In this paper, we develop a rank-mapping algorithm for an icosahedral grid system on a massive parallel computer with the 3-D torus network topology, specifically on the K computer. Our aim is to improve the weak scaling performance of the point-to-point communications for exchanging grid-point values between adjacent grid regions on a sphere. We formulate a new rank-mapping algorithm to reduce the maximum number of hops for the point-to-point communications. We evaluate both the new algorithm and the standard ones on the K computer, using the communication kernel of the Nonhydrostatic Icosahedral Atmospheric Model (NICAM), a global atmospheric model with an icosahedral grid system. We confirm that, unlike the standard algorithms, the new one achieves almost perfect performance in the weak scaling on the K computer, even for 10,240 nodes. Results of additional experiments imply that the high scalability of the new rank-mapping algorithm on the K computer is achieved by reducing network congestion in the links between adjacent nodes.