Exact value of Tammes problem for N=10

Abstract

Let $C_i$ ($i=1,\dots ,N$) 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 $i=2,\dots ,N-1$, that $M_i$ is set on the perimeter of $C_{i-1}$, and that each area of the set $({\cup }_{\nu =1}^{i-1}{C}_{\nu })\cap C_i$ becomes maximum. In this paper, for $N=10$, 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 value $r_{10}$ for $N=10$.

arXiv comment5 pages, 1 figure