International
Tables for Crystallography Volume A Spacegroup symmetry Edited by Th. Hahn © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. A, ch. 9.1, pp. 745749
Section 9.1.8. Delaunay reduction^{a}Universität Erlangen–Nürnberg, RobertKochStrasse 4a, D91080 Uttenreuth, Germany, and ^{b}Institut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D91054 Erlangen, Germany 
Further classifications use reduction theory. There are different approaches to the reduction of quadratic forms in mathematics. The two most important in our context are
The investigations by Gruber (cf. Chapter 9.3 ) have shown the common root of both crystallographic approaches. As in Chapters 9.2 and 9.3 the Niggli reduction will be discussed in detail, we shall discuss the Delaunay reduction here.
We start with a lattice basis . This basis is extended by a vector All scalar products are considered. The reduction is performed minimizing the sum It can be shown that this sum can be reduced as long as one of the scalar products is still positive. If e.g. the scalar product is still positive, a transformation can be performed such that the sum of the transformed is smaller than : In the twodimensional case, holds.
If all the scalar products are less than or equal to zero, the three shortest vectors forming the reduced basis are contained in the set which corresponds to the maximal set of faces of the Dirichlet domain (14 faces).
For practical application, it is useful to classify the patterns of the resulting scalar products regarding their equivalence or zero values. These classes of patterns correspond to the reduced bases and result in `Symmetrische Sorten' (Delaunay, 1933) that lead directly to the conventional crystallographic cells by fixed transformations (cf. Patterson & Love, 1957; Burzlaff & Zimmermann, 1993). Table 9.1.8.1 gives the list of the 24 `symmetrische Sorten'. Column 1 contains Delaunay's symbols, column 2 the symbol of the Bravais type. For monoclinic centred lattices, the matrix of the last column transforms the primitive reduced cell into an Icentred cell, which has to be transformed to A or C according to the monoclinic standardization rules, if necessary. operates on the basis in the following form: where denotes the matrix of the basis vectors of the reduced cell and the matrix of the conventional cell.

Column 3 gives metrical conditions for the occurrence of certain Voronoi types. Column 5 indicates the relations among the scalar products of the reduced vector set. In some cases, different Selling patterns are given for one `symmetrische Sorte'. This procedure avoids a final reduction step (cf. Patterson & Love, 1957) and simplifies the computational treatment significantly. The number of `symmetrische Sorten', and thus the number of transformations which have to be applied, is smaller than the number of lattice characters according to Niggli. Note that the introduction of reduced bases using shortest lattice vectors causes complications in more than three dimensions (cf. Schwarzenberger, 1980).
References
Burzlaff, H. & Zimmermann, H. (1993). Kristallsymmetrie–Kristallstruktur, pp. 90–105. Erlangen: R. Merkel.Delaunay, B. N. (1933). Neuere Darstellung der geometrischen Kristallographie. Z. Kristallogr. 84, 109–149.
Patterson, A. L. & Love, W. E. (1957). Remarks on the Delaunay reduction. Acta Cryst. 10, 111–116.
Schwarzenberger, R. L. E. (1980). Ndimensional crystallography. San Francisco: Pitman.
Selling, E. (1874). Über binäre und ternäre quadratische Formen. Crelles J. Reine Angew. Math. 77, 143ff.
Wolff, P. M. de, Belov, N. V., Bertaut, E. F., Buerger, M. J., Donnay, J. D. H., Fischer, W., Hahn, Th., Koptsik, V. A., Mackay, A. L., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1985). Nomenclature for crystal families, Bravaislattice types and arithmetic classes. Report of the International Union of Crystallography Ad hoc Committee on the Nomenclature of Symmetry. Acta Cryst. A41, 278–280.