A Course of Pure Geometry by E. Askwith

By E. Askwith

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To date, not a single quantitative analysis of a QCS, comparable in its quality to the determination of a regular crystal structure, has been published. The first steps on this way have been done by the application of higher dimensional Patterson methods and contrast variation techniques (isomorphous replacement). One big problem, the lack of single crystals suitable for X - ray structure analysis, has been overcome recently with the detection of the new copper based stable icosahedral phases. Now the way is open for accurate single - crystal X - ray and even neutron diffraction experiments, including inelastic scattering for the study of the dynamical properties of quasi - crystals.

Analogously, the perpendicular-space shapes for a decoration with atoms on the mid-points of the faces follow as the intersection of three triacontahedra, one on the origin, one displaced over ~l and one displaced over ~i, resulting in a rhombic dodecahedron (RD). The perpendicular-space shape for an atom in the middle of a PR and a OR, respectively, is obtained as the intersection of four triacontahedra in both cases. The first is obtained from triacontahedra shifted over ~l, ~i, and ~l, respectively, defining a PR in perpendicular space.

The same as Fig. 5,0,0,0,0,0). 46 in Table 2. 2, as is to be expected. This again gives experimental evidence for the correctness of the 6D structure model presented here. DISCUSSION AND CONCLUSIONS In this paper the decorated 3D Penrose tiling as model for the structure of icosahedral quasicrystals is considered. The 6D atoms and their positions in the 6D unit cell were derived. This lead to the definition of five fundamental perpendicular-space shapes and nine secundary shapes (Table 1). The five fundamental shapes are completely determined by the requirement that the 3D structure must be a decorated Penrose tiling.

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