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The main feature of this Appendix is Table A1.1, a list of the densest known packings in En for n = 1 through 24, and for a few selected n larger than 24. The Table is essentially that in [83] updated by material from [112], [113] and [95]. It differs from that in [83], however, in that density, instead of center density is used. (Recall that the density, ρ, of a packing in En is the fraction of En within the spheres of the packing, while the center density,δ, is ρ divided by Vn, the volume of a sphere in the packing.) The latter is the number of centers of unit spheres per unit n-dimensional volume.
Several other comments on Table A1.1 are in order.
Under the column marked “Type,” B indicates that both a lattice and a nonlattice packing with these parameters are known. L indicates that at present only a lattice packing is known, and N that only a nonlattice packing is known. A indicates a local arrangement of spheres touching one sphere.
N. J. A. Sloane kindly provided the sources [112], [113] and [95]. Specifically, the packing P10c in E10 appears in [113, p. 31], the packings P11c in E11 and one in E36 with no designation appear in [112, p. 118], and new bounds for contact numbers appear in [95].
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