Abstract
Categories of the generalized crystallography, where structures are tiled into a large number of identical cells, include quasi-identity and quasi-equivalence. The hierarchy of organization levels including the n-dimensional space is considered. In the theory of proportions, irrational numbers, such as e, π, τ, etc., play an important role. The golden ratio τ is basic to the geometry of structures with five- or tenfold symmetry that are called quasi-crystals. The irrational nature of these numbers probably means that in the Euclidean space E3, we deal with projections of the fundamental polyhedra from a higher-dimensional space. Thus, the structures of quasi-crystals in the three-dimensional space represent slices of much more complex assemblies in the higher-dimensional space.
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Dedicated to Academician S. M. Aldoshin on the occasion of his 60th birthday (“Ubi materia — ubi geometria” (Johannes Kepler)).
Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 2, pp. 0269–0274, February, 2013.
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Shevchenko, V.Y., Zhizhin, G.V. & Mackay, A.L. On the structure of quasi-crystals in the higher-dimensional space. Russ Chem Bull 62, 265–269 (2013). https://doi.org/10.1007/s11172-013-0039-8
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DOI: https://doi.org/10.1007/s11172-013-0039-8