Sphere Packings, Lattices and Groups by J. H. Conway, N. J. A. Sloane (auth.)

By J. H. Conway, N. J. A. Sloane (auth.)

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Extra info for Sphere Packings, Lattices and Groups

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43 can be packed into a sphere of radius I in 100-dimensional space? Ov) Two- and three-dimensional packings have many applications. For example the circles in a two-dimensional packing may represent optical fibers as seen in the cross-section of a cable [KinIl. Three-dimensional packings have applications in chemistry and physics [Berll], [HoaIl, [Hoa2], [Kit4], [O'KeIl, [SI017], [SI019], [Teol]-[Te03], [WeI2]-[WeIS], [ZimIl; biology [RiU], [TamS]; antenna design [Strl]; choosing directions for X-ray tomography [She3], [She4], [Smi7]; and in performing statistical analysis on spheres [Wat24].

In three dimensions, for example, there are non lattice packings that are just as dense as the face-centered cubic lattice. This happens because the fcc can be built up by layers, beginning with one layer of spheres placed in the hexagonal lattice arrangement of Fig. 3b, with the centers at the points marked 'a'. There are then two (equivalent) ways to place the second Chapter 1 8 layer: the spheres can be placed above the positions marked 'b' or above those marked 'c'. Suppose we place them at 'b'.

N Name of packing o Ao I ;A I AI Center density 0 Attained Bound Density 6. fi. fi. fi. 6 A'rrJ. 03516 Aj{1. fi. fi. fi. 3 shows that in dimensions above 1000 these lattices are denser than any presently known. 3. Sphere packings in more than 24 dimensions. 12 Bound Kissing no. 6 8,lOc 8,6 found. Many generalizations and extensions of (28) have been found, although no essential improvement is known for large n. In its general form this bound is known as the Minkowski-Hlawka theorem. We still do not know how to construct packings that are as good as (28).

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