346 VIII The Geometry of Numbers
Proof Letb 1 ,...,bmbe the facet vectors ofΛand put
Λ′={x=β 1 b 1 +···+βmbm:β 1 ,...,βm∈Z}.EvidentlyΛ′is a subgroup ofRnand actually a discrete subgroup, sinceΛ′⊆Λ.IfΛ′
were contained in a hyperplane ofRnany point on the line through the origin orthog-
onal to this hyperplane would belong to the Voronoi cellVofΛ, which is impossible
becauseVis bounded. HenceΛ′containsnlinearly independent vectors.
ThusΛ′is a sublattice ofΛ. It follows that the Voronoi cellVofΛis contained in
the Voronoi cellV′ofΛ′.Butify∈V′,then
‖y‖≤‖bi−y‖,(i= 1 ,...,m)and hencey∈V. ThusV′=V. Hence theΛ′-translates ofVand theΛ-translates of
Vare both tilings ofRn.SinceΛ′⊆Λ, this is possible only ifΛ′=Λ.
Since every integral linearcombination of facet vectors is in the lattice, Proposi-
tion 16 implies
Corollary 17Distinct lattices inRnhave distinct Voronoi cells.
Proposition 16 does not say that the lattice has a basis of facet vectors. It is known
that every lattice inRnhas a basis of facet vectors ifn≤6, but ifn>6 this is still an
open question. It is known also that every lattice inRnhas a basis of minimal vectors
whenn≤4 but, whenn>4, there are lattices with no such basis. In fact a lattice may
have no basis of minimal vectors, even though every lattice vector is an integral linear
combination of minimal vectors.
Lattices and their Voronoi cells have long been used in crystallography. An
n-dimensionalcrystalmay be defined mathematically to be a subset ofRnof the form
F+Λ={x+y:x∈F,y∈Λ},whereFis a finite set andΛa lattice. Crystals may be studied by means of their
symmetry groups.
AnisometryofRnis an invertible affine transformation which leaves unaltered the
Euclidean distance between any two points. For example, any orthogonal transforma-
tion is an isometry and so is a translation by an arbitrary vectorv. Any isometry is the
composite of a translation and an orthogonal transformation. Thesymmetry groupof a
setX⊆Rnis the group of all isometries ofRnwhich mapXto itself.
We d e fi n e a nn-dimensionalcrystallographic groupto be a groupGof isometries
ofRnsuch that the vectors corresponding to translations inGform ann-dimensional
lattice. It is not difficult to show that a subset ofRnis ann-dimensional crystal if and
only if it is discrete and its symmetry group is ann-dimensional crystallographic group.
It was shown by Bieberbach (1911) that a groupGof isometries ofRnis a crys-
tallographic group if and only if it is discrete and has a compact fundamental domain
D,i.e.thesets{g(D):g∈ G}form a tiling ofRn. He could then show that the
translations in a crystallographic group form a torsion-free abelian normal subgroup
of finite index. He showed later (1912) that two crystallographic groupsG 1 ,G 2 are
isomorphic if and only if there exists an invertible affine transformationAsuch that