Classification of lattices: A new step

From the classification of (three-dimensional) lattices into the 14 Bravais types, the finer classifications into the 44 Niggli characters and 24 Delaunay sorts are considered. The last two divisions are mutually incompatible and the Niggli characters show a disturbing 'asymmetry' with respect to the conventional parameters. The aim of the paper is to find a common subdivision of both the Niggli characters and Delaunay sorts that reveals no 'asymmetry' and is crystallographically meaningful. The first attempt based on separating the non-sharp inequalities (less than or equal to) into sharp inequalities (<) and equalities (=) in the systems defining the Niggli characters removed only the 'asymmetry', whereas the incompatibility with the Delaunay sorts remained. The second approach may be called the hyperfaces idea. To any lattice there are attached several points in E-5, its Buerger points. These Buerger points lie in two convex five-dimensional hyperpolyhedra Omega(+), Omega(-). The division of lattices into classes is determined by the distribution of their Buerger points along the vertices, edges, faces, three- and four-dimensional hyperfaces and the interior of Omega(+) and Omega(-). The resulting classes are called genera. There are 127 of them. They form a subdivision of both the Delaunay sorts and the Niggli characters (and, consequently, also of the Bravais types) and their parameter ranges are open. Genera stand for a remarkably strong bond between lattices. The lattices belonging to the same genus agree in a series of important crystallographic properties. Genera are explicitly described by systems of linear inequalities. The five-dimensional geometrical objects obtained in this way are illustrated by their three-dimensional cross sections. From these illustrations, a suitable notation of the genera was derived. Extensive tables enable the determination of the genus of a given lattice.