The structures arising in spaces of various dimensions with simultaneous normal partitioning of spaces and their hierarchical fillings are considered. The conditions for the appearance of translational symmetry in these structures are investigated. It is shown that simultaneous hierarchical filling and normal tiling in three-dimensional spaces do not lead to the formation of translational symmetry. Such consistent transformations lead to many elements of translational symmetry in spaces of higher dimension. The higher the dimension of space, the more complex the emerging structure and the more symmetry the elements.
TopDiffractions Patterns Of Quasicrystals
Finding in 1982 of ordered structures deprived (as it seemed) of translational symmetry (Shechtman et al., 1984), next called “quasi - crystals”, had marked the beginning of numerous cycles of papers and books devoted to the experimental and theoretical study of these unusual materials. Later it was found that the diffraction patterns of quasicrystals have a latent periodicity (Zhizhin, 2014), if we consider the diffraction pattern as a projection of a structure from a space of higher dimension. Figure 1 shows a typical diffraction pattern of intermetallic compounds.
Figure 1. Electron diffraction pattern of compound Al72Ni20Co8(Eiji Abe et al., 2004)
Similar structures have other intermetallic compounds involving transition metals, for example: Al6Mn (Shechtman et al., 1984), Al70Fe20W10 (Mukhopadhyay et al., 1993), Ti54Zr26Ni20 (Zhang & Kelton, 1993).
Figure 10.1 shows that the luminous points, which are a reflection of light from the impact of the electron beam, form five families of parallel lines oriented with respect to each other at angles of a multiple of 72 degrees. The distances between the parallel lines and the angles are determined by the golden section. A geometrical model of the structure of the diffraction patterns of quasi - crystals was constructed (Shevchenko, Zhizhin & Mackey, 2013a; Shevchenko, Zhizhin & Mackey, 2013b; Zhizhin, 2014; Zhizhin & Diudea, 2016). It was shown that the elementary cell of this geometric structure is a polytope of dimension 4, which was called a gold hyper - rhombohedron. This cell is plotted on the diffraction pattern in Figure 1 with solid segments of light lines. I can see that it passes through the luminous points of the diffraction pattern observing its geometry. This cell fills the entire space with the translation reflected by the diffraction pattern. To determine the regularities of this cell, it is depicted in Figure 2 on an enlarged scale.
Figure 2. Golden hyper - rhombohedron
The polytope in Figure 10.2 consists of 8 three-dimensional polyhedrons whose faces are determined by construction with a golden section. Namely, all two - dimensional faces represent the same diamonds with angles