A Brief Survey of Paradigmatic Fractals from a Topological Perspective
Abstract
:1. Introduction
2. Key Dimension Numbers and the Classification of Fractals
3. Survey of Paradigmatic Fractals from Topological Perspectives
3.1. Self-Avoiding Koch Curves
3.2. Sierpiński Gasket Shape (Network versus Self-Avoiding Curve) and PCF Fractals
3.3. The Standard Sierpiński Carpets
3.4. The Space-Filling Curves
3.5. Cantor Set and Totally Disconnected Fractals
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Husain, A.; Nanda, M.N.; Chowdary, M.S.; Sajid, M. Fractals: An Eclectic Survey, Part-I. Fractal Fract. 2022, 6, 89. [Google Scholar] [CrossRef]
- Husain, A.; Nanda, M.N.; Chowdary, M.S.; Sajid, M. Fractals: An Eclectic Survey, Part-II. Fractal Fract. 2022, 6, 379. [Google Scholar] [CrossRef]
- Balankin, A.S. Fractional space approach to studies of physical phenomena on fractals and in confined low-dimensional systems. Chaos Solitons Fractals 2020, 32, 109572. [Google Scholar] [CrossRef]
- Mandelbrot, B.B. Les Objets Fractals: Forme, Hasard et Dimension; Flammarion: Paris, France, 1975. [Google Scholar]
- Weierstrass, K. Über Continuirliche Functionen Eines Reellen Arguments, die für Keinen Werth des Letzteren Einen Bestimmten Differentialquotienten Besitzen. In Ausgewählte Kapitel aus der Funktionenlehre. Teubner-Archiv zur Mathematik; Siegmund-Schultze, R., Ed.; Springer: Vienna, Austria, 1988; Volume 9. [Google Scholar] [CrossRef]
- Barański, K.; Bárány, G.; Romanowska, J. On the dimension of the graph of the classical Weierstrass function. Adv. Math. 2014, 265, 32–59. [Google Scholar] [CrossRef] [Green Version]
- Cantor, G. Uber unendliche, lineare Punktmannigfaltigkeiten V. Math. Ann. 1883, 21, 545–591, Reprinted in Cantor, G. Gesammelte Abhandlungen Mathematischen und Philosophischen Inhalts; Zermelo, E., Ed.; Springer: New York, NY, USA, 1980; pp. 165–209. [Google Scholar] [CrossRef] [Green Version]
- Smith, H.J.S. On the integration of discontinuous functions. Proc. Lond. Math. Soc. 1875, 1, 140–153. [Google Scholar] [CrossRef]
- Conway, J.B. A Course in Point Set Topology; Springer: New York, NY, USA, 2014. [Google Scholar] [CrossRef]
- Peano, G. Sur une courbe, qui remplit toute une aire plane. Math. Ann. 1890, 36, 157–160, Reprinted in Arbeiten zur Analysis und zur mathematischen Logik. Teubner-Archiv zur Mathematik; Springer: Vienna, Austria, 1990; Volume 13. [Google Scholar] [CrossRef]
- Humke, P.D.; Huynh, K.V. Finding keys to the Peano curve. Acta Math. Hungar. 2022, 167, 255–277. [Google Scholar] [CrossRef]
- Hilbert, D. Uber die stetige Abbildung einer Linie auf ein Flăchenstuck. Math. Ann. 1891, 38, 459–460, Reprinted in Dritter Band: Analysis Grundlagen der Mathematik Physik Verschiedenes; Springer: Berlin/Heidelberg, Germany, 1935. [Google Scholar] [CrossRef] [Green Version]
- Sagan, H. On the geometrization of the Peano curve and the arithmetization of the Hilbert curve. Int. J. Math. Education Sci. Tech. 1992, 23, 403–411. [Google Scholar] [CrossRef]
- Nisha, P.S.A.; Hemalatha, S.; Sriram, S.; Subramanian, K.G. P Systems for Patterns of Sierpinski Square Snowflake Curve. Punjab Univ. J. Math. 2020, 52, 11–18. Available online: http://journals.pu.edu.pk/journals/index.php/pujm/article/viewArticle/3679 (accessed on 13 July 2023).
- Sierpiński, W. Sur une nouvelle courbe continue qui remplit toute une aire plane. Bull. Acad. Sci. Crac. Sci. Math. Nat. Ser. A 1912, 462–478. [Google Scholar]
- von Koch, H. Sur un e courbe continue sans tangente, obtenue par une construction geometrique elementaire. Ark. Mat. Astron. Och Fys. 1904, 1, 681–704. [Google Scholar]
- Edgar, G. Classics on Fractals; Addison Wesley: Reading, MA, USA, 1993. [Google Scholar]
- Sierpiński, W. Sur une courbe dont tout point est un point de ramification. C. R. Acad. Paris 1915, 160, 302–305. [Google Scholar]
- Stewart, I. Four Encounters with Sierpinski’s Gasket. Math. Intell. 1995, 17, 52–64. [Google Scholar] [CrossRef]
- Sierpiński, W. Sur une Corbe Cantorienue qui contient une image biunivoquet et continué detoute Corbe doné. C. R. Acad. Paris 1916, 162, 629–632. [Google Scholar]
- Franz, A.; Schulzky, C.; Tarafdar, S.; Hoffmann, K.H. The pore structure of Sierpinski carpets. J. Phys. A Math. Gen. 2001, 34, 8751–8765. [Google Scholar] [CrossRef]
- Menger, K.; Brouwer, L.E.J. “Allgemeine Räume und Cartesische Räume”. Zweite Mitteilung: “Ueber umfassendste n~dimensionale Mengen”. In Selecta Mathematica; Springer: Vienna, Austria, 2002; pp. 89–92. [Google Scholar] [CrossRef]
- Balankin, A.S.; Bory-Reyes, J.; Luna-Elizarraras, M.E.; Shapiro, M. Cantor-type sets in hyperbolic numbers. Fractals 2016, 24, 1650051. [Google Scholar] [CrossRef]
- Broadbent, S.R.; Hammersley, J.M. Percolation processes. I. Crystals and mazes. Proc. Camb. Philos. Soc. 1957, 53, 629–641. [Google Scholar] [CrossRef]
- Stauffer, D. Scaling theory of percolation clusters. Phys. Rep. 1979, 57, 1–74. [Google Scholar] [CrossRef]
- Hausdorff, F. Dimension und äußeres Maß. Math. Ann. 1918, 79, 157–179. [Google Scholar] [CrossRef]
- Carathéodory, C. Über das lineare mass von punktmengen-eine verallgemeinerung das längenbegriffs. Nachrichten Ges. Der Wiss. Gött. 1914, 1914, 404–426. Available online: http://eudml.org/doc/58921 (accessed on 13 July 2023).
