• 1 Grodno, Belarus, Faculty of physics and technology – Yanka Kupala State University of Grodno, Belarus,
  • 2 Molder, ltd – Grodno, Belarus,
  • 3 Faculty of Innovative Technologies of Mechanical Engineering – Yanka Kupala State University of Grodno, Belarus,


The physical properties of nanoparticles as modifiers are depend on their geometrical characteristics. These objects have 5- and 10-axis symmetry which is forbidden for crystals. The 3×3 matrices-generators of point groups of rotation in crystallographical and crystallophysics basises have as matrix elements 0 and ±1, except groups hexa- and trigonal in H-basis crystals. But these lattices too have 0 and ±1 as matrix elements for matrical representation of point moving in crystallographical basis. For describing the point groups of pentagonal and decagonal symmetries, instead of crystal lattices the so-named general regular lattices (GRL). The two dimensional GRL is known as Penrose’s sets. For 3-d pentagonal sets there are 14 groups of point symmetry, which are not crystallographic because their elements may be golden ratio.


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