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FAMILY of GENERALIZED TRIADIC KOCH FRACTALS: DIMENTIONS and FOURIER IMAGES 1,2 Galina V

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FAMILY of GENERALIZED TRIADIC KOCH FRACTALS: DIMENTIONS and FOURIER IMAGES 1,2 Galina V FRACTALS IN PHYSICS 81 FAMILY OF GENERALIZED TRIADIC KOCH FRACTALS: DIMENTIONS AND FOURIER IMAGES 1,2 Galina V. Arzamastseva, 1 Mikhail G. Evtikhov, 1 Feodor V. Lisovsky, 1 Ekaterina G. Mansvetova 1Kotel’nikov Institute of Radio-Engineering and Electronics, Fryazino Branch, Russian Academy of Science, http://fire.relarn.ru 141120 Fryazino, Moscow region, Russian Federation [email protected], [email protected], [email protected], [email protected] 2Modern Humanitarian Academy, http://www.muh.ru 109029 Moscow, Russian Federation Abstract. Fourier images of generalized triadic Koch fractals (curves and snowflakes) with variable vertex angle of generator were obtained by digital methods. The comparison of different methods of fractal dimensions determination using Fraunhofer diffraction patterns was made. Analysis of the size ratio dependence of central (fractal) and peripheral (lattice) parts of diffraction pattern both upon vertex angle at a fixed value of prefractal generation number and upon prefractal generation number at a fixed value of vertex angle was made. The features of Koch curves and Koch snowflakes Fourier images are discussed. Keywords: annular zones method, box-counting method, digital methods, Fourier image, fractal dimension, Fraunhofer diffraction, Koch curve, Koch fractal, Koch snowflake, method of circles, scaling factor, scaling invariance, symmetry. UDC 51.74; 535.42 Bibliography – 27 references Recieved 17.11.2015 RENSIT, 8(1):81-90 DOI: 10.17725/rensit.2016.08.081 CONTENTS: 1. INTRODUCTION 1. INTRODUCTION (81) Externally similar to the snowflake outline 2. DECLARATION OF PROCEDURE FOR KOCH fractal (grandparent of fractals family which FRACTALS FOURIER IMAGES OBTAINING (82) will be discussed below) was named in honor 3. METHODS OF KOCH FRACTALS DIMENSION of Swedish mathematician Helge von Koch, DETERMINATION (83) who first described it in published in 1904 year 4. FOURIER IMAGES OF KOCH CURVES WITH article the title of which in French sounds like VARIABLE VERTEX ANGLE OF GENERATOR "Of nontangent continuous curve to obtain an (83) elementary geometric constructions" [1]. 5. DEPENDENCE OF KOCH CURVES DIMENSION The Koch fractals were quite a complicated ON VERTEX ANGLE OF GENERATOR (84) genealogy. It turned out that a year before the 6. PROPERTIES OF KOCH CURVES FOURIER Japanese mathematician Takagi Teiji published IMAGES (85) an article with almost the identical title "Simple 7. PROPERTIES OF KOCH SNOWFLAKES FOURIER example of continuous function without IMAGES (87) derivatives" [2], where he described the so- 8. PROPERTIES OF NODIFIED KOCH CURVES called "blancmange" curve (currently referred FOURIER IMAGES (88) to as Takagi-Landsberg curve), which was the 9. CONCLUSION (88) nearest relative of the Koch snowflake. Then REFERENCIES (89) the family of curves under consideration was expanded in 1933 by Hildebrandt [3] and in 1957 RENSIT | 2016 | Vol. 8 | No. 1 82 GALINA V. ARZAMASTSEVA, MIKHAIL G. EVTIKHOV, FEODOR V. LISOVSKY, EKATERINA G. MANSVETOVA FRACTALS IN PHYSICS by de Rham [4]. The last extension was the most mode. Fourier images, obtained by digitized representative: the individual cases of curves de- pictures of domain structures, correspond Frame are such wellknown classical fractals like both the results of the theoretical calculation the Koch snowflake, spacefilling Peano curves, and the experimentally observed diffraction curves of Cesaro-Faber, Takagi-Landsberg, patterns [7]. In addition, similar numerical Levi, etc., see [5]. methods have been used for a number of other In the present work we studied the properties fractal objects [8], for which the phenomenon obtained by numerical methods Fourier images of diffraction have been well studied by and Hausdorff dimension spectra of the triad other authors [9-11]. The agreement obtained geometric fractals family with generator in the Fourier images (diffraction patterns) with the form of a symmetrical four-segment broken results of these experimental and theoretical line with an arbitrary angle between the central studies was quite good. Note that numerical units and with the initiator (broken line) in methods are quite simple and do not require the form of a straight line (Koch curve) or an analytical description of fractals, that was in the form of an equilateral triangle (Koch used, for example, the authors of the works snowflake). When constructing a fractal, at [12, 13]. each step the segments of the initial broken The first step in the implementation of used line are divided symmetrically relative to the numerical methods was to generate black-and- center into three equal (for α = 60°) or unequal white raster image of selected fractal objects (for α ≠ 60°) parts (hence "triad" in the name with the help of specially designed programs. of a family). Further images of fractals were approximated Used in the present