Shape limit in Voronoi tilings for Bernoulli spirals

Y. Yamagishi¹, T. Sushida², A. Hizume¹

¹Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu 520-2194, Japan

²Organization for the Strategic Coordination of Research and Intellectual Property, Meiji University, 4-21-1 Nakano, Tokyo 164-8525, Japan

yg@rins.ryukoku.ac.jp

Let  be a Bernoulli spiral set in the complex plane, generated by , . We studied the geometry and topology of triangular tilings with the vertex set  in [1], and the shape limit of triangular tiles as  in [2].  In the phyllotaxis theory,  is called the plastochrone ratio, and  the divergence angle.  Here we consider the Voronoi tiling with the site set .   The parastichy number, i.e. the number of spirals consisting of contact Voronoi cells, is obtained by the continued fraction expansion of the . The Voronoi tiling is a quadrilateral tiling if it has two parastichies, or hexagonal tiling if it has three parastichies. 

Suppose that  is a quadratic irrational number.  If we only consider the quadratic Voronoi tilings, then the limit set of the shapes of the quadrilateral tiles as  is a finite set of rectangles.  In particular, if  is linearly equivalent to the golden section , the limit is the square [3].

Rothen and Koch [4] observed the shape invariance under compression with the golden section divergence angle, in the linear lattice model.  Our work is an extension to the cylindrical model.  The shape limit in the linear lattice model was studied in [5].

Figure 1. A quadrilateral Voronoi tiling with similarity symmetry, generated by , .  A global view and a local view.  Each site point is indexed by an integer.  The tiles are close to squares.

1. T. Sushida, A. Hizume, Y. Yamagishi, J. Phys. A: Math. Theor., 45, (2012), 235203.

2. T. Sushida, A. Hizume, Y. Yamagishi, Acta Phys. Polonica A, 126, (2014), 633-636.

3. Y. Yamagishi, T. Sushida, A. Hizume, Nonlinearity, 28, (2015),1077-1102.

4. F. Rothen, A.-J. Koch, J. Physique France, 50, (1989), 633-657.

5. T. Sushida, A. Hizume, Y. Yamagishi, RIMS Kokyuroku Bessatsu, B47, (2014), 023-032.

This work is partially supported by JSPS Kakenhi Grant 24654029, 15K05011, and the Joint Research Center for Science and Technology of Ryukoku University. We would like to thank Smith College for the hospitality extended to Yamagishi during his stay in 2015-2016.