- Besicovitch, A.S. Sets of Fractional Dimensions (IV): On Rational Approximation to Real Numbers. J. Lond. Math. Soc. 2019, s1–s9, 126–131. [Google Scholar] [CrossRef]
- Besicovitch, A.S.; Ursell, H.D. Sets of Fractional Dimensions (V): On Dimensional Numbers of Some Continuous Curves. J. Lond. Math. Soc. 1937, s1–s12, 18–25. [Google Scholar] [CrossRef]
- Falconer, K.S. Techniques in Fractal Geometry; Wiley: Chichester, UK, 1997. [Google Scholar]
- Falconer, K.S. Fractal Geometry: Mathematical Foundations and Applications; Wiley: New York, NY, USA, 1990. [Google Scholar]
- Mandelbrot, B.B. The Fractal Geometry of Nature; Freemann: New York, NY, USA, 1982. [Google Scholar]
- Mandelbrot, B.B. How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 1967, 156, 636–638. [Google Scholar] [CrossRef] [Green Version]
- Mandelbrot, B.B. Chance, and Dimension; Freemann: New York, NY, USA, 1977. [Google Scholar]
- Hutchinson, J.E. Fractals and self-similarity. Indiana Univ. Math. J. 1981, 30, 713–747. Available online: https://www.jstor.org/stable/24893080 (accessed on 13 July 2023). [CrossRef]
- Barnsley, M.F. Fractals Everywhere; Academic Press: Orlando, FL, USA, 1988. [Google Scholar]
- Manin, Y. The notion of dimension in geometry and algebra. Bull. Amer. Math. Soc. 2006, 43, 139–161. [Google Scholar] [CrossRef]
- Kinsner, W. A Unified Approach to fractal Dimensions. J. Cogn. Inform. Nat. Intell. 2007, 1, 26–46. [Google Scholar] [CrossRef] [Green Version]
- Fernández-Martínez, M. A survey on fractal dimension for fractal structures. Appl. Math. Nonlin. Sci. 2016, 1, 437–472. [Google Scholar] [CrossRef]
- Bhattacharjee, S.M. What is dimension. In Topology and Condensed Matter Physics; Texts and Readings in Physical Sciences; Bhattacharjee, S., Mj, M., Bandyopadhyay, A., Eds.; Springer: Singapore, 2017; Volume 19. [Google Scholar] [CrossRef]
- Lehrbäck, J.; Tuominen, H. A note on the dimensions of Assouad and Aikawa. J. Math. Soc. Japan 2013, 66, 343–356. [Google Scholar] [CrossRef]
- Stanley, H.E. Application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media. J. Stat. Phys. 1984, 36, 843–860. [Google Scholar] [CrossRef] [Green Version]
- Nakayama, T.; Yakubo, K. Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations. Rev. Mod. Phys. 1994, 66, 381–443. [Google Scholar] [CrossRef]
- Barnsley, M.; Vince, A. Developments in fractal geometry. Bull. Math. Sci. 2013, 3, 299–348. [Google Scholar] [CrossRef] [Green Version]
- Balankin, A.S.; Patiño, J.; Patiño, M. Inherent features of fractal sets and key attributes of fractal models. Fractals 2022, 30, 2250082. [Google Scholar] [CrossRef]
- Havlin, S.; Ben-Avraham, D. Diffusion in disordered media. Adv. Phys. 2002, 51, 187–292. [Google Scholar] [CrossRef]
- McGreggor, K.; Kunda, M.; Goel, A. Fractals and Ravens. Artif. Intell. 2014, 215, 1–23. [Google Scholar] [CrossRef]
- Nicolás-Carlock, J.R.; Carrillo-Estrada, J.L.; Dossetti, V. Fractality à la carte: A general particle aggregation model. Sci. Rep. 2016, 6, 19505. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Halberstam, N.; Hutchcroft, T. What are the limits of universality? Proc. R. Soc. A 2022, 478, 2259. [Google Scholar] [CrossRef]
- Cruz, M.-Á.M.; Ortiz, J.P.; Ortiz, M.P.; Balankin, A.S. Percolation on Fractal Networks: A Survey. Fractal Fract. 2023, 7, 231. [Google Scholar] [CrossRef]
- Mandelbrot, B.B. Self-Affine Fractals and Fractal Dimension. Phys. Scr. 1985, 32, 257–260. [Google Scholar] [CrossRef]
- Edgar, G.A. Measure, Topology and Fractal Geometry, 2nd ed.; Springer: New York, NY, USA, 2008. [Google Scholar]
- Balka, R.; Buczolich, Z.; Elekes, M. A new fractal dimension: The topological Hausdorff dimension. Adv. Math. 2015, 274, 881–927. [Google Scholar] [CrossRef] [Green Version]
- Balankin, A.S. The topological Hausdorff dimension and transport properties of Sierpinski carpets. Phys. Lett. A 2017, 381, 2801–2808. [Google Scholar] [CrossRef]
- Balankin, A.S.; Mena, B.; Martinez-Cruz, M.A. Topological Hausdorff dimension and geodesic metric of critical percolation cluster in two dimensions. Phys. Lett. A 2017, 381, 2665–2672. [Google Scholar] [CrossRef]
- Lotfi, H. The µ-topological Hausdorff dimension. Extr. Math. 2019, 34, 237–254. [Google Scholar] [CrossRef]
- Ben Mabrouk, A.; Selm, B. On the topological Billingsley dimension of self-similar Sierpiński carpet. Eur. Phys. J. Spec. Top. 2021, 230, 3861–3871. [Google Scholar] [CrossRef]
- Vannimenus, J. On intrinsic properties of fractal lattices and percolation clusters. J. Phys. Lett. 1984, 45, L1071–L1076. [Google Scholar] [CrossRef] [Green Version]
- Gouyet, J.F. Physics and Fractal Structures; Springer: New York, NY, USA, 1996. [Google Scholar]
- Balankin, A.S. A continuum framework for mechanics of fractal materials I: From fractional space to continuum with fractal metric. Eur. Phys. J. B 2015, 88, 90. [Google Scholar] [CrossRef]
- Kigami, J. Harmonic calculus on p.c.f. self-similar sets. Trans. Am. Math. Soc. 1993, 355, 721–755. [Google Scholar] [CrossRef]
- Lindstrøm, T. Brownian motion on nested fractals. Mem. Am. Math. Soc. 1990, 83, 1–128. [Google Scholar] [CrossRef] [Green Version]
- Balankin, A.S. Effective degrees of freedom of a random walk on a fractal. Phys. Rev. E 2015, 92, 062146. [Google Scholar] [CrossRef]
