Topographic maps begin to be recognized as one of the major computational structures underlying neural computation in the brain. They provide dimension-reducing projections between feature spaces that seem to be established and maintained under the participation of selforganizing, adaptive processes. In this contribution, we investigate how well the structure of such maps can be replicated by simple adaptive processes of the kind proposed by Kohonen. We will particularly address the important issue, how the dimensionality of the input space affects the spatial organization of the resulting map.