work, the term "Fourier by a grid function on a uniform grid with image" actually refers to the 2D-graphic number of nodes n1 × n2, where the values n1 representation of distribution of the squared и n2 were chosen sufficiently large (up to 4096) modules of the component of the Fourier in order to ensure adequate approximation of transform for images of fractals (see figures in the smallest details and have the opportunity text). The resulting picture is close to that obtained to explore generation of prefractals with high by Fraunhofer diffraction of electromagnetic numbers of generations. For digitized pictures waves on fractals, which makes possible to run using fast Fourier transform we determined verification field experiments. the values of the Fourier component squared modules, i.e. the spectral distribution of 2. DECLARATION OF PROCEDURE intensity I of diffracted radiation in the zone FOR KOCH FRACTALS FOURIER of Fraunhofer. To display the intensity I of the IMAGES OBTAINING diffraction maxima on the 2D-plane, one can Used in the present work a procedure for use either a linear (or logarithmic) scale intensity determination of fractal object picture using different gray levels, or representation Fourier images (i.e., diffraction patterns in the of I values in the form of circles with radius Fraunhofer zone) using numerical methods, proportional to intensity (or logarithm of it), previously described and applied to study the where coefficient of proportionality is chosen fractallike domain structures in magnetic films for reasons of optimal illustrativity of images. [6], later was applied to the real test objects This article used the second method, modified (mono- and biperiodical domain structures), by additional Gaussian blur of circles displaying when it was possible in the visible range of the values of I. wavelengths to observe directly and photograph In the following sections, according to the diffraction pattern by the "transmission" made in the introduction comments of, No. 1 | Vol. 8 | 2016 | RENSIT FAMILY OF GENERALIZED TRIADIC KOCH FRACTALS: 83 FRACTALS IN PHYSICS DIMENTIONS AND FOURIER IMAGES − we'll use for Koch fractal with the generator I( r) = IeDf , (2) in the form of symmetrical relative to the k 0 from which it follows that the value of fractal middle fourbar broken line with an arbitrary dimension D equals the modulo of the angular angle between the central bars name "Koch f coefficient of the straight line, approximating curve", if the initiator is a line segment, and the dependence Ī(r ) in double logarithmic the name "Koch snowflake", if the initiator is k scale. an equilateral triangle (see Fig. 1). Note that to preserve the self-similarity of prefractal In the second method (hereinafter, method by iterations all the segments of generator of the annular zones) the diffraction plane in a form of broken line must have the same is divided into annular regions by concentric k length. When α → 0°, the broken line becomes system of circles with radii rk = r0m , where delta-shaped, when α → 180°, it merges with a r0 – is an initial radius and m > 1 is a constant initiator line; case α = 60° corresponds to the multiplier. Then the average resulting intensity classical Koch curve. of the diffracted radiation inside of each ring is calculated k+1 3. METHODS OF KOCH FRACTALS 2π mr0 1 Ir = Iρϕ,, ρ d ρ d ϕ DIMENSION DETERMINATION ( k ) 22k 2 ∫∫ ( ) π mr0 ( m−1) 0 mrk (3) Analysis of the diffraction pattern allows to 0 determine the Hausdorff dimension of fractals after that the dependence Ī(rk) in double studied by two different numerical methods. The logarithmic scale is built, and module of the slope first method (hereinafter, method of circles) based of the linear section of which determines the on a numerical determination of the average fractal dimension Df.. For self-similar fractals, resultant intensity of the diffracted radiation to discussed in this work fractals belong, the 2π rk value of coincides with the scaling factor of 1 Ir( ) = I(ρϕ, ) ρ d ρ d ϕ the elements in the transition between adjacent k π r 2 ∫∫ (1) k 00 hierarchical levels of the fractal, and the value in the circles of variable radius r = r + kδ k 0 r of r0 must be chosen so that each of the annular centered at the position of the main diffraction zones contained only congruent fragments of maximum (ρ and φ – polar coordinates in the the dif-fraction patterns [9-13]. plane of diffraction, r and δ – are the initial 0 r The dimension of the fractals can be radius and increment of radius as well = 0, 1, k determined also by nonspectral box counting 2, ...) and then using the formula technique, which uses a built in double logarithmic scale the dependence N(εk) = f(1/ εk), where N(εk) – needed to fully cover the fractal is the number of squares with side length ε k → 0 [14-19]. The value of dimension D k→∞ f is determined with an angular coefficient of dependence N(εk). 4. FOURIER IMAGES OF KOCH CURVES WITH VARIABLE VERTEX ANGLE OF GENERATOR In Fig. 2 and Fig. 3 the Fourier images (diffraction pattern) of Koch curves with Fig.
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