Fractal images of quasicrystals as an example of Penrose lattice

Yudin, V. V.; Karygina, Yu. A.

doi:10.1134/1.1420819

Cited by 10 publications

(19 citation statements)

References 2 publications

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“…In our papers [9][10][11][12] a set of studies dedicated to searching the general representation method of quasi-crystalline lattices and cellular systems of mesodefects of quartz glasses and amorphous films was carried out. The formalism for mapping these objects into coordination tree Cayley graphs was introduced [10,11].…”

Section: Informodynamical Methods For Studying Of Cellular Lattice Symentioning

confidence: 99%

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Penrose tiling fractality in coordination Cayley’s tree graphs representation

Mihalyuk

Titov

Yudin

2010

Physica A: Statistical Mechanics and its Applications

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Section: Informodynamical Methods For Studying Of Cellular Lattice Symentioning

confidence: 99%

“…It is necessary to give such representation of lattice, cellular structures, where both ''tiles'', and coordination components would be peer. Generalized tree Cayley graphs (TCG) [9][10][11] are the correct such objects. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Penrose tiling fractality in coordination Cayley’s tree graphs representation

Mihalyuk

Titov

Yudin

2010

Physica A: Statistical Mechanics and its Applications

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“…And as the axioms are now perceived assertion that mosaics are two-dimensional analogs of quasicrystals, and to the last with any quasisymmetry can be put into correspondence associated fractal structures [43,44]. Methods of fractal geometry are now used to describe models of quasi-crystalline structures, and tree-graph procedure -to originate them [44,45].…”

Section: Discussionmentioning

confidence: 99%

Foirier Images Quasisymmetry of Generalized Triadic Koch Prefractals

Arzamastseva¹,

Evtikhov²,

Lisovsky³

et al. 2016

RENSIT

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Abstract. The study of the Fourier-images properties was made by numerical methods for the family of flat triadic geometric prefractals with generator in the form of symmetric four-stage broken line with an arbitrary angle at the apex between the central units and the initiator in the form of a straight line (Koch curve) or in the form of an equilateral triangle (the Koch snowflake). To obtain the Fourier images the pictures of fractals were approximated by a grid function on a uniform grid with cells small enough for adequate mapping of high generation prefractal details, and then were digitized in order to use fast Fourier transform for determination the values of the squared modules of the Fourier component, that is, the spectral intensity distribution of diffraction maxima in the Fraunhofer region. An analysis showed that for the values of the vertex or base angles equal to the integer fraction of 180 degrees, Fourier images are the same as for the perfect crystals with the symmetry axes of the 2-nd, 3-th, 4-th and 6-th order, or as for parcuet mosaics or quasicrystals with the axes of quasisymmetry of any order. Really in the Fourier images of the Koch curves with the initiator in the form of a straight line was observed axis of quasisymmetry from 3rd to 9 th and 11-th order. Similar to the above-described properties are also inherent to Fourier images of the Koch snowflake with the initiator in the form of an equilateral triangle. The configuration of the observed Fourier images can be approximately regarded as a radial-annular, at that in the peripheral ("lattice") of the images is dominated by the radial nature of the frequency distribution of diffraction reflections along the radius, and in the central ("fractal") -a ring with self-similarity. The lattice part has a kind of clustering: all the rays have a strong central chain of reflexes along the radii and parallel to it the weaker satellites on both sides. All Fourier images had the center of symmetry, which is an integral attribute of the diffraction patterns in the Fraunhofer zone for any objects, however, the rotational symmetry was not perfect: the positions of the diffraction reflexes when rotating images at angles that correspond to the order of the symmetry axis remain unchanged, but their intensity could vary. The cause of the observed features is that prefractals, unlike crystals, are not a continuum of point objects but two-dimensional set of equal length line segments with different orientation in space. In this set for the considered configurations of the generator it is possible to allocate several two-dimensional subsets with equally oriented segments, each of which contains a number of partial one-dimensional diffraction gratings formed by segments located along the same line. These parallel lattice in the general case contain a different number of segments, and the degree of filling and the distance between adjacent segments, determining the intensity and diffraction reflections distribution structure along the line, depend on the orie...

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“…This means that the stochastic operator undergoes a stretching or contraction in accordance with the norm of the distribution. Analysis of the percolation problem in entropic and divergent terms constitutes the infodynamical part of our theory [11], [13], [15], [30], [32], [34]. We believe that just the percolation dependences of these infodynamical functionals and accordingly their critical indices will help to obtain the general method of solving the problem of identifying the order type of grid structures and lattices.…”

Section: Percolation Of Infodynamical Functionals On Ctgsmentioning

confidence: 99%

“…This implies that the [2q×2p] quartet alphabet has certain linguistic functions. The above brief analysis allows proposing a hypothesis that the Penrose tiling can be interpreted as a two-dimensional text [31] on a three-level dual alphabet [32] with a tree-graph grammar. We end this section by considering a key point related to the entropy percolation on the Penrose CTG.…”

Section: Percolation Of the Entropy Functional On A Quartet Alphabet mentioning

confidence: 99%

Entropic measure of the order-disorder character in lattice systems in the representation of coordination Cayley tree graphs

2010

Self Cite

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We systematically expound the infodynamical method for analyzing lattice and grid systems. We establish the logic and algorithm for mapping given objects to coordination Cayley tree graphs and present their main properties. Tree graphs of grid systems are complicated objects, and the principle of cluster-type simplicial decomposition can be used to study them. Based on a simplicial decomposition, we construct the enumerating structures, from which we construct entropy-type functionals. We pose the percolation problem on Cayley tree graphs in a nonconventional sense, which may be considered for both enumerating structures and their entropies. The corresponding entropy percolational dependences and their critical indices can be considered sufficiently universal measures of order in lattice systems. The simpliciality also implies an analogy with the fractality principle. We introduce three types of fractal characteristics and give analytic expressions for fractal dimensions for the tangential and streamer representations and for the Mandelbrot shell.

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Fractal images of quasicrystals as an example of Penrose lattice

Cited by 10 publications

References 2 publications

Penrose tiling fractality in coordination Cayley’s tree graphs representation

Penrose tiling fractality in coordination Cayley’s tree graphs representation

Foirier Images Quasisymmetry of Generalized Triadic Koch Prefractals

Entropic measure of the order-disorder character in lattice systems in the representation of coordination Cayley tree graphs

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