Koch fractals in physical optics and their Fraunhofer diffraction patterns

Horváth, P.; Šmı́d, Petr; Vašková, Ivana; Hrabovský, M.

doi:10.1016/j.ijleo.2008.06.010

Cited by 21 publications

(6 citation statements)

References 9 publications

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“…So, when the things are enlarged, an accurate similarity exists among their elements and this similarity continues in a limited or unlimited end [29]. One of the most famous fractals is called Koch, which is designed by Helge von Koch [8]. In this fractal a segment of unit length is considered and then it is divided into three parts.…”

Section: Fractalmentioning

confidence: 99%

“…So, each shape in fractal is the ratio of the logarithm of the number of the shapes of the self-similarity to the enlargement factor. The fractal dimension is calculated according to the equation (1) [8].…”

Section: Fractalmentioning

confidence: 99%

“…Fractals use self-similarity and repetitions to create shapes in unlike the Euclid shapes [7]- [8]. The fractal shapes are erratic and also the disorder rate is fixed for any scale.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of the Fractal Koch Method in Computer Games Development

Gharehchopogh¹,

Maleki²,

Sadouni³

2014

IJCGA

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Lately according to the software and hardware progress, the development of the computer games has been so much. As the development of the computer games is one of the most favorite projects of computer sciences. A requirement for development of the computer games is the computer graphics programming science and the mathematical calculations. A calculation method is fractal Koch which uses the selfsimilarity specifications of itself. Formation of fractals takes place by the repetitions, so it is possible to

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Section: Fractalmentioning

confidence: 99%

Section: Fractalmentioning

confidence: 99%

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Analysis of the Fractal Koch Method in Computer Games Development

Gharehchopogh¹,

Maleki²,

Sadouni³

2014

IJCGA

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“…The features of Fraunhofer diffraction for other fractal objects (Vicsek fractal, Cantor bar, Sierpinski carpet, etc. [19][20][21][22][23][24]) were also investigated.…”

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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“…The model presented in this paper is intended for serving the educational process of engineering students taking a Heat and Waves Physics course. The models used in this paper are well known, but its simplification for computational simulation purposes as well as the application of the simplified model on educational processes, are still an area of interest .…”

Section: Introductionmentioning

confidence: 99%