Vol. 8, №1, 2016

FRACTALS IN PHYSICS

FAMILY OF GENERALIZED TRIADIC KOCH FRACTALS: DIMENTIONS AND FOURIER IMAGES
^1,2Galina V. Arzamastseva, ¹Mikhail G. Evtikhov, ¹Feodor V. Lisovsky, ¹Ekaterina 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
²Modern Humanitarian Academy, http://www.muh.ru
109029 Moscow, Russian Federation
arzamastseva@mail.ru, emg20022002@mail.ru, lisf@df.ru, mansvetova_eg@mail.ru

Received 17.11.2015

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

RENSIT, 2016, 8(1):81-90 DOI: 10.17725/rensit.2016.08.081

REFERENCES

Koch H. von. Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire. Arkiv för Matematik, Astronomi och Fysikk, 1904, 1:681-704.
Takagi T. A simple example of the continuous function without derivative. Proc. Phys.-Math. Japan, 1903, 1:176–177.
Hildebrandt TH. A simple continuous function with a finite derivative at no point. Amer. Math. Monthly, 1933, 40(9):547-548.
de Rham G. Sur un exemple de fonction continue sans dériveée. Enseign. Math., 1957, 3:71-72.
Vepstas L. A gallery of de Rham curves. linas@linas.org http://www.linas.org/math/de_Rham.pdf, 2006.
Arzamastseva GV, Evtikhov MG, Lisovsky FV, Lukashenko LI. Komp'yuternoe modelirovanie difraktsii sveta na fraktal'nykh domennykh strukturakh [Computer simulation of light diffraction on fractal domain structures]. Trudy XIX Mezhdunar. sckoly-seminara “Novye magnitnye materialy mikroelektroniki”, Moscow, Lomonosov MSU, 2004:632-634 (in Russ.).
Arzamastseva GV, Evtikhov MG, Lisovsky FV, Mansvetova EG, Temiryazeva MP. Amorfizatsiya biperiodicheskikh domennykh struktur v kvaziodnoosnykh magnitnykh plenkakh kriticheskoy tolschiny [Amorphization of periodic domain structures in quasi uniaxial magnetic films of the critical thickness]. ZHETF, 2008, 134(2):282-290 (in Russ.).
Arzamastseva GV, Evtikhov MG, Lisovsky FV, Mansvetova EG. Furie obrazy fraktal'nykh obektov [The Fourier images of fractal objects]. Izvestiya RAN, ser. fiz., 2010, 74(10):1430-1432 (in Russ.).
Alain C, Cloitre M. Optical diffraction on fractals. Phys. Rev. B, 1986, 33(5):3566-3569.
Sakurada Y., Uozumi J., Asakura T. Diffraction fields of fractally bounded apertures. Optical review, 1994, 1(1):3-7.
Uozumi J, Kimura H, Asakura T. Fraunhofer diffraction by Koch fractals. J. Mod. Optics, 1990, 37(6):1011-1031.
Zinchik AA, Muzychenko YaB, Stafeev SK. O printsipakh amplitudnoy i amplitudno-fazovoy prostranstvennoy fil'tratsii [On the principles of the amplitude and phase-amplitude spatial filtering]. Izv. VUZov. Priborostroenie, 2007, 50(7):46-52 (in Russ.).
Zinchik AA, Muzychenko YaB, Smirnov AV, Stafeev SK. Raschet fraktal'noy razmernosti regulyarnykh fraktalov po kartine difraktsii v dal'ney zone [Calculation of fractal dimension of the regular fractals on the diffraction pattern in the far zone]. Nauchno-tekhnichesky vestnik SPbGU ITMO, 2009, 2(60):17-24 (in Russ.).
Mandelbrot BB. The fractal geometry of nature. New York, WH Freeman, 1983.
Pietronero L, Tosatti E (eds.) Fractals in physics, North-Holland, Amsterdam, 1986.
Feder J. Fractals. N.Y.-London, Plenum Press, 1988.
Schroeder M. Fractals, chaos, power laws. Minutes from an infinite paradise. N.Y., WH Freeman, 1992.
Stoyan D, Stoyan H. Fractals, random shapes and point fields. Chichester, John Wiley and Sons, 1994, 399 p.
Peitgen H-O, Jürgens H, Saupe D. Chaos and fractals. New frontiers of science. New York, Springer-Verlag, 2004, 864 p.
Gouyet J-F. Physics and fractal structures. Paris, Ecole Polytechnique, 1995, 247 p.
Arzamastseva GV, Evtikhov MG, Lisovsky FV, Mansvetova EG. Komp'yuternoe modelirovanie difraktsii Fraungofera na H-fraktalakh i krivykh Peano [Computer simulation of Fraunhofer diffraction at the H-fractals and Peano curves]. Elektromagnitnye volny i elektronnye sistemy, 2012, 7:8-58 (in Russ.).
Cohen N. Fractal antennas: Part 1. Communications Quarterly, Summer 1995:7-22.
Cohen N. Self-similarity and the geometric requirements for frequency independence in antennae. Fractals, 1999, 7(1):79-84.
Best SR. The Koch fractal monopole antenna: The significance of fractal geometry in determin-ing antenna performance. Proceedings of the 2001 Аntenna Аpplications Symposium. Allerton Park Monticello, Illinois, 2000.–www.ecs.umass.edu/ece/allerton/papers2001/2001-p194.pdf.
Vinoy KJ, Abraham JK, Varadan VK. On the relationship between fractal dimension and the performance of multi-resonant dipole antennas using Koch curves. IEEE Transactions on Antennas and Propagation, 2003, 51(9):2296-2303.
Karaboikis M, Soras C, Tsachtsiris G, Makios V. Four-element printed monopole antenna systems for diversity and MIMO terminal devices. Proceedings of the 17th International Conference on Applied Electromagnetics and Communications, Dubrovnik, 2003:193-196.
Tsachtsiris G, Soras C, Karaboikis M, Makios V. A printed folded Koch monopole antenna for wireless devices. Microwave and Optical Technology Letters, 2004, 40(5):374–378.

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