- Alexander, S.; Orbach, R. Density of states on fractals: Fractions. J. Phys. Lett. 1982, 43, L623–L631. [Google Scholar] [CrossRef] [Green Version]
- Orbach, R. Dynamics of fractal networks. Science 1986, 231, 814–819. [Google Scholar] [CrossRef] [PubMed]
- Mosco, U. Invariant field metrics and dynamical scalings on fractals. Phys. Rev. Lett. 1997, 79, 4067–4070. [Google Scholar] [CrossRef]
- Telcs, T. The Art of Random Walks; Springer: New York, NY, USA, 2006. [Google Scholar]
- Haynes, C.P.; Roberts, A.P. Generalization of the fractal Einstein law relating conduction and diffusion on networks. Phys. Rev. Lett. 2009, 103, 020601. [Google Scholar] [CrossRef] [Green Version]
- Burioni, R.; Cassi, D. Random walks on graphs: Ideas, techniques and results. J. Phys. A Math. Gen. 2005, 38, R45–R78. [Google Scholar] [CrossRef]
- Balankin, A.S.; Mena, B.; Martínez-González, C.L.; Morales-Matamoros, D. Random walk in chemical space of Cantor dust as a paradigm of superdiffusion. Phys. Rev. E 2012, 86, 052101. [Google Scholar] [CrossRef]
- Ungar, S. The Koch Curve: A Geometric Proof. Am. Math. Month. 2007, 114, 61–66. [Google Scholar] [CrossRef]
- Sagan, H. The taming of a monster: A parametrization of the von Koch Curve. Int. J. Math. Educ. Sci. Technol. 1994, 25, 869–877. [Google Scholar] [CrossRef]
- Paramanathan, P.; Uthayakumar, R. Fractal interpolation on the Koch Curve. Comp. Math. Appl. 2010, 59, 3229–3233. [Google Scholar] [CrossRef] [Green Version]
- Epstein, M.; Sniatycki, J. The Koch curve as a smooth manifold. Chaos Solitons Fractals 2008, 38, 334–338. [Google Scholar] [CrossRef]
- Milosevic, N.T.; Ristanovic, D. Fractal and nonfractal properties of triadic Koch curve. Chaos Solitons Fractals 2007, 34, 1050–1059. [Google Scholar] [CrossRef]
- Jia, B. Bounds of the Hausdorff measure of the Koch curve. Appl. Math. Comp. 2007, 190, 559–565. [Google Scholar] [CrossRef]
- Carpinteri, A.; Pugno, N.; Sapora, A. Free vibration analysis of a von Koch beam. Int. J. Solids Struct. 2010, 47, 1555–1562. [Google Scholar] [CrossRef] [Green Version]
- Ding, J.; Fan, L.; Zhang, S.; Zhang, H.; Yu, W.W. Simultaneous realization of slow and fast acoustic waves using a fractal structure of Koch curve. Sci. Rep. 2018, 8, 1481. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- McCartney, M. The area, centroid and volume of revolution of the Koch curve. Int. J. Math. Educ. Sci. Technol. 2021, 52, 782–786. [Google Scholar] [CrossRef]
- Yang, G.; Yang, X.; Wang, P. Hölder Derivative of the Koch Curve. J. Appl. Math. Phys. 2023, 11, 101–114. [Google Scholar] [CrossRef]
- Darst, R.B.; Palagallo, J.A.; Price, T.E. Generalizations of the Koch curve. Fractals 2008, 16, 267–274. [Google Scholar] [CrossRef]
- Rani, M.; Haq, R.U.; Verma, D.K. Variants of Koch curve: A review. Int. J. Comput. Appl. 2012, 2, 20–24. [Google Scholar]
- McCartney, M. Four variations on a fractal theme. Int. J. Math. Educ. Sci. Tech. 2022. [Google Scholar] [CrossRef]
- Yang, X.; Yang, G. Arithmetic-analytical expression of the Koch-type curves and their generalizations (I). Acta Math. Appl. Sin. Engl. Ser. 2015, 31, 1167–1180. [Google Scholar] [CrossRef]
- Cantrell, M.; Palagallo, J. Self-intersection points of generalized Koch curves. Fractals 2008, 19, 213–220. [Google Scholar] [CrossRef]
- Essex, C.; Davison, M.; Schulzky, C.; Franz, A.; Hoffmann, K.H. The differential equation describing random walks on the Koch curve. J. Phys. A Math. Gen. 2001, 34, 8397–8406. [Google Scholar] [CrossRef]
- Maritan, A.; Stella, A. Spectral dimension of a fractal structure with long-range interactions. Phys. Rev. B 1986, 34, 456–459. [Google Scholar] [CrossRef] [PubMed]
- Kusuoka, S. A diffusion process on a fractal. In Probabilistic Methods on Mathematical Physics; Proceedings of the Taniguchi International Symposium, Katata and Kyoto, 1985; Ito, K., Ikeda, N., Eds.; Kinokuniya: Tokyo, Japan, 1987; pp. 251–274. [Google Scholar]
- Goldstein, S. Random Walks and Diffusions on Fractals. In Percolation Theory and Ergodic Theory of Infinite Particle Systems; Kesten, H., Ed.; Springer: New York, NY, USA, 1987. [Google Scholar] [CrossRef]
- Cao, S.; Qiu, H. Some Properties of the Derivatives on Sierpinski Gasket Type Fractals. Constr. Approx. 2017, 46, 319–347. [Google Scholar] [CrossRef] [Green Version]
- Kigami, J. Effective resistances for harmonic structures on p.c.f. self-similar sets. Math. Proc. Camb. Phil. Soc. 1994, 115, 291–303. [Google Scholar] [CrossRef]
- Akiyama, S.; Dorfer, G.; Thuswaldner, J.M.; Winkler, R. On the fundamental group of the Sierpiński-gasket. Topol. Appl. 2009, 156, 1655–1672. [Google Scholar] [CrossRef] [Green Version]
- Lacan, F.; Tresser, C. Fractals as objects with nontrivial structures at all scales. Chaos Solitons Fractals 2015, 75, 218–242. [Google Scholar] [CrossRef]
- Reiter, C.; With, J. 101 ways to build a Sierpinski triangle. ACM SIGAPL APL Quote Quad. 1997, 27, 8–16. [Google Scholar] [CrossRef]
- Bader, M. Sierpinski Curves. In Space-Filling Curves; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar] [CrossRef]
- Magrone, P. Sierpinski’s curve: A (beautiful) paradigm of recursion. Slov. Časopis Pre Geom. Graf. 2020, 17, 17–28. Available online: http://ssgg.sk/G/Abstrakty/G_cisla/G33.pdf#page=17 (accessed on 13 July 2023).
