Gruber P. Convex and Discrete Geometry

Extension to Crystallographic Groups?
21 Lattices 355
Most results in the geometry of numbers and part of the results in discrete geometry
rest on the notion of lattices, that is, discrete groups of translations in E d . Considering
this, it is surprising that there is no equally elaborate theory for other crystallographic
groups, although there are some pertinent results. We mention a result of
Delone [258] on the number of facets of a stereohedron, i.e. a space filler by means
of a crystallographic group, the negative solution of Hilbert’s 18th problem, a version
of Blichfeldt’s theorem due to Schmidt [896], p. 30 based on a crystallographic
structure, and a result on ball packings with crystallographic groups of Horváth and
Molnár [522]. A starting point for research in this direction could be Engel’s article
[297]. See also the short chapter on crystallography in Erdös, Gruber and Hammer
[307] and the books of Senechal [925] and Engel, Michel and Senechal [301].
The crystallographer Peter Engel [300] has reservations about a substantial parallel
theory for crystallographic groups – in spite of his important pertinent contributions.
His argument is that the context of general crystallographic groups is so
complicated that one may not expect a lot of non-trivial results.
All convex bodies in this chapter are proper.
21 Lattices
The notion of lattice already appeared implicitly in the work of Kepler [576, 577],
who used it in the context of packing of balls. Crystallographers such as Haüy [483]
in the eighteenth century and many crystallographers in the nineteenth century based
their investigations on lattices, although experimental proof that lattices are underlying
crystals was given only in the early twentieth century by von Laue and father
and son Bragg, for which all three got the Nobel Prize. A different source for lattices
is number theory. Here the classical reference is Gauss [364], who seems to have
first seen the relation between positive definite quadratic forms and lattice packing
of balls. Lattices and convex bodies are the main ingredients of the geometry of
numbers. Results dealing with lattices are often the starting point for more general
investigations in discrete geometry.
In this section, basic notions related to lattices and some of the fundamental properties
of lattices are presented, as needed in the context of the geometry of numbers.
For additional information on lattices, mainly from the viewpoint of the geometry
of numbers, see Cassels [195], Gruber and Lekkerkerker [447] and Lagarias [625].
While we study relations between lattices in general, special lattices, theta series
and codes are treated in the book of Gruber and Lekkerkerker. More information
on these topics is presented by Ebeling [282] and Conway and Sloane [220]. For
relations to crystallography, see Erdös, Gruber and Hammer [307], Engel [296, 297]
and Lagarias [625]. Engel, Michel and Senechal [301] treat lattices from a crystallographic
viewpoint. For algorithmic problems and results on lattices compare the
references in Sect. 28.