Number Theory: An Introduction to Mathematics

5 Densest Packings 347

G 2 = A−^1 G 1 A. With the aid of results of Minkowski and Jordan it follows that,
for a given dimensionn, there are only finitely many non-isomorphic crystallographic
groups. These results provided a positive answer to the first part of the 18th Problem
of Hilbert (1900).
The structure of physical crystals is analysed by means of the corresponding
3-dimensional crystallographic groups. Astronger concept than isomorphism is useful
for such applications. Two crystallographic groupsG 1 ,G 2 may be said to beproperly
isomorphicif there exists an orientation-preserving invertible affine transformationA
such thatG 2 = A−^1 G 1 A. An isomorphism class of crystallographic groups either
coincides with a proper isomorphism class or splits into two distinct proper isomor-
phism classes.
Fedorov (1891) showed that there are 17 isomorphism classes of 2-dimensional
crystallographic groups, none of which splits. Collating earlier work of Sohncke
(1879), Schoenflies (1889) and himself, Fedorov (1892) also showed that there are 219
isomorphism classes of 3-dimensional crystallographic groups, 11 of which split. More
recently, Brownet al.(1978) have shown that there are 4783 isomorphism classes of
4-dimensional crystallographic groups, 112 of which split.

5 DensestPackings............................................

The result of Hermite, mentioned at the beginning of the chapter, can be formulated
in terms of lattices instead of quadratic forms. For any real non-singular matrixT,the
matrix

A=TtT

is a real positive definite symmetric matrix. Conversely, by a principal axes transfor-
mation, or more simply by induction, it may be seen that any real positive definite
symmetric matrixAmay be represented in this way.
LetΛbe the lattice

Λ={y=Tx∈Rn:x∈Zn}

and put

γ(Λ)=m(Λ)/d(Λ)^2 /n,

where d(Λ)is the determinant andm(Λ)the minimum ofΛ.Thenγ(ρΛ)=γ(Λ)
for anyρ>0. Hermite’s result that there exists a positive constantcn, depending only
onn, such that 0<xtAx≤cn(detA)^1 /nfor somex∈Znmay be restated in the form

γ(Λ)≤cn.

Hermite’s constantγnis defined to be the least positive constantcnsuch that this
inequality holds for allΛ⊆Rn.
It may be shown thatγnnis a rational number for eachn. It follows from Proposi-
tion 2 thatlimn→∞γn/n≤ 2 /πe. Minkowski (1905) showed also that