- Luna-Elizarrarás, M.E.; Shapiro, M.; Balankin, A.S. Fractal-type sets in the four-dimensional space using bicomplex and hyperbolic numbers. Anal. Math. Phys. 2020, 10, 13. [Google Scholar] [CrossRef]
- Aslan, N.; Saltan, M.; Demir, B. On Topological Conjugacy of Some Chaotic Dynamical Systems on the Sierpinski Gasket. Filomat 2021, 35, 2317–2331. [Google Scholar] [CrossRef]
- David, C. Laplacian, on the Arrowhead Curve. Proc. Int. Geometry Center 2020, 13, 19–49. [Google Scholar] [CrossRef]
- Riane, N.; David, C. Sierpinski Gasket versus Arrowhead Curve. Comm. Nonlin. Sci. Numer. Sim. 2020, 89, 105311. [Google Scholar] [CrossRef]
- Hilfer, R.; Blumen, A. Renormalisation on Sierpinski-type fractals. J. Phys. A Math. Gen. 1984, 17, L537–L545. [Google Scholar] [CrossRef]
- Brzezinska, M.; Cook, A.M.; Neupert, T. Topology in the Sierpinski-Hofstadter problem. Phys. Rev. B 2018, 98, 205116. [Google Scholar] [CrossRef] [Green Version]
- Chen, J.; He, L.; Wang, Q. The Eccentric Distance Sum of Sierpinski Gasket and Sierpinski Network. Fractals 2019, 27, 1950016. [Google Scholar] [CrossRef]
- Ri, S. Fractal functions on the Sierpinski Gasket. Chaos Solitons Fractals 2020, 138, 110142. [Google Scholar] [CrossRef]
- Pai, S.; Prem, A. Topological states on fractal lattices. Phys. Rev. B 2019, 100, 155135. [Google Scholar] [CrossRef] [Green Version]
- Landry, T.M.; Lapidus, M.L.; Latrémolière, F. Metric approximations of spectral triples on the Sierpiński gasket and other fractal curves. Adv. Math. 2021, 385, 107771. [Google Scholar] [CrossRef]
- Nakajima, Y. Dimensions of slices through the Sierpiński gasket. J. Differ. Eq. Appl. 2022, 28, 429–456. [Google Scholar] [CrossRef]
- Padmapriya, P.; Mathad, V. Topological indices of sierpinski gasket and Sierpinski gasket rhombus graphs. J. Appl. Eng. Math. 2022, 12, 136–148. Available online: http://jaem.isikun.edu.tr/web/images/articles/vol.12.no.1/12.pdf (accessed on 13 July 2023).
- Zhou, O.; Deng, Y. Generating Sierpinski gasket from matrix calculus in Dempster–Shafer theory. Chaos Solitons Fractals 2023, 166, 112962. [Google Scholar] [CrossRef]
- Pollicott, M.; Slipantschuk, J. Sierpinski Fractals and the Dimension of Their Laplacian Spectrum. Math. Comput. Appl. 2023, 28, 70. [Google Scholar] [CrossRef]
- Vicsek, T. Fractal Growth Phenomena; World Scientific: Singapore, 1989. [Google Scholar]
- Blumen, A.; von Ferber, C.; Jurjiu, A.; Koslowski, T. Generalized Vicsek Fractals: Regular Hyperbranched Polymers. Macromolecules 2004, 37, 638–650. [Google Scholar] [CrossRef]
- Deng, J.; Ye, Q.; Wang, Q. Weighted average geodesic distance of Vicsek network. Phys. A 2019, 527, 121327. [Google Scholar] [CrossRef]
- Lim, J.; Eiber, C.D.; Sun, A.; Maples, A.; Powley, T.L.; Ward, M.P.; Lee, H. Fractal Microelectrodes for More Energy-Efficient Cervical Vagus Nerve Stimulation. Adv. Healthc. Mat. 2023, 12, 2202619. [Google Scholar] [CrossRef] [PubMed]
- Cormick, C.; Ermann, L. Ground state of composite bosons in low-dimensional graphs. Phys. Rev. A 2023, 107, 043324. [Google Scholar] [CrossRef]
- Cristea, L.L.; Steinsky, B. Connected generalised Sierpiński carpets. Topol. Appl. 2010, 157, 1157–1162. [Google Scholar] [CrossRef] [Green Version]
- Rani, M.; Goel, S. Categorization of new fractal carpets. Chaos Solitons Fractals 2009, 41, 1020–1026. [Google Scholar] [CrossRef]
- Lau, K.S.; Luo, J.J.; Rao, H. Topological structure of fractal squares. Math. Proc. Camb. Phil. Soc. 2013, 155, 73–86. [Google Scholar] [CrossRef]
- Cristea, L.L.; Steinsky, B. Mixed labyrinth fractals. Topol. Appl. 2017, 229, 112–125. [Google Scholar] [CrossRef]
- Luo, J.; Yao, X.T. A Note on Topology of Fractal Squares with Order Three. Fractals 2021, 28, 2150005. [Google Scholar] [CrossRef]
- Cristea, L.L. A geometric property of the Sierpiński carpet. Quaest. Math. 2005, 28, 251–262. [Google Scholar] [CrossRef]
- Luo, J.J.; Liu, J.C. On the classification of fractal squares. Fractals 2016, 24, 1650008. [Google Scholar] [CrossRef]
- Rao, F.; Wang, X.; Wen, S. On the topological classification of fractal squares. Fractals 2017, 25, 1750028. [Google Scholar] [CrossRef]
- Rao, F.; Wen, S. Remarks on Quasisymmetric Rigidity of Square Sierpiński Carpets. Fractals 2018, 26, 1850060. [Google Scholar] [CrossRef]
- Ruan, H.J.; Wang, Y. Topological invariants and Lipschitz equivalence of fractal squares. J. Math. Anal. Appl. 2017, 451, 327–344. [Google Scholar] [CrossRef]
- Ma, J.; Zhang, Y. Topological Hausdorff dimension of fractal squares and its application to Lipschitz classification. Nonlinearity 2020, 33, 6053. [Google Scholar] [CrossRef]
- Zhang, Y.F. A lower bound of topological Hausdorff dimension of fractal squares. Fractals 2020, 28, 2050115. [Google Scholar] [CrossRef]
