` CLOSED FRACTAL FORMS IN QUASICRYSTALLINE STRUCTURES WITH ROTATIONAL SUMMETRY OF 6-TH, 3-RD, 2-ND AND 1-ST ORDER
A.I.Lazarev
Institute of Organometallic Chemistry by G.A.Razuvaev of RAS, Tropinin street, 49, GSP-445, Nizhny Novgorod, 603600, Russia, E-mail: lazar@imoc.sinn.ru
The formation of structures in a nature and in the technological processes, artificially created by the people, happens in more often to non-equilibrium conditions and a corollary it is an aperiodicity of these structures. The quasicrystals were by an impressive example of aperiodic structures1. As now it has become clear, at least from a mathematical point of view, in models of flat quasicrystal structures (QCS) the rotational symmetries of any N-fold symmetry are possible with various of self-similarity or inflation - deflation factors (Q). Together with G.A.Domrachev and collaborators in ours recent works2,3 we have begun to consider models of various types of fractal structures with Q =(n +ml1/2)/k, where n, m, l, k - any numbers and have tried to apply such algebraic concepts as "group", "ring" and "field" for modeling various types of aperiodic structures, in particular, quasicrystals4. Earlier we have paid attention to origin of the closed fractal shapes (CFS) in QCS and have shown a possibility of their classification for rotation symmetries of any folds5-10. Examples of constructing the fractal QCS and origin of CFS in them for structures with rotation symmetries 1-st, 2-nd, 3-rd and 6-th order and Q = 3 were considered by us in works6,7. By the present message we again pay attention to necessity of study of the CFS for structures with rotational symmetry 1-st, 2-nd, 3-rd and 6-th orders. There are some reasons: a) they are conventional crystallographic symmetries, but they, on our sight, were not considered from a quasicrystalline point of view; b) it is possible to receive symmetries of this type which are less complicated, and QCS with CFS in them may be tiled in a plane by only type of figures by equilateral rhombs with acute angles between sides being equal to 2p/6 radians; c) the factors of self-similarity in structures can be submitted by natural numbers (Q = 3, 4, 5...), in contrast to more complicated irrational or even transcendental Q; d) if in QCS the symmetry is expressed with a natural number N, divided by 6 without a remainder, in such structures the local symmetries 1, 2, 3 and 6-th are possible (for example, in a series of numbers..., 1997, 1998, 1999..., numbers 1997 and 1999 are prime numbers - "twins", and average value between prime numbers - "twins", number 1998, is composite and is also divided by 1, 2, 3, 6...)10. Let's remind principles of constructing the fractal QCS for rotational symmetries 1-st, 2-nd, 3-rd and 6-th orders6,7 (for example, for a deflation model): I) the equilateral rhombs on triangular or hexagonal lattices can be connected with one another by only limited number of modes (five); II) the global symmetry center of a structure is selected; III) then, one mode of connection of rhombs is selected from these five modes, that is the symmetry for constructing the QCS is selected; IV) a factor of self-similarity Q is set; V) the decorating the rhomb sides of the greater size by Kox lines, for example, segments of polygonal lines with lengths equal 1/Q (Q = 3, 4, 5,... is produced); VI) the chosen number of generations of fractal frames is created; VII) "pores (emptiness)" in fractal frames are filled in without hollows and overlapping with rhombs of junior generations by one from possible modes; VII) the procedure of constructing is finished by tiling a structure with rhombs of minimum sizes (Fig.1a.).
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Fig.1. On Fig.1a the
deflation model of three generations of fractal QCS from
self-similar rhombs connected by obtuse tops in a global symmetry
center of a structure is shown. Sides of rhombs are decorated by
Kox lines. Factor of self-similarity Q = 5. On Fig.1a in the same
structure, as on Fig.1b instead of rhomb sides the short
diagonals of these rhombs are shown. The development of the CFS
on several generations of a fractal QCS is observed.
In the structures we can see the CFS derivated from rhombs, connected among themselves in a defined sequence. So, if instead of rhombs sides (Fig.1a) to mark short diagonals, the structure accepts other feature (Fig.1b.) and the CFS already remind models like to "chemical branched molecules". The initial triangular translation lattice, on which the construction begins, can be considered as a sum of two fractal aperiodic sublattices: one of which is derivated by rhomb sides (Fig.1a), and other (Fig.1b) by short diagonals of the same rhombs. In that and other case it is possible to select the CFS. The studying of the CFS in the QCs will help to understand deeply general principles of structural self-organization of condensed matter. The CFS in particular can serve as structure models in various non-equilibrium processes of solid phase formation.
I am grateful to G.A.Domrachev, A.Yu. Sukhanov, A.P.
Matveyev for discussion of results, as well as to A.A.Zaytsev
(graduate student). This work was supported by the RFBR
(96-15-97455).