A Rejoinder to Hales’s Article

Conclusion

This then is an item-by-item reality check on the mathematical content of their objections to my article [2], on which they have based their assessment on the status of the Kepler conjecture. It should now be clear from the responses that their objections are without any mathematical foundations.

Let me conclude this article with a few remarks on the proof of Kepler’s conjecture.

• (i) Although the definition of close neighbors to be spheres with center-distance at most 2.18 is notcanonical, it plays a useful role in simplifying proofs in many places.

• (ii) The locally averaged density introduced in [2] §3 is a specific weighted average of the local densities of a central sphere and its close neighbors. It enables us to reduce the proof of Kepler’s conjecture to the optimal estimation of such alocal invariant (see [2], Lemma 1 and Theorem 2).

• (iii) Lower bound volume estimation of the local cells in various geometric settings is, of course, the basic technique involved in theoptimal estimations of both thelocal density and thelocally averaged density (see [2], Theorems 1 and 2). A system of volume estimation techniques tailored for this purpose are developed in §§4, 5, and 6 of [2].

• (iv) The cluster of spheres consisting of a central sphere and itsclose neighbors is called a core packing. The geometry of core packings with 12 or 13 close neighbours are essentially the only ones whose local geometry plays a critical role in the proof of Kepler’s conjecture. Techniques in spherical geometry are developed in order to analyze the relevant aspects of the geometry of these types of core packings (see [2], §§7 and 8).

• (v) Technically, the volume estimation of the resulting local cell of a given type of local extension is much simpler than that involved in the optimal estimation of the local density when there is no restriction on the local extension at all. Here, the control on the bucklingeffect and the understanding of the geometry of core packings with 12 close neighbors (see [2], §8) plays a major role.

• (vi) Kepler’s conjecture is a natural and fundamental problem about three-dimensional Euclidean space (the space in which we live) which involves the arrangement of infinitely many identical spheres. It is natural that spherical geometry should play the major role throughout the proof of Kepler’s conjecture. In retrospect, I find my understanding of spherical geometry considerably enriched through working on this problem. I think a truly interested and sufficiently patient reader of [2] will not only understand the proof of Kepler’s conjecture but also share this valuable experience. Spherical geometry is the “orthogonal part” of Euclidean geometry, where many of the deeper properties of Euclidean space reside. One may say that the study of the sphere packing problem enables us to understand E³ better. The techniques in spherical geometry developed for the purpose of proving Kepler’s conjecture may well find further use in the geometry and physics of solids.