*Forma,* Vol. 27 (No. 1), pp. 45–53, 2012

## Application of Voronoi Tessellation of Spherical Surface to Geometrical Models of Skeleton Forms of Spherical Radiolaria

Takashi Yoshino^{1}*, Atsushi Matsuoka^{2}, Toshiyuki Kurihara^{3}, Naoto Ishida^{4}, Naoko Kishimoto^{5}, Katsunori Kimoto^{6} and Shu Matsuura^{7}

^{1}Department of Mechanical Engineering, Toyo University, Japan

^{2}Department of Geology, Faculty of Science, Niigata University, Japan

^{3}Graduate School of Science and Technology, Niigata University, Japan

^{4}Venture Business Laboratory, Niigata University, Japan

^{5}Department of Mechanical Engineering, Setsunan University, Japan

^{6}Research Institute for Global Change (RIGC), Japan Agency for Marine-Earth Science and Technology (JAMSTEC), Japan

^{7}Faculty of Education, Tokyo Gakugei University, Japan

*E-mail address: tyoshino@toyo.jp

(Received September 20, 2011; Accepted June 20, 2012)

**Abstract. **
We investigated two geometrical models of skeleton forms of spherical radiolaria. Both models are based on Voronoi tessellation of given points on a sphere. We allocate a given number of points called "generators", which can be related to pore frames, and obtained their Voronoi tessellation and approximated polyhedron. The first model is based on random allocation of generators, and the second one is based on global minimization of a potential function whose value is calculated from a generator distribution. Depending on the types of these generator distributions, we obtained different types of approximated polyhedrons. Using these polyhedrons, we analyzed the frequency distributions of the number of vertices of the polygons and the total edge lengths. We then compared the polyhedrons derived by the tessellation with real radiolaria. A comparison of frequency distributions revealed that the first model is not sufficient for mesh-like radiolaria. However, the second model had similar distribution to that of another type of spherical radiolaria which has almost regular structure. Under the condition of same number of generators, the second model produces approximately 6 percent smaller total edge length than the first model.

**Keywords: **
Spherical Radiolaria, Voronoi Tessellation, Geometrical Model, Skeleton Form

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