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Arithmetic-analytical expression of the Koch-type curves and their generalizations (I)

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Abstract

In this paper, by constructing the one-to-one correspondence between its IFS and the quaternary fractional expansion, the n-th iteration analytical expression and the limit representation of the family of the Koch-type curves with arbitrary angles are obtained. The distinction between our method and that of H. Sagan is that we provide the generation process analytically and represent it as a graph of a series function which looks like the Weierstrass function. With these arithmetic expressions, we further analyze and prove some of the fractal properties of the Koch-type curves such as the self-similarity, the Hölder exponent and with the property of continuous everywhere but differentiable nowhere. Then, we will show that the Koch-type curves can be approximated by different constructed generators. Based on the analytic transformation of the Koch-type curves, we also constructed more continuous but nowhere-differentiable curves represented by arithmetic expressions. This result implies that the analytical expression of a fractal has theoretical and practical significance.

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Authors and Affiliations

  1. Statistics and Mathematics College, Yunnan University of Finance and Economics, Kunming, 650221, China

    Xiao-ling Yang

  2. Mathematics and Statistics College, Yunnan University, Kunming, 650091, China

    Guang-jun Yang

Authors
  1. Xiao-ling Yang
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  2. Guang-jun Yang
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Correspondence to Xiao-ling Yang.

Additional information

Supported partly by National Natural Science Foundation of China (No. 60962009).

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Yang, Xl., Yang, Gj. Arithmetic-analytical expression of the Koch-type curves and their generalizations (I). Acta Math. Appl. Sin. Engl. Ser. 31, 1167–1180 (2015). https://doi.org/10.1007/s10255-015-0522-0

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  • DOI: https://doi.org/10.1007/s10255-015-0522-0

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