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Voronoi tiling and circle packing on spiral lattices with rotational symmetry

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Abstract

It is shown that the bifurcation diagram of circle packings on logarithmic spiral lattices with rotational symmetry is graph-theoretically dual to the bifurcation diagram of Voronoi tessellations, by using the relative metric. If the rotation parameter (called divergence angle) is badly approximable, then the aspect ratio of the quadrilateral Voronoi cells is bounded. If the divergence angle is linearly equivalent to the golden section, then the shape of the quadrilateral cells tend to square as the plastochron ratio tends to 1.

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Acknowledgements

The authors would like to thank Matti Vuorinen for the references on the relative metric. We also thank the referee for helpful comments and advice to refine the paper in various aspects. This work was partially supported by JSPS Kakenhi Grant 18K13452.

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Authors and Affiliations

  1. Mars Networks Corporation, 3-1-1 Sakae-cho, Futyu, Tokyo, 183-0051, Japan

    Takuro Uezono

  2. Department of Computer Science and Technology, Salesian Polytechnic, 4-6-8 Oyamagaoka, Machida, Tokyo, 194-0215, Japan

    Takamichi Sushida

  3. Applied Math and Info Course, Ryukoku University, Seta, Otsu, 520-2194, Japan

    Yoshikazu Yamagishi

Authors
  1. Takuro Uezono
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  2. Takamichi Sushida
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  3. Yoshikazu Yamagishi
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Correspondence to Yoshikazu Yamagishi.

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Uezono, T., Sushida, T. & Yamagishi, Y. Voronoi tiling and circle packing on spiral lattices with rotational symmetry. Japan J. Indust. Appl. Math. 40, 709–736 (2023). https://doi.org/10.1007/s13160-022-00552-9

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  • DOI: https://doi.org/10.1007/s13160-022-00552-9

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