The mathematical tools which have been developed in the first two chapters of Part I will have useful applications here in Part II; since crystallographic studies require both analytical treatment as well as geometric visualization. Geometric models, perspective drawings, or projections of frames of symmetry, of crystal structures, and of complicated molecules are very instructive. However, often models are difficult to build, perspective drawings become confusing, and projections suffer from loss of information. In addition, distances and angles may be distorted, and it is sometimes not easy to see the important geometric relations.
Analytical methods, e.g. the matrix formalism, provide instruments which are often only slightly dependent on or even independent of the complexity of the subject. In many cases they can be applied using computers. Moreover, there are internal tests which enable the user to check the results of the calculations for inner consistency. Such methods are indispensable in particular in crystal-structure determination and evaluation. Only very simple crystal structures can be considered without them.
Crystallographic symmetry and its applications have been investigated and developed by mineralogists, mathematicians, physicists, and chemists from different countries over several centuries. The result is the beautiful and still rapidly growing tree of contemporary crystallography. However, it is not necessary to know the whole of this field of knowledge in order to apply and to take advantage of it. The crystallographic tools necessary for the exploration of matter and for solid state research can be taken from the volumes of the International Tables for Crystallography series. Symmetry is described in Vol. A of this series. By the manuscript on hand, the reader shall be enabled to use and exploit the contents of that volume A, abbreviated IT A in this manuscript.
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