Abstract
Based on fractal super fibers and binary fractal fibers, the following objectives are approached in this paper: First, the concept of multiple-cell elements is induced and abstracted. Second, through multiple-cell elements, the constructability of regular multifractals with strict self-similarities is confirmed, and the universality of the construction mode for regular multifractals is proved. Third, through the construction mode and multiple-cell elements, regular multifractals are demonstrated to be equivalent to generalized regular single fractals with multilayer fine structures. On the basis of such equivalence, the dimension formula of the regular single fractal is extended to that of the regular multifractal, and the geometry of regular single fractals is extended to that of regular multifractals. Fourth, through regular multifractals, a few golden fractals are constructed.
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Yin, Yajun, Zhang, Tong, Yang, Fan, and Qiu, Xinming. Geometric conditions for fractal super carbon nanotubes with strict self-similarities. Chaos, Solitons and Fractals 37(5), 1257–1266 (2008)
Yin, Yajun, Yang, Fan, Zhang, Tong, and Fan, Qinshan. Growth condition and growth limit for fractal super fibers and fractal super tubes. International Journal of Nonlinear Sciences and Numerical Simulations 9(1), 96–102 (2008)
Yin, Yajun, Yang, Fan, Fan, Qinshan, and Zhang, Tong. Cell elements, growth modes and topology evolutions of fractal supper fibers. International Journal of Nonlinear Sciences and Numerical Simulation 10(1), 1–12 (2009)
Yin, Yajun, Yang, Fan, and Fan, Qinshan. Isologous fractal super fibers or fractal super lattices. International Journal of Electrospun Nanofibers and Applications 2(3), 193–201 (2008)
Fan, Jie, Liu, Junfang, and He, Jihuan. Hierarchy of wool fibers and fractal dimensions. International Journal of Nonlinear Sciences and Numerical Simulation 9(3), 293–296 (2008)
He, Jihuan, Ren, Zhongfu, Fan, Jie, and Xu, Lan. Hierarchy of wool fibers and its interpretation using E-infinity theory. Chaos, Solitons and Fractals 41(4), 1839–1841 (2009)
Yin, Yajun, Yang, Fan, Li, Ying, and Fan, Qinshan. The fractal geometry and topology abstracted from hair fibers. Applied Mathematics and Mechanics (English Edition) 30(8), 983–990 (2009) DOI 10.1007/S10483-009-0804-5
Huang, L. J. and Ding, J. R. The current status of multifractal approach (in Chinese). Progress in Physics 11(3), 69–330 (1991)
Mandelbrot, B. B. On the intermittent free turbulence. Turbulence of Fluids and Plasmas (ed. Weber, E.), Interscience, New York, 483–492 (1969)
Grassberger, P. Generalized dimensions of strange attractors. Physics Letters A 97(6), 227–230 (1983)
Hentschel, H. and Procaccia, I. The infinite number of generalized dimensions of fractals and strange attractors. Physica D 8(3), 435–444 (1983)
Frisch, U. and Parisi, G. On the singularity structure of fully developed turbulence. Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics (eds. Ghil, M., Benzi, R., and Parisi, G.), North-Holland, Amsterdam, 84–87 (1985)
Benzi, R., Paladin, G., Parisi G., and Vulpiani, A. On the multifractal nature of fully developed turbulence and chaotic systems. J. Phys. A 17(18), 3521–3531 (1984)
Halsey, T. C., Jensen, M. H., Kadanoff, L. P., Procaccia I., and Shraiman B. I. Fractal measures and their singularities: the characterization of strange sets. Phys. Rev. A 33(2), 1141–1151 (1986)
Bensimon, D., Jensen, M. H., and Kadanoff, L. Renormalization-group analysis of the global structure of the period-doubling attractor. Phys. Rev. A 33(5), 3622–3624 (1986)
Feigenbaum, M., Jensen, M., and Procaccia, I. Time ordering and the thermodynamics of strange sets: theory and experimental tests. Phys. Rev. Lett. 57(13), 1503–1506 (1986)
Feigenbaum, M. Some characterizations of strange sets. J. Stat. Phys. 46(5–6), 919–925 (1987)
Cheng, Lingzhong, Feng, Jingsheng, Feng, Ziqiang, and Zhong, Cuiping. The Color Illustrated Handbook for Histology (in Chinese), The People’s Health Press, Beijing (2000)
Zhou, W. X., Wang, Y. J., and Yu, Z. H. Geometrical characteristics of singularity spectra of multifractals (I): classical Renyi definition (in Chinese). Journal of East China University of Science and Technology 26(4), 385–389 (2000)
Zhou, W. X., Wang, Y. J., and Yu, Z. H. Geometrical characteristics of singularity spectra of multifractals (II): partition function definition (in Chinese). Journal of East China University of Science and Technology 26(4), 390–395 (2000)
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Communicated by Zhe-wei ZHOU
Project supported by the National Natural Science Foundation of China (No. 10872114) and the Natural Science Foundation of Jiangsu Province (No. BK2008370)
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Yin, Yj., Li, Y., Yang, F. et al. Multiple-cell elements and regular multifractals. Appl. Math. Mech.-Engl. Ed. 31, 55–65 (2010). https://doi.org/10.1007/s10483-010-0106-2
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DOI: https://doi.org/10.1007/s10483-010-0106-2
Key words
- binary fractal fibers
- binary cell elements
- regular binary fractals
- multiple-cell elements
- regular multifractals