- Huang, L.; Rao, H. A dimension drop phenomenon of fractal cubes. J. Math. Anal. Appl. 2021, 497, 124918. [Google Scholar] [CrossRef]
- Montiel, M.E.; Aguado, A.S.; Zaluska, E. Topology in fractals. Chaos Solitons Fractals 1996, 7, 1187–1207. [Google Scholar] [CrossRef]
- Jia, B. Maximum density for the Sierpinski carpet. Comp. Math. Appl. 2009, 57, 1615–1621. [Google Scholar] [CrossRef] [Green Version]
- Hülse, M. From Sierpinski Carpets to Directed Graphs. Complex Syst. 2010, 19, 45–71. [Google Scholar] [CrossRef]
- Manning, A.; Simon, K. Dimension of slices through the Sierpinski carpet. Trans. Am. Math. Soc. 2013, 365, 213–250. [Google Scholar] [CrossRef] [Green Version]
- Bailey, D.H.; Borwein, J.M.; Crandall, R.E.; Rose, M.G. Expectations on fractal sets. Appl. Math. Comp. 2013, 220, 695–721. [Google Scholar] [CrossRef]
- van Veen, E.; Yuan, S.; Katsnelson, M.I.; Polini, M.; Tomadin, A. Quantum transport in Sierpinski carpets. Phys. Rev. B 2016, 93, 115428. [Google Scholar] [CrossRef]
- Zhao, L.; Wang, S.; Xi, L. Average geodesic distance of Sierpinski carpet. Fractals 2017, 25, 1750061. [Google Scholar] [CrossRef]
- D’Angeli, D.; Donno, A. Metric compactification of infinite Sierpiński carpet graphs. Discrete Math. 2016, 339, 2693–2705. [Google Scholar] [CrossRef]
- Wang, S.; Xi, L.; Xu, H.; Wang, L. Scale-free and small-world properties of Sierpinski networks. Phys. A 2017, 465, 690–700. [Google Scholar] [CrossRef]
- Canning, S.; Walker, A.J.; Roach, P.A. The Effectiveness of a Sierpinski Carpet-Inspired Transducer. Fractals 2017, 25, 1750050. [Google Scholar] [CrossRef] [Green Version]
- Balankin, A.S.; Martínez-Cruz, M.A.; Susarrey-Huerta, O.; Damian-Adame, L. Percolation on infinitely ramified fractal networks. Phys. Lett. A 2018, 382, 12–19. [Google Scholar] [CrossRef]
- Balankin, A.S.; Martínez-Cruz, M.A.; Álvarez-Jasso, M.D.; Patiño-Ortiz, M.; Patiño-Ortiz, J. Effects of ramification and connectivity degree on site percolation threshold on regular lattices and fractal networks. Phys. Lett. A 2019, 383, 957–966. [Google Scholar] [CrossRef]
- Balankin, A.S.; Ramírez-Joachin, J.; González-López, G.; Gutíerrez-Hernández, S. Formation factors for a class of deterministic models of pre-fractal pore-fracture networks. Chaos Solitons Fractals 2022, 162, 112452. [Google Scholar] [CrossRef]
- Barlow, M.T.; Bass, R.F. Brownian Motion and Harmonic Analysis on Sierpinski Carpets. Can. J. Math. 1999, 51, 673–744. [Google Scholar] [CrossRef] [Green Version]
- Sergeyev, Y.D. Evaluating the exact infinitesimal values of area of Sierpinski’s carpet and volume of Menger’s sponge. Chaos Solitons Fractals 2009, 42, 3042–3046. [Google Scholar] [CrossRef] [Green Version]
- Wu, W.; Xi, L. Dimensions of slices through a class of generalized Sierpinski sponges. J. Math. Anal. Appl. 2013, 399, 514–523. [Google Scholar] [CrossRef]
- Herrmann, R. A fractal approach to the dark silicon problem: A comparison of 3D computer architectures—Standard slices versus fractal Menger sponge geometry. Chaos Solitons Fractals 2015, 70, 38–41. [Google Scholar] [CrossRef] [Green Version]
- Molina-Abril, H.; Real, P.; Nakamura, A.; Klette, R. Connectivity calculus of fractal polyhedrons. Pattern Recognit. 2015, 48, 1150–1160. [Google Scholar] [CrossRef]
- Yang, Y.; Feng, Y.; Yu, Y. The generalization of Sierpinski carpet and Menger sponge in n-dimensional space. Fractals 2017, 25, 1750040. [Google Scholar] [CrossRef] [Green Version]
- Bickle, A. MegaMenger Graphs. Coll. Math. J. 2018, 49, 20–26. [Google Scholar] [CrossRef]
- Cicalò, S. The Construction of the Origami Level-n Menger Sponge Complement by the PJS Technique. Crystals 2021, 11, 468. [Google Scholar] [CrossRef]
- Panagiotopoulos, A.; Solecki, S. A combinatorial model for the Menger curve. J. Topol. Anal. 2022, 14, 203–229. [Google Scholar] [CrossRef] [Green Version]
- Rosenzweig, R.; Shavi, U. The laminar flow field at the interface of a Sierpinski carpet configuration. Water Resour. Res. 2007, 43, W10402. [Google Scholar] [CrossRef] [Green Version]
- Monceau, P.; Lévy, J.C.S. Spin waves in deterministic fractals. Phys. Lett. A 2010, 374, 1872–1879. [Google Scholar] [CrossRef]
- Jian-Hua, L.; Bo-Ming, Y.U. Tortuosity of Flow Paths through a Sierpinski Carpet. Chin. Phys. Lett. 2011, 28, 034701. [Google Scholar] [CrossRef]
- Khabbazi, A.E.; Hinebaugh, J.; Bazylak, A. Analytical tortuosity–porosity correlations for Sierpinski carpet fractal geometries. Chaos Solitons Fractal 2015, 78, 124–133. [Google Scholar] [CrossRef]
- Aarao Reis, F.D.A. Scaling relations in the diffusive infiltration in fractals. Phys. Rev. E 2016, 94, 052124. [Google Scholar] [CrossRef] [Green Version]
