*Forma,* Vol. 22 (No. 2), pp. 157-175, 2007

*Original Paper*

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

Teruhisa Sugimoto^{1}* and Masaharu Tanemura^{1,2}

^{1}The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan

^{2}Department 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 *r*_{10} for *N* = 10.

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