through a leisurely and quite readable discussion of some of the pitfalls encountered when one attempts to define the density of both a fixed packing in Euclidean space, and the optimum density of a packing built from a fixed, finite set of tiles. In particular, he points out how aperiodic tilings show that the optimally dense packing may be fundamentally non-unique.
In the second part of the paper, he switches to hyperbolic geometry, and first observes that the technique used to prove the existence of optimal packings in Euclidean spaces fails here; the reason being that the huge size of the boundary of a set in hyperbolic space means that boundary effects cannot be neglected. On the other hand, the author discusses Károly Böröczky’s disk packing from [K. Böröczky, Mat. Lapok 25, 265–306 (1974); see also Acta Math. Acad. Sci. Hung. 32, 243–261 (1978; Zbl 0422.52011)] which convinced the packing community that “there could not in fact be a consistent theory of densest packings in hyperbolic spaces”, basically because there is no consistent way to define the density of Böröczky’s packing with respect to a tesselations of the hyperbolic plane by squares.
Next, Radin surveys the results obtained in [Lewis Bowen and Charles Radin, Geom. Dedicata 104, 37–59 (2004; Zbl 1069.52019)] on how to use an extension of Nevo’s ergodic theorem [Amos Nevo and Elias M. Stein, Ann. Math. (2) 145, No. 3, 565–595 (1997; Zbl 0884.43004)] to define the optimal density of certain packings of balls of fixed radius in hyperbolic space, this time with respect to “expanding” spheres instead of squares. Finally, he mentions the result obtained by himself and Bowen that for all but countably many radii \(R>0\), the densest packing of \(\mathbb H^d\) by spheres of radius \(R\) is not unique, and in fact aperiodic for most fixed radii.
The paper concludes with the suggestion that “the nonuniqueness associated with aperiodicity could be eliminated by reformulating the optimization problem as having its solutions be invariant measures, rather than packings which reproduce such measures.”
For the entire collection see [Zbl 1085.51002].