- Balankin, A.S.; Morales-Ruiz, L.; Matías-Gutierres, S.; Susarrey, O.; Samayoa, D.; Patiño-Ortiz, J. Comparative study of gravity-driven discharge from reservoirs with translationally invariant and fractal pore networks. J. Hydrol. 2018, 565, 467–473. [Google Scholar] [CrossRef]
- Voller, V.R.; Aarão-Reis, F.A.D. Determining effective conductivities of fractal objects. Int. J. Therm. Sci. 2021, 159, 106577. [Google Scholar] [CrossRef]
- Aguilar-Madera, C.G.; Herrera-Hernández, E.C.; Espinosa-Paredes, G.; Briones-Carrillo, J.A. On the effective diffusion in the Sierpiński carpet. Comp. Geosci. 2021, 25, 467–473. [Google Scholar] [CrossRef]
- Kushwaha, B.; Dwivedi, K.; Ambekar, R.S.; Pal, V.; Jena, D.P.; Mahapatra, D.R.; Tiwary, C.S. Mechanical and Acoustic Behavior of 3D-Printed Hierarchical Mathematical Fractal Menger Sponge. Adv. Eng. Mater. 2021, 23, 2001471. [Google Scholar] [CrossRef]
- Viet, N.V.; Karathanasopoulos, N.; Zaki, W. Effective stiffness, wave propagation, and yield surface attributes of Menger sponge-like pre-fractal topologies. Int. J. Mech. Sci. 2022, 227, 107447. [Google Scholar] [CrossRef]
- Moore, E.H. On certain crinkly curves. Trans. Am. Math. Soc. 1990, 1, 72–90. [Google Scholar] [CrossRef]
- Alsina, J. The Peano curve of Schoenberg is nowhere differentiable. J. Approx. Theor. 1981, 33, 28–42. [Google Scholar] [CrossRef] [Green Version]
- Sagan, H. An Elementary Proof that Schoenberg’s Space-Filling Curve Is Nowhere Differentiable. Math. Mag. 1992, 65, 125–128. [Google Scholar] [CrossRef]
- Martínez-Abejón, A. Nowhere Differentiability Conditions of Composites on Peano Curves. Bull. Malays. Math. Sci. Soc. 2022, 45, 101–111. [Google Scholar] [CrossRef]
- Mokbel, M.F.; Aref, W.G.; Kamel, I. Analysis of multi-dimensional space-filling curves. GeoInformatica 2003, 7, 179–209. [Google Scholar] [CrossRef]
- Mokbel, M.F.; Aref, W.G. Irregularity in high-dimensional space-filling curves. Distrib. Parallel Databases 2011, 29, 217–238. [Google Scholar] [CrossRef]
- Haverkort, H.J. How many three-dimensional Hilbert curves are there? J. Comput. Geom. 2017, 8, 206–281. [Google Scholar] [CrossRef]
- de Freitas, J.E.; de Lima, R.F.; dos Santos, D.T. The n-dimensional Peano Curve. São Paulo J. Math. Sci. 2019, 13, 678–688. [Google Scholar] [CrossRef] [Green Version]
- Paulsen, W. A Peano-based space-filling surface of fractal dimension three. Chaos Solitons Fractals 2023, 168, 113130. [Google Scholar] [CrossRef]
- Massopust, P.R. Fractal Peano curves. J. Geom. 1989, 34, 127–138. [Google Scholar] [CrossRef]
- Shchepin, E.V. On fractal Peano curves. Proc. Steklov. Inst. Math. 2004, 247, 272–280. Available online: https://www.mathnet.ru/eng/tm27 (accessed on 13 July 2023).
- Bauman, K.E. The Dilation Factor of the Peano–Hilbert Curve. Math. Notes 2006, 80, 609–620. [Google Scholar] [CrossRef]
- Wilder, R.L. Evolution of the topological concept of “connected”. Am. Math. Month. 1978, 85, 720–726. [Google Scholar] [CrossRef]
- Schoenflies, A. Beiträge zur Theorie der Punktmengen I. Math. Ann. 1903, 58, 195–234. [Google Scholar] [CrossRef] [Green Version]
- Fleron, J.F. A Note on the History of the Cantor Set and Cantor Function. Math. Mag. 1994, 67, 136–140. [Google Scholar] [CrossRef]
- Hocking, J.G.; Young, G.S. Topology; Dover Publications: New York, NY, USA, 1988. [Google Scholar]
- Álvarez-Samaniego, B.; Álvarez-Samaniego, W.P.; Ortiz-Castro, J. Some existence results on Cantor sets. J. Egypt. Math. Soc. 2017, 25, 326–330. [Google Scholar] [CrossRef]
- Athreya, J.S.; Reznick, B.; Tyson, J.T. Cantor Set Arithmetic. Am. Math. Month. 2019, 126, 4–17. [Google Scholar] [CrossRef]
- Bula, W. On compact Hausdorff spaces having finitely many types of open subsets. Colloq. Math. 1979, 41, 211–214. Available online: http://eudml.org/doc/265278 (accessed on 13 July 2023). [CrossRef]
- Benyamini, Y. Applications of the Universal Surjectivity of the Cantor Set. Am. Math. Month 1998, 105, 832–839. Available online: http://www.jstor.org/stable/2589212 (accessed on 13 July 2023). [CrossRef]
- Ávila, F.; Urenda, J.; Zaldívar, A. On the Cantor and Hilbert cube frames and the Alexandroff-Hausdorff theorem. J. Pure Appl. Algebra 2022, 226, 106919. [Google Scholar] [CrossRef]
- Willard, S. General Topology; Addison Wesley Publishing Company: Reading, MA, USA, 1970. [Google Scholar]
- Rudin, W. Functional Analysis, 2nd ed.; McGraw-Hill: New York, NY, USA, 1991. [Google Scholar]
- Mihaila, I. The Rationals of the Cantor Set. Coll. Math. J. 2004, 35, 251–255. [Google Scholar] [CrossRef]
- Raut, S.; Datta, D.P. Analysis on a fractal set. Fractals 2009, 17, 45–52. [Google Scholar] [CrossRef]
- Raut, S.; Datta, D.P. Non-archimedean scale invariance and cantor sets. Fractals 2010, 18, 111–118. [Google Scholar] [CrossRef] [Green Version]
