Symmetry and features of structure in materials described in the space higher than three dimensions.

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This project has been carried out since the end of Oct. in 1990, and proceeded for 17 months. The results obtained in this period are as follows :1. the results by individual works of the members and by their cooperative works, and 2. the discussion meetings held in the above period with the investigators related to the proicclIn the former, three categories of studies have been carried out : (1) structural studies on incommensurate phases, studies on quasicrystals and on amorphous materials, (2) interpretation of feature of the structures and their properties in higher dimensions, and (3) studies on symmetries in the space higher than three dimensions. In category (1) , the structures of Ca_2CoSi_2O_7, of high Tc superconductors in Bi-oxide series and of zirconium titanates have been analyzed. As fundamental contributions, a computer program for structure determination of composite crystals was newly developed and a method to determine four-dimensional space group by convergent-beam e … More lectron diffraction was also established. Structures of quasicrystals in Al-Ni-Co and Al-Cu-Co systems were clarified by high-resolution electron microscopy and those in Al-Co-Ni and Al-Cu-Li systems were elucidated by X-ray diffraction method. The coordinations of transition metals in Ca (Mg, M) (Si, Al)_2O_6 (M : Mn, Fe, Co, Ni) glasses were studied by X-ray diffraction and EXAFS methods. A method to determine fluctuation of solutions by small angle X-ray diffraction was also established and was successfully applied to mixtures of alcoholic solutions and water. In category (2), a new concept "a graphical geodesic" was proposed. This was applied to characterize intrinsic regularity of the triangulated Penrose tilings which are closely related to the structures of qusicrystals. In (3), Studies on super space groups of composite crystals, on non-characteristic orbits of space groups, on statistics of point groups of crystals and on characters of packing of ellipses have been carried out.In the latter, the first discussion meeting was held in January, 1991, and the second one in February, 1992. The member of this project together with the investigators related to the project attended the meeting and the results of their studies were presented. The discussions were fruitful. Less