7 Further Remarks 359An interesting subset of all crystallographic groups consists of those generated by
reflections in hyperplanes, since Stiefel (1941/2) showed that they are in 1-1 corre-
spondence with the compact simply-connected semi-simple Lie groups. See the ‘Note
historique’ in Bourbaki [8].
There has recently been considerable interest in tilings ofRnwhich, although not
lattice tilings, consist of translates of finitely manyn-dimensional polytopes. The first
example, inR^2 , due to Penrose (1974), was explained more algebraically by de Bruijn
(1981). A substantial generalization of de Bruijn’s construction was given by Katz
and Duneau (1986), who showed that many such ‘quasiperiodic’ tilings may be ob-
tained by a method of cut and projection from ordinary lattices in a higher-dimensional
space. The subject gained practical significance with the discovery by Shechtmanet al.
(1984) that the diffraction pattern of an alloy of aluminium and magnesium has icosa-
hedral symmetry, which is impossible for a crystal. Many other ‘quasicrystals’ have
since been found. The papers referred to are reproduced, with others, in Steinhardt and
Ostlund [56]. The mathematical theory of quasicrystals is surveyed in Leet al.[37].
Skubenko [54] has given an upper bound for Hermite’s constantγn. Somewhat
sharper bounds are known, but they have the same asymptotic behaviour and the proofs
are much more complicated. A lower bound forγnwas obtained with a new method
by Ball [2].
For the densest lattices inRn(n≤ 8 ), see Ryshkov and Baranovskii [49]. The
enumeration of all root lattices is carried out in Ebeling [18]. (A more general prob-
lem is treated in Chap. 3 of Humphreys [30] and in Chap. 6 of Bourbaki [8].) For the
Voronoi cells of root lattices, see Chap. 21 of Conway and Sloane [14] and Moody and
Patera [43]. For theDynkin diagramsassociated with root lattices, see also Reiten [47].
Rajan and Shende [46] characterize root lattices as those lattices for which every
facet vector is a minimal vector, but their definition of root lattice is not that adopted
here. Their argument shows that if every facet vector of a lattice is a minimal vector
then, after scaling to make the minimal vectors have square-norm 2, it is a root lattice
in our sense.
There is a fund of information about lattice packings of balls in Conway and
Sloane [14]. See also Thompson [58] for the Leech lattice and Coxeter [16] for the
kissing number problem.
We have restricted attention to lattice packings and, in particular, to lattice pack-
ings of balls. Lattice packings of other convex bodies are discussed in the books on
geometry of numbers cited above. Non-lattice packings have also been much studied.
The notion of density is not so intuitive in this case and it should be realized that the
density is unaltered if finitely many sets are removed from the packing.
Packings and coverings are discussed in the texts of Rogers [48] and
Fejes T ́oth [23], [24]. For packings of balls, see also Zong [62]. Sloane [55] and
Elkies [20] provide introductions to the connections between lattice packings of balls
and coding theory.
The third part of Hilbert’s 18th problem, which is surveyed in Milnor [41], deals
with the densest lattice ornon-lattice packing of balls inRn. It is known that, for
n=2, the densest lattice packing is also a densest packing. The original proof by
Thue (1882/1910) was incomplete, but a complete proof was given by L. Fejes T ́oth
(1940). The famousKepler conjectureasserts that, also forn=3, the densest lattice