- Datta, D.P.; Raut, S.; Raychoudhuri, A. Ultrametric Cantor sets and growth of measure. P-Adic. Num. Ultrametr. Anal. Appl. 2011, 3, 7–22. [Google Scholar] [CrossRef]
- Čiklová-Mlíchová, M. Li–Yorke sensitive minimal maps II. Nonlinearity 2009, 22, 1569. [Google Scholar] [CrossRef]
- Nobel, L. Polynomial hulls and envelopes of holomorphy of subsets of strictly pseudoconvex boundaries. Int. J. Math. 2012, 23, 1250107. [Google Scholar] [CrossRef]
- Wallin, R. The Elements of Cantor Sets: With Applications; Willey: New Jersey, NJ, USA, 2013. [Google Scholar]
- Kraft, R. What’s the difference between Cantor sets? Amer. Math. Mon. 1994, 101, 640–650. [Google Scholar] [CrossRef]
- Soltanifar, M. A Different Description of a Family of Middle-α Cantor Sets. Am. J. Undergrad. Res. 2006, 5, 9–12. Available online: http://www.ajuronline.org/uploads/Volume%205/Issue%202/52C-SoltanifarArt.pdf (accessed on 31 July 2023). [CrossRef]
- Khan, S.I.; Islam, M.S. An exploration of the generalized Cantor set. Int. J. Sci. Technol. Res. 2013, 2, 50–54. Available online: https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=50006f37d8e6c3416f100d780415e2429a7e4295 (accessed on 31 July 2023).
- Leary, C.C.; Ruppe, D.A.; Hartvigsen, G. Fractals, average distance and the Cantor set. Fractals 2010, 18, 327–341. [Google Scholar] [CrossRef]
- Camerlo, R. Continua and their σ-ideals. Topol. Appl. 2005, 150, 1–18. [Google Scholar] [CrossRef] [Green Version]
- Crovisier, S.; Rams, M. IFS attractors and Cantor set. Topol. Appl. 2006, 153, 1849–1859. [Google Scholar] [CrossRef] [Green Version]
- Beardon, A.F.; Rowat, C. Efficient sets are small. J. Math. Econom. 2013, 49, 367–374. [Google Scholar] [CrossRef] [Green Version]
- Cabrelli, C.; Hare, K.; Molter, U. Classifying Cantor sets by their fractal dimensions. Proc. Am. Math. Soc. 2010, 138, 3965–3974. [Google Scholar] [CrossRef] [Green Version]
- Barov, S.; Dijkstra, J.J.; van der Meer, M. On Cantor sets with shadows of prescribed dimension. Topol. Appl. 2012, 159, 2736–2742. [Google Scholar] [CrossRef] [Green Version]
- Krushkal, V. Sticky Cantor sets in Rd. J. Topol. Anal. 2018, 10, 477–482. [Google Scholar] [CrossRef]
- Das, S.K.; Mishra, J.; Nayak, S.R. Generation of Cantor sets from fractal squares: A mathematical prospective. J. Interdiscip. Math. 2022, 25, 863–880. [Google Scholar] [CrossRef]
- Fujita, T. A fractional dimension, selfsimilarity and a generalized diffusion operator. Probabilistic Methods Math. Phys. Proc. Taniguchi Symp. 1987, pp. 83–90. Available online: https://cir.nii.ac.jp/crid/1570854174508692864 (accessed on 31 July 2023).
- Fujita, T. Some asymptotic estimates of transition probability densities for generalized diffusion processes with self-similar speed measures. Publ. Res. Inst. Math. Sci. 1990, 26, 819–840. Available online: https://www.jstage.jst.go.jp/article/kyotoms1969/26/5/265819/article (accessed on 13 July 2023). [CrossRef] [Green Version]
- Evans, S.N. Local properties of Lévy processes on a totally disconnected group. J. Theor. Probab. 1989, 2, 209–259. [Google Scholar] [CrossRef]
- Lobus, J.U. Constructions and generators of one-dimensional quasidiffusions with applications to self-affine diffusions and Brownian motion on the Cantor set. Stoch. Stoch. Rep. 1993, 42, 93–114. [Google Scholar] [CrossRef]
- Aldous, D.; Evans, S.N. Dirichlet forms on totally disconnected spaces and bi-partite Markov chains. J. Theor. Probab. 1999, 12, 839–857. [Google Scholar] [CrossRef]
- Freiberg, U. Spectral asymptotics of generalized measure geometric Laplacians on Cantor like sets. Forum Math. 2005, 17, 87–104. [Google Scholar] [CrossRef]
- SBhamidi, S.N.; Evans, R.; Peled, P. Ralph, Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan. Inst. Math. Stat. Collect. 2008, 2, 42–75. [Google Scholar] [CrossRef] [Green Version]
- Karwowski, W. Diffusion processes with ultrametric jumps. Rep. Math. Phys. 2007, 60, 221–235. [Google Scholar] [CrossRef]
- Takahashi, H.; Tamura, Y. Homogenization on disconnected selfsimilar fractal sets in R. Tokyo J. Math. 2005, 28, 127–238. [Google Scholar] [CrossRef]
- Del Muto, M.; Figa-Talamanca, A. Anisotropic diffusion on totally disconnected abelian groups. Pac. J. Math. 2006, 225, 221–229. [Google Scholar] [CrossRef]
- Parvate, A.; Gangal, A.D. Calculus on fractal subsets of real line-I: Formulation. Fractals 2009, 17, 53–81. [Google Scholar] [CrossRef]
- Pearson, J.; Bellissard, J. Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets. J. Noncommut. Geom. 2009, 3, 447–480. [Google Scholar] [CrossRef] [Green Version]
- Datta, D.P.; Raut, S.; Chaudhuri, A.R. Diffusion in a class of fractal sets. Int. J. Appl. Math. Stat. 2012, 30, 37–50. Available online: http://www.ceser.in/ceserp/index.php/ijamas/article/view/684 (accessed on 13 July 2023).
