This thesis studies optimal distribution of N points on a 3-sphere by exploring both analytic and numerical solutions of iterative method. Optimal distribution of n points on a 3-sphere is defined as the maximum sum of the mutual distances among the points. Also, if the points are considered as charged particles, the optimal distribution of the points is defined as the particles located in an equilibrium state according to Coulomb potential. Hence, the optimal distributions become maximum distance and minimum coulomb potential problems. The numerical calculations are done by iterative method using various software packages including C, Java, Mathematica, and Matlab. For N=2, 3, 4, the analytic solutions are hand calculated and proved. Both unique solutions of N= 2, 3, 4, 6, 8, 12 and non-unique solutions when n is not one of those numbers (e.g., n=20) are explored. Local extrema and global extrema of the maximum distance and minimum coulomb potential are discussed. Furthermore, the non-unique solutions of N=60 and the structure of carbon 60, Buckyball, are studied by using tabular and graphic display of the numerical results. The symmetry of Buckyball which has relevance to its stability is discussed. The main results are software development for numerical solutions, and unique and non-unique solutions of placing n points uniformly on a 3-sphere.
