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    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 nontrivial arcs have infinite length, fractals where all nontrivial 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 paper 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.

    This paper is dedicated to Christian Krattenthaler on the occasion of his 60th birthday.


    • 1. L. L. Cristea and B. Steinsky , Curves of infinite length in 4 × 4-labyrinth fractals, Geom. Dedicata 141 (2009) 1–17. Crossref, ISIGoogle Scholar
    • 2. L. L. Cristea and B. Steinsky , Curves of infinite length in labyrinth fractals, Proc. Edinb. Math. Soc. Ser. (2) 54(2) (2011) 329–344. Crossref, ISIGoogle Scholar
    • 3. L. L. Cristea and B. Steinsky , Mixed labyrinth fractals, Topology Appl. 229 (2017) 112–125. CrossrefGoogle Scholar
    • 4. L. L. Cristea and G. Leobacher , On the length of arcs in labyrinth fractals, Monatsh. Math. 185(4) (2018) 575–590. Crossref, ISIGoogle Scholar
    • 5. L. L. Cristea and G. Leobacher, Supermixed labyrinth fractals, J. Fractal Geom. Accepted for publication (2018), Google Scholar
    • 6. K.-S. Lau, J. J. Luo, and H. Rao , Topological structure of fractal squares, Math. Proc. Cambridge Philos. Soc. 155(1) (2013) 73–86. Crossref, ISIGoogle Scholar
    • 7. K. Falconer , Fractal Geometry, Mathematical Foundations and Applications, 3rd edn. (John Wiley & Sons, Hoboken, NJ, 2014). Google Scholar
    • 8. R. D. Mauldin and S. C. Williams , Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309(2) (1988) 811–829. Crossref, ISIGoogle Scholar
    • 9. U. Freiberg, B. M. Hambly and J. E. Hutchinson , Spectral asymptotics for V-variable Sierpinski gaskets, Ann. Inst. Henri Poincaré Probab. Stat. 53(4) (2017) 2162–2213. Crossref, ISIGoogle Scholar
    • 10. Z. Zhu, Y. Xiong and L. Xi , Lipschitz equivalence of self-similar sets with triangular pattern, Sci. China Math. 54(5) (2011) 1019–1026. Crossref, ISIGoogle Scholar
    • 11. M. Samuel, A. V. Tetenov and D. A. Vaulin , Self-similar dendrites generated by polygonal systems in the plane, Sib. Èlektron. Mat. Izv. 14 (2017) 737–751. Google Scholar
    • 12. A. A. Potapov, V. A. German and V. I. Grachev , Fractal labyrinths as a basis for reconstruction planar nanostructures, in 2013 Int. Conf. Electromagnetics in Advanced Applications (ICEAA), Turin, Italy, 2013, pp. 949–952, CrossrefGoogle Scholar
    • 13. 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 Int. Radar Symp. (IRS), Vol. 2, Dresden, Germany, 2013, pp. 941–946, ISSN: 2155–5745. Google Scholar
    • 14. A. A. Potapov and W. Zhang , Simulation of new ultra-wide band fractal antennas based on fractal labyrinths, in 2016 CIE Int. Conf. Radar (RADAR), China, Guangzhou, 2016, pp. 1–5, CrossrefGoogle Scholar
    • 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, J. Int. Sci. Publ. Mater. Methods Technol. 11 (2017) 492–512. Google Scholar
    • 16. A. Jana and R. Edwin García , Lithium dendrite growth mechanisms in liquid electrolytes, Nano Energy 41 (2017) 552–565. Crossref, ISIGoogle Scholar
    • 17. A. Giri, M. D. Choudhury, T. Dutta and S. Tarafdar , Multifractal growth of crystalline nacl aggregates in a gelatin medium, Crystal Growth Design 13(1) (2013) 341–345. Crossref, ISIGoogle Scholar
    • 18. S. Tarafdar, A. Franz, C. Schulzky and K. H. Hoffmann , Modelling porous structures by repeated Sierpinski carpets, Physica A 292(1–4) (2001) 1–8. Crossref, ISIGoogle Scholar
    • 19. S. Seeger, K. H. Hoffmann and C. Essex , Random walks on random Koch curves, J. Phys. A: Math. Theor. 42(22) (2009) 11. Crossref, ISIGoogle Scholar
    • 20. G. Edgar , Measure, Topology and Fractal Geometry, 2nd edn. (Springer, New York, NY, 2008). CrossrefGoogle Scholar
    • 21. G. A. Edgar and R. D. Mauldin , Multifractal decompositions of digraph recursive fractals, Proc. Lond. Math. Soc. (3) 65(3) (1992) 604–628. CrossrefGoogle Scholar
    • 22. K. Kuratowski , Topology, Vol. 2 (Academic Press, 1968). Google Scholar
    • 23. C. Tricot , Curves and Fractal Dimension, With a foreword by Michel Mendès France (Springer-Verlag, New York, NY, 1995), CrossrefGoogle Scholar
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