Symmetry and hierarchical structure described in the space higher than three dimension

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This project has been carried out since April in 1992, and proceeded for two years. The results obtained in this period are as follows : 1.the results obtained by research of individual members and of their co-operation, and 2, the discussion meetings held in the above period with the members of the project and investigators related to the title.In the former, three categories of studies have been carried out : (1)structural studies on incommensurate phases, on quasicrystals and on amorphous materials, (2)studies on symmetries in the space higher than three dimensions, and (3)studies on physical properties. In category (1), the structures of Ca_2CoSi_2O_7 of high Tc superconductors in Bi-oxide series, of mica polytypes and of BaZnGeO_4 have been investigated. As fundamental contributions, a method to determine higher-dimensional super space group of composite crystals was established. Structures of quasicrystals in Al-Pb-Mn, Mg-Ga-Al-Zn, Al-Cu-Co and Al-Ni-Fe systems were studied by el … More ectron microscopy and electron diffraction methods. Structures of Ca-silicate glass, Ag_3AsS_3 glass, glasses of Y_2O_3-Al_2O_3-SiO_2 and tellurite glasses were investigated by X-ray diffraction and EXAFS methods. The mode of fluctuation of ethanol and water system was also clarified. In category (2), graphical geodesic of Penrose tiling was studied. Determination of point groups and space groups of incommensurate phases in (3+1)-dimension was newly proposed. A method of determination of super space-group of composite crystals was also established. In (3), studies on fluctuation of mixed solutions with diffraction data together with thermodynamic parameters were carried out and new and adequate interpretation of fluctuation was presented. Studies on mechanism of pattern formation at interface between different phase were also undertaken.In the latter, the discussion meeting on amorphous structures was held in January, 1993, and many specialists on the problem attended the meeting and discussed eagerly. The second discussion meeting was held in November, 1993, and the attendants reported results of their research and also discussed on the future development of the studies on symmetry and structure of matter in the space higher than three dimension. Less