About the paper

coauthor Ligia-Loreta Cristea
language English
published in Fractals, 27, No. 08 (2019)
page 1950131
DOI 10.1142/S0218348X19501317
supported by FWF, project P28991-N35

Abstract

We define and study a class of fractal dendrites called triangular labyrinth fractals. For the construction we use triangular labyrinth pattern systems, consisting of two triangular patterns: a white and a yellow one. Correspondingly, we have two fractals: a white and a yellow one. The fractals studied here are self-similar, and fit into the framework of graph directed constructions. The main results consist in showing how special families of triangular labyrinth patterns systems, which are defined based on some shape features, can generate exactly three types of dendrites: labyrinth fractals where all non-trivial arcs have infinite length, fractals where all non-trivial arcs have finite length, or fractals where the only arcs of finite lengths are line segments parallel to a certain direction. We also study the existence of tangents to arcs. The article is inspired by research done on labyrinth fractals in the unit square that have been studied during the last decade. In the triangular case, due to the geometry of triangular shapes, some new techniques and ideas are necessary in order to obtain the results.

References

  1. D.H.N. Anh, K.H. Hoffmann, S. Seeger, S. Tarafdar, Diffusion in disordered fractals, EPL (Europhysics Letters), 70 (2005), 109—115.
  2. L.L. Cristea, G. Leobacher, On the length of arcs in labyrinth fractals, Monatsh. Math., 185 (2018), 575—590.
  3. L.L. Cristea, G. Leobacher, Supermixed labyrinth fractals, J. Fractal Geom., zur Veröffentlichung angenommen.
  4. L.L. Cristea, B. Steinsky, Curves of infinite length in 4×4-labyrinth fractals, Geom. Dedicata, 141 (2009), 1—17.
  5. L.L. Cristea, B. Steinsky, Curves of infinite length in labyrinth fractals, Proc. Edinb. Math. Soc., II. Ser., 54 (2011), 329—344.
  6. L.L. Cristea, B. Steinsky, Mixed labyrinth fractals, Topology Appl., 229 (2017), 112—125.
  7. G. Edgar, Measure, topology, and fractal geometry. 2nd ed., New York, NY: Springer, 2008.
  8. G Edgar, R.D. Mauldin, Multifractal decompositions of digraph recursive fractals, Proc. Lond. Math. Soc. (3), 65 (1992), 604—628.
  9. K. Falconer, Fractal geometry. Mathematical foundations and applications. 3rd ed., Hoboken, NJ: John Wiley & Sons, 2014.
  10. U. Freiberg}, B. M. Hambly, J.~E. Hutchinson, Spectral asymptotics for V-variable Sierpinski gaskets, Ann. Inst. Henri Poincar&aacut;, Probab. Stat., 53 (2017), 2162—2213.
  11. A. Giri, M. Dutta Choudhury, T. Dutta, S. Tarafdar, Multifractal growth of crystalline nacl aggregates in a gelatin medium, Crystal Growth & Design}, 13 (2013), 341-—345.
  12. A.Jana, R.E. Garc&iacut;a, Lithium dendrite growth mechanisms in liquid electrolytes, Nano Energy, 41 (2017), 552—565.
  13. K. Kuratowski, Topology, Volume II, Academic Press, 1968.
  14. R.D. Mauldin, S.C. Williams, Hausdorff dimension in graph directed constructions, Trans. Am. Math. Soc., 309 (1988), 811—829.
  15. A. Potapov and V. Potapov, Fractal radioelement's, devices and systems for radar and future telecommunications: Antennas, capacitor, memristor, smart 2d frequency-selective surfaces, labyrinths and other fractal metamaterials, Journal of International Scientific Publications: Materials, Methods & Technologies, 11 (2017), 492—512.
  16. A.A. Potapov, V.A. German, and V.I. Grachev, Fractal labyrinths as a basis for reconstruction planar nanostructures, in: 2013 International Conference on Electromagnetics in Advanced Applications (ICEAA), Torino, 2013, 949—952.
  17. A.A. Potapov, V.A. German, and V.I. Grachev, "Nano -" and radar signal processing: Fractal reconstruction complicated images, signals and radar backgrounds based on fractal labyrinths, in: 2013 14th International Radar Symposium (IRS) volume~2, Dresden, 2013, 941—946.
  18. A.A. Potapov, W. Zhang, Simulation of new ultra-wide band fractal antennas based on fractal labyrinths, in: 2016 CIE International Conference on Radar (RADAR), Guangzhou, 2016, 1—5.
  19. M. Samuel, A.V. Tetenov, D.A. Vaulin, Self-similar dendrites generated by polygonal systems in the plane, Sib. Èlektron. Mat. Izv., 14 (2017), 737—751.
  20. S. Seeger, K.H. Hoffmann, C. Essex, Random walks on random Koch curves, J. Phys. A, Math. Theor., 42 (2009), 225002.
  21. S. Tarafdar, A. Franz, C. Schulzky, K.H. Hoffmann, Modelling porous structures by repeated Sierpinski carpets, Physica A, 292 (2001), 1—8.
  22. C. Tricot, Curves and fractal dimension. With a foreword by Michel Mendès France. Transl. from the French, New York, NY: Springer, 1995.
  23. Z. Zhu, Y. Xiong, L. Xi, Lipschitz equivalence of self-similar sets with triangular pattern, Sci. China, Math., 54 (2011), 1019—1026.

Links

Fractals
Austrian Science Foundation (FWF)
Homepage of the project P28991