Modern Geometric Calculations in Crystallography

The aim of mathematical crystallography is the classification of periodic structures by means of different equivalence relationships, yielding the well known crystallographic classes and Bravais lattices [ 1 ]. Periodicity (crys-tallinity) has been the paradigm of classical crystallography. Recently, more systematical attention has been paid to structures which are not orthodox crystals. For example, some generalizations, involving curved spaces with non-Euclidean metrics, were developed for the understanding of random and liquid crystalline structures [ 2 ]. However, the first step away from orthodox crystalline order, represented by the 230 crystallographic space groups, was motivated by the appearance of quasicrystals in 1984 [ 3 ]. Since then, crystallography has been the subject of deep revisions. Quasicrystals are metallic alloys whose diffraction patterns exhibit sharp spots (like a crystal) but non-crystallographic symmetry. This means that the lattice underlying the atomic structure cannot be periodic. So, crystallography faces a non-crystalline but perfectly ordered structure. There are also many other directions in which classical crystallography can be generalized, by relaxing or altering various requirements, to include structures which are ordered but do not follow the exact paradigm of crystallization [ 4 ].