Packing and Minkowski Covering of Congruent Spherical Caps on a Sphere, II: Cases of N = 10, 11, and 12

Teruhisa Sugimoto¹* and Masaharu Tanemura^1,2

¹The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan
²Department of Statistical Science, The Graduate University for Advanced Studies, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan
*E-mail address: sugimoto@ism.ac.jp

(Received August 6, 2007; Accepted October 3, 2007)

Keywords: Spherical Cap, Sphere, Packing, Covering, Tammes Problem

Abstract. Let C_i (i = 1, ..., N) be the i-th open spherical cap of angular radius r and let M_i be its center under the condition that none of the spherical caps contains the center of another one in its interior. We consider the upper bound, r_N, (not the lower bound!) of r of the case in which the whole spherical surface of a unit sphere is completely covered with N congruent open spherical caps under the condition, sequentially for i = 2, ..., N - 1, that M_i is set on the perimeter of C_i-1, and that each area of the set (∪ C_v) ∩ C_i becomes maximum. In this paper, for N = 10, 11, and, 12, we found out that the solutions of our sequential covering and the solutions of the Tammes problem were strictly correspondent. Especially, we succeeded to obtain the exact closed form of r₁₀ for N = 10.