We have investigated an optimum form of the modified icosahedral grid that is generated by applying the spring dynamics to the standard icosahedral grid system. The spring dynamics can generate a more homogeneous grid system than the standard icosahedral grid system by tuning the natural spring lenght: as the natural spring length becomes longer, the ratio of maximum grid interval to minimum one becomes closer to unit. When the natural spring length is larger than a critical value, however, the spring dynamic system does not have a stable equilibrium. By setting the natural spring length to be the marginally critical value, we can obtain the most homogeneous grid system, which is most efficient in terms of the CFL condition. We have analyzed eigenmodes involved in the initial error of the geostrophic balance problem [test case 2 of D. L. Williamson et al. (1992, J. Comput. Phys. 102, 211)]. Since the balance state in the discrete system differs slightly from the exact solution of the analytic system, the initial error field includes both the gravity wave mode and the Rossby wave mode. As the results of the analysis are based on Hough harmonics decompositions, we detected Rossby and gravity wave modes with zonal wavenumber 5, which are asymmetric against the equator. These errors are associated with icosahedral grid structure. The symmetric gravity wave mode with zonal wavenumber 0 also appears in the error field. To clarify the evolution of Rossby waves, we introduce divergence damping to reduce the gravity wave mode. From the simulated results of the geostrophic problem with various grid systems, we found that the spuriously generated Rossby wave mode is eliminated most effectively when the most homogeneously distributed grid system is used. It is therefore, concluded that the most homogeneous grid system is the best choice from the viewpoint of numerical accuracy as well as computational efficiency.