In diffraction analysis, crystal structures are usually assumed to be ideal 3-dimensional periodic objects. Since real crystals are never perfect, what we obtained is just an averaged image of the real structure over a large number of unit cells. However a knowledge on the average structure is not enough for understanding the properties of many solid state materials. Therefore an important task for methods of solving crystal structures is to extend their scope to include real crystals. Modulated crystal structures belong to that kind of crystal structures, in which atoms suffer from certain substitutional and/or positional fluctuation. If the period of fluctuation matches that of the three-dimensional unit cell then a superstructure results, otherwise an incommensurate modulated structure is obtained. Incommensurate modulated phases can be found in many important solid state materials. In many cases, the transition to the incommensurate modulated structure corresponds to a change of certain physical properties. Hence it is important to know the structure of incommensurate modulated phases in order to understand the mechanism of the transition and properties in the modulated state. Many incommensurate modulated structures were solved by trial-and-error methods. With these methods it is necessary to make assumption on the modulation in advance. This leads often to difficulties and erroneous results. In view of diffraction analysis, it is possible to phase the reflections directly and solve the structure objectively without relying on any assumption about the modulation wave.
        A common feature of incommensurate modulated structures is that they do not have 3-dimensional periodicity. However incommensurate modulated structures can be regarded as the 3-dimensional hypersection of a 4- or higher-dimensional periodic structure. Obviously direct analysis of incommensurate modulated structures would better be implemented in multi-dimensional space. For this purpose we need firstly a theory on multi-dimensional symmetry and secondly a method to solve directly the multi-dimensional phase problem. The work of Janner and co-workers (see Janssen, T., Janner, A., Looijenga-Vos, A. and De Wolff, P. M. (1992), in Wilson, A. J. C. Ed. International Tables for Crystallography, Vol. C, Kluwer Academic Publishers, Dordrecht, pp. 797-835; 843-844.) has treated the first problem, while the multidimensional direct methods developed in our group are aiming at the second.