Vol. 8, №2, 2016

FRACTALS IN PHYSICS

DETECTION AND INVESTIGATION OF ANOMALOUS (UNDAMPED) THERMAL WAVES
Galina V. Arzamastseva, Mikhail G. Evtikhov, Feodor V. Lisovsky, Ekaterina G. Mansvetova
Kotel`nikov Institute of Radioengineering and Electronics, Fryazino Branch, Russian Academy of Sciences, http://fire.relarn.ru
1, Vvedensky sq., 141120 Fryazino, Moscow region, Russian Federation
arzamastseva@mail.ru, emg20022002@mail.ru, lisf@df.ru, mansvetova_eg@mail.ru

Received 06.12.2016

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 orientation of the lattice and the generation number of prefractal.

Keywords: digital methods, Fourier image, Fraunhofer diffraction, generator, initiator, Koch curve, Koch fractal, Koch snowflake, parquet mosaic, prefractal, quasiqristal, quasisymmetry, scaling invariance, self-similarity, symmetry

UDC 51.74; 535.42

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RENSIT, 2016, 8(2):207-214 DOI: 10.17725/rensit.2016.08.207

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