Crystallography of quasicrystals: The problem of restoration of broken symmetry

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In the paper, the concept of fuzzy packings of the structure units have been introduced which throw a new light on the crystallography of quasicrystals. With this concept one constructs the sructure of 2D-quasicrystal on a base of pentragrid from the rhombi on one kind which constitute a lacy cover of the plane. The space symmetry of such a structure is a positional colour groups isomorphous to wreath product of two generalized groups of the prototype phase $$\begin{gathered}(T^1 \times T^5 \times T^{5^2 } \times T^{5^3 } \times T^{5^4 } ) \ltimes \bar 5m = (T_1 \times T_2 )^{C_5 } \ltimes C_{10v} = T^1 wr_q C_{10v} , \hfill \\T^1 = T_1 \times T_2 ,T^5 = T_2 \times T_3 ,T^{5^2 } = T_3 \times T_4 ,T^{5^3 } = T_4 \times T_5 ,T^{5^4 } = T_5 \times T_1 \hfill \\\end{gathered}$$ describling the symmetry of a long and a short orders of the pentagrid correspondingly. The same is true for the symmetry group of the Penrose pattern which is dual to those pentagrid, its transformation being positional rigid translations and rotations accompanied by the appropriate local transformations of the inflation-deflation types rearranging the internal structures of the “physical points”, the smallest domains of the pattern pocessing by the pentagonal colour point symmetry group. It is shown that colour and nD approaches constitute the isomorphic languages of description of the icosahedral quasicrystal phenomenon in the frame of common V (n) = V (3)E⊕ V (d)11