In this paper, a self-organization framework for swarm systems in three dimensions is presented. The framework uses artificial potential functions(APFs) to direct the robots toward the goal as well as to keep them in a swarm system. This research extends conventional APFs used for self-organizations in two dimension environment to three dimensions. In three dimension environment, the ground potential for the boundary surfaces that commonly appear in three dimension environments is proposed. Accordingly, the comparison between the paths without and with the ground potentials shows the necessity and effect of ground potentials. Extensive simulations are given to show the effectiveness of the extended potentials and various properties in three dimension environments.