Abstract
Categories of the generalized crystallography, where structures are tiled by a large number of identical cells, include quasi-identity and quasi-equivalence. The hierarchy of the organization levels is considered, involving the n-D space. In the theory of proportions, the irrational numbers, such as e, ί, π, and τ, play an important role. The golden-ratio number τ is fundamental for the geometry of structures with five- or tenfold symmetry, eventually called quasicrystals. The fact that these numbers are irrational suggests that in the Euclidean space E3, we observe the projections of the fundamental polyhedra from a higher-dimensional space. Thus, complex crystal (or quasicrystal) structures are only cells of some much more complex assemblies of higher dimensionality.
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Shevchenko, V.Y., Zhizhin, G.V., Mackay, A.L. (2013). On the Structure of Quasicrystals in a Higher-Dimensional Space. In: Diudea, M., Nagy, C. (eds) Diamond and Related Nanostructures. Carbon Materials: Chemistry and Physics, vol 6. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6371-5_17
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DOI: https://doi.org/10.1007/978-94-007-6371-5_17
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