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 Sugimoto1* and Masaharu Tanemura1,2

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:

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

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

Abstract. Let Ci (i = 1, ..., N) be the i-th open spherical cap of angular radius r and let Mi 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, rN, (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 Mi is set on the perimeter of Ci-1, and that each area of the set (∪ Cv) ∩ Ci 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.

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