Exact value of Tammes problem for N=10

TL;DR

This paper determines the exact angular radius for covering a sphere with 10 caps, linking a sequential covering problem to the Tammes problem and providing a precise solution for N=10.

Contribution

It establishes a correspondence between a sequential spherical covering problem and the Tammes problem, and computes the exact radius for N=10.

Findings

Exact value of r_10 for N=10 obtained

Sequential covering solutions match Tammes problem solutions

Provides a new precise solution for sphere covering with 10 caps

Abstract

Let $C_{i}$ ($\,i=1,\ldots ,N\,$) 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 $i=2,\ldots ,N-1\,$, 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 $N = 10$.