Nuclei undergoing fission can be described by a multi-dimensional potential-energy surface that guides the nuclear shape evolution--from the ground state, through intermediate saddle points and finally to the configurations of separated fission fragments. Until now, calculations have lacked adequate exploration of the shape parameterization of sufficient dimensionality to yield features in the potential-energy surface (such as multiple minima, valleys, saddle points and ridges) that correspond to characteristic observables of the fission process. Here we calculate and analyse five-dimensional potential-energy landscapes based on a grid of 2,610,885 deformation points. We find that observed fission features--such as the distributions of fission fragment mass and kinetic energy, and the different energy thresholds for symmetric and asymmetric fission--are very closely related to topological features in the calculated five-dimensional energy landscapes.