- Kigami, J. Transitions on a noncompact Cantor set and random walks on its defining tree. Ann. Inst. Henri Poincaré Probab. Stat. 2013, 49, 1090–1129. [Google Scholar] [CrossRef]
- Sokolov, I. What is the alternative to the Alexander–Orbach relation? J. Phys. A Math. Theor. 2016, 49, 095003. [Google Scholar] [CrossRef]
- Golmankhaneh, A.K.; Ashrafi, S.; Baleanu, D.; Fernandez, A. Brownian Motion on Cantor Sets. Int. J. Nonlin. Sci. Numer. Sim. 2020, 21, 275–281. [Google Scholar] [CrossRef]
- Iomin, A.; Sandev, T. Fractional diffusion to a cantor set in 2d. Fractal Fract. 2020, 4, 52. [Google Scholar] [CrossRef]
- Golmankhaneh, A.K.; Sibatov, R.T. Fractal stochastic processes on thin cantor-like sets. Mathematics 2021, 9, 613. [Google Scholar] [CrossRef]
- Heo, J.; Kang, S.; Lim, Y. Dirichlet forms and ultrametric Cantor sets associated to higher-rank graphs. J. Aust. Math. Soc. 2021, 110, 194–219. [Google Scholar] [CrossRef] [Green Version]
- Arkashov, N.S.; Seleznev, V.A. Geometric model of the formation of superdiffusion processes. Theor. Math. Phys. 2022, 210, 376–385. [Google Scholar] [CrossRef]
- Takahashi, H.; Tamura, Y. Diffusion processes in Brownian environments on disconnected selfsimilar fractal sets in R. Stat. Prob. Lett. 2023, 193, 109694. [Google Scholar] [CrossRef]
- Balankin, A.S.; Mena, B. Vector differential operators in a fractional dimensional space, on fractals, and in fractal continua. Chaos Solitons Fractals 2023, 168, 113203. [Google Scholar] [CrossRef]
- Bressan, A.; Mazzola, M.; Nguyen, K.T. Diffusion approximations of Markovian solutions to discontinuous ODEs. J. Dyn. Diff. Equat. 2023. [Google Scholar] [CrossRef]
- Karwowski, W. Random Processes on Non-Archimedean Spaces. In Quantum and Stochastic Mathematical Physics: Springer Proceedings in Mathematics & Statistics 377; Hilbert, A., Mastrogiacomo, E., Mazzucchi, S., Rüdiger, B., Ugolini, S., Eds.; Springer: Berlin/Heidelberg, Germany, 2023. [Google Scholar] [CrossRef]
- Golmankhaneh, A.K.; Balankin, A.S. Sub-and super-diffusion on Cantor sets: Beyond the paradox. Phys. Lett. A 2018, 382, 960–967. [Google Scholar] [CrossRef]
- Freiberg, U.; Zahle, M. Harmonic calculus on fractals—A measure geometric approach I. Potential Anal. 2002, 16, 265–277. [Google Scholar] [CrossRef]
- Bakhtin, Y. Self-similar Markov processes on Cantor set. arXiv 2008, arXiv:0810.3260v1. [Google Scholar]
- Martin, H.W. Strongly rigid metrics and zero dimensionality. Proc. Am. Math. Soc. 1977, 67, 157–161. [Google Scholar] [CrossRef]
- Nekvinda, A.; Zindulka, O. A Cantor set in the plane that is not σ-monotone. Fund. Math. 2011, 213, 221–230. [Google Scholar] [CrossRef]
- Steinhurst, B. Uniqueness of Locally Symmetric Brownian Motion on Laakso Spaces. Potential Anal. 2013, 38, 281–298. [Google Scholar] [CrossRef] [Green Version]
- Oblakova, A.I. Isometric embeddings of finite metric spaces. Moscow Univ. Math. Bull. 2016, 71, 1–6. [Google Scholar] [CrossRef]
- Zezula, P.; Amato, G.; Dohnal, V.; Batko, M. Similarity Search: The Metric Space Approach; Springer: New York, NY, USA, 2006. [Google Scholar] [CrossRef]
- Julien, A.; Savinien, J. Embeddings of self-similar ultrametric Cantor sets. Topol. Appl. 2011, 158, 2148–2157. [Google Scholar] [CrossRef] [Green Version]
Basic Dimension Numbers | Parameters | Dimension Numbers | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
25 | 1 | 1 | 1.8614 | 1.9746 | 1.951 | 0.0497 | 0.470 | 4/5 | 1.9723 | 1.892 |
9 | 1 | 1 | 1.6309 | 1.8928 | 1.806 | 0.2031 | 0.384 | 2/3 | 1.877 | 1.771 |
16 | 4 | 1 | 1.5 | 1.7935 | 1.659 | 0.3684 | 0.303 | 1/2 | 1.75 | 1.792 |
25 | 9 | 1 | 1.4307 | 1.7227 | 1.572 | 0.4683 | 0.259 | 2/5 | 1.6584 | 1.829 |
36 | 16 | 1 | 1.3869 | 1.6719 | 1.516 | 0.5333 | 0.232 | 1/3 | 1.5912 | 1.860 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Patiño Ortiz, J.; Patiño Ortiz, M.; Martínez-Cruz, M.-Á.; Balankin, A.S. A Brief Survey of Paradigmatic Fractals from a Topological Perspective. Fractal Fract. 2023, 7, 597. https://doi.org/10.3390/fractalfract7080597
Patiño Ortiz J, Patiño Ortiz M, Martínez-Cruz M-Á, Balankin AS. A Brief Survey of Paradigmatic Fractals from a Topological Perspective. Fractal and Fractional. 2023; 7(8):597. https://doi.org/10.3390/fractalfract7080597
Chicago/Turabian StylePatiño Ortiz, Julián, Miguel Patiño Ortiz, Miguel-Ángel Martínez-Cruz, and Alexander S. Balankin. 2023. "A Brief Survey of Paradigmatic Fractals from a Topological Perspective" Fractal and Fractional 7, no. 8: 597. https://doi.org/10.3390/fractalfract7080597