Résumé : Hexagonal close-packings are the most efficient way to pack unit disks in R^2. But what do we mean by the definite article "the" in the previous sentence? Are hexagonal close-packings the only optimal packings? If we measure optimality by density, the answer is, No. In fact, there are far too many density-optimal packings to classify in any meaningful way. This suggests that density is too coarse a notion to capture everything that we mean (or should mean!) by "efficient". To study efficiency, we study deficiency, and seek ways to quantify defects in a regular packing. An obstacle here is that common kinds of defects inhabit disparate scales (e.g., point defects are infinitesimal compared to line defects, which in turn are infinitesimal compared to the bulk). This suggests we turn to extensions of the real numbers that include infinitesimal elements (or rather, as turns out to be more helpful, infinite elements). We use a regularization trick to make sense of these ideas (starting in one dimension). This enables us to sharpen our notion of optimal packing so that the optimal disk-packings are provably the hexagonal close-packings and no others. A side-benefit is a natural but apparently new finitely additive, non-Archimedean measure in Euclidean n-space; it agrees with n-dimensional volume when applied to finite regions, but some infinite regions are "more infinite" than others. For slides related to an earlier version of this talk, see http://jamespropp.org/brown18a.pdf
[Slides.pdf] [arXiv] [article] [vidéo]
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