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Spiral patterns of color symmetry from dynamics

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Abstract

This paper explores the esthetics of a kind of logarithmic spiral tilings that has not been investigated. It possesses the form similar to the structure of spiral galaxies, which globally displays the cyclic symmetry. The paper first studies the symmetry group associated with the spiral tiling. Then, using the generators of the group, the construction method of such tilings is analyzed in detail. To create colorful patterns on spiral tilings, a special dynamical system which is compatible with its symmetry group is designed. To promote the esthetic appeal of spiral patterns, a simple but practical strategy for generating patterns of color symmetry is presented. Based on the resulting patterns, several interesting methods are proposed to construct more types of derived spiral patterns.

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Acknowledgements

The author thanks the anonymous referees for their careful reading, helpful comments and many precious suggestions that improved the manuscript. For the pursuit of beauty, most of the esthetic patterns demonstrated in this article have relatively high resolution, which are produced under the VC++ 6.0 programming environment with the aid of powerful OpenGL. The basic spiral patterns (such as patters shown in Figs. 3, 4) are created in the square region \([-\,\mathrm{e}^{\left( \frac{2\pi }{n}+\pi \right) \cot \phi },\mathrm{e}^{\left( \frac{2\pi }{n}+\pi \right) \cot \phi }]\times [-\,\mathrm{e}^{\left( \frac{2\pi }{n}+\pi \right) \cot \phi },\mathrm{e}^{\left( \frac{2\pi }{n}+\pi \right) \cot \phi }]\). Such basic spiral patterns are then used to construct derived spiral patterns (i.e., patterns shown in Figs. 5, 6, 7, 8, 9, 10). We thank Adobe and Microsoft for their friendly technical support. This work was supported by the Natural Science Foundation of China (Nos. 11461035, 11761038 and 11761039), Science and Technology Plan Project of Jiangxi Provincial Education Department (Nos. GJJ160749 and GJJ160758).

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Authors and Affiliations

  1. School of Mathematics and Physics, Jinggangshan University, Qingyuan, China

    Peichang Ouyang, Xiaosong Tang & Tao Yu

  2. Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong

    Kwokwai Chung

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  1. Peichang Ouyang
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  2. Xiaosong Tang
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  3. Kwokwai Chung
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  4. Tao Yu
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Correspondence to Xiaosong Tang.

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Ouyang, P., Tang, X., Chung, K. et al. Spiral patterns of color symmetry from dynamics. Nonlinear Dyn 94, 261–272 (2018). https://doi.org/10.1007/s11071-018-4357-0

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