Physical dynamics divide naturally into the dissipative and conservative extremes, in which friction either dominates or becomes irrelevant. Otto's work heralded a breakthrough in our ability to realize dissipative dynamical systems as infinite dimensional Riemannian gradient flows. Here we exploit this point of view to analyze the long time behaviour of the nonlinear (fast) diffusion equation, used to model heat transport, population spreading, fluid seepage, curvature flows, and avalanches in sandpiles --- as a prototype from the dissipative regime. The spectrum of the entropy is explicitly determined, and the dynamics are found to undergo a phase transition in which rotational symmetry is broken as the strength of the nonlinearity is varied.