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. 8.1, p. 720
Section 8.1.1. Introduction^{a}Institut für Kristallographie, Universität, D76128 Karlsruhe, Germany 
The aim of this part is to define and explain some of the concepts and terms frequently used in crystallography, and to present some basic knowledge in order to enable the reader to make best use of the spacegroup tables.
The reader will be assumed to have some familiarity with analytical geometry and linear algebra, including vector and matrix calculus. Even though one can solve a good number of practical crystallographic problems without this knowledge, some mathematical insight is necessary for a more thorough understanding of crystallography. In particular, the application of symmetry theory to problems in crystal chemistry and crystal physics requires a background of group theory and, sometimes, also of representation theory.
The symmetry of crystals is treated in textbooks by different methods and at different levels of complexity. In this part, a mainly algebraic approach is used, but the geometric viewpoint is presented also. The algebraic approach has two advantages: it facilitates computer applications and it permits statements to be formulated in such a way that they are independent of the dimension of the space. This is frequently done in this part.
A great selection of textbooks and monographs is available for the study of crystallography. Only Giacovazzo (2002) and Vainshtein (1994) will be mentioned here.
Surveys of the history of crystallographic symmetry can be found in Burckhardt (1988) and LimadeFaria (1990).
In addition to books, many programs exist by which crystallographic computations can be performed. For example, the programs can be used to derive the classes of point groups, space groups, lattices (Bravais lattices) and crystal families; to calculate the subgroups of point groups and space groups, Wyckoff positions, irreducible representations etc. The mathematical program packages GAP (Groups, Algorithms and Programming), in particular CrystGap, and Carat (Crystallographic Algorithms and Tables) are examples of powerful tools for the solution of problems of crystallographic symmetry. For GAP, see http://www.gapsystem.org/ ; for Carat, see http://wwwb.math.rwthaachen.de/carat/ . Other programs are provided by the crystallographic server in Bilbao: http://www.cryst.ehu.es/ .
Essential for the determination of crystal structures are extremely efficient program systems that implicitly make use of crystallographic (and noncrystallographic) symmetries.
In this part, as well as in the spacegroup tables of this volume, `classical' crystallographic groups in three, two and one dimensions are described, i.e. space groups, plane groups, line groups and their associated point groups. In addition to threedimensional crystallography, which is the basis for the treatment of crystal structures, crystallography of two and onedimensional space is of practical importance. It is encountered in sections and projections of crystal structures, in mosaics and in frieze ornaments.
There are several expansions of `classical' crystallographic groups (groups of motions) that are not treated in this volume but will or may be included in future volumes of the IT series.

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