Fourier images of fractal objects

Abstract. An experimental investigation of diffraction of the collimated light beam (with a wavelength of 0.63 µm) in the zone of Fraunhofer by obtained with computer images of fractals, which are transferred onto a transparent film using an imagesetter with a resolution of 1333 points/cm (3386 dpi) and the spot size of 7.5 µm. Diffraction pattern were visually observed on the screen and recorded using a digital camera and then transferred to the computer for processing. The objects of study were various prefractals of 4th-9th generations: the "classic" (the Sierpinski carpet, a fractal of Vicsek, the Koch snowflake and less known (eg. of L-systems). Observed in experiments diffraction patterns were compared with digitally obtained ones that is, with the Fourier-images of the prefractal pictures, approximated by a mesh function on a uniform square grid at different values of the used in cflculation parameter p, determining the ratio between the ratio of the grid period and the smallest prefractal element size. The relation between the values of the parameters p and the degree of compliance of a digital diffraction patterns to the observed experimentally was determined.

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“…Afterwards similar numerical methods have been used for many other fractal objects [10][11][12][13][14][15].…”

Section: Procedures For Fourier-images Determinination Of Prefractal Pmentioning

confidence: 99%

Light Diffraction by Fractals: Comparison of Experimental Data With the Obtained by Numerical Methods Fourier Images of the Object Pictures

Arzamastseva¹,

Evtikhov²,

Lisovsky³

et al. 2017

RENSIT

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“…Using digitized images were obtained Fourier images for many flat geometric and algebraic fractals: the Sierpinski carpet, H-fractals, Peano curves, bifractals, L-system fractals, Mandelbrot sets [28,32,[33][34][35].…”

Section: Obtaining a Fourier-images Of The Koch Curves And Snowflakesmentioning

confidence: 99%

“…It is not necessary to ignore the aesthetic factor: often talk about the beauty of fractals [27], but the beauty of their Fourier images is just as impressive [28].…”

Section: Introductionmentioning

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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