We show how phyllotaxis (the arrangement of leaves on plants) and the deformation configurations seen on plant surfaces can be understood as the preferred nonlinear buckling pattern of a thin shell on an elastic foundation. The key new idea is the recognition that the optimal configuration is one which minimizes the strain energy as measured by the product of the Airy stress and Guassian curvature of the deformed plant surface. The nonlinear (cubic) component of this contribution is largest negative on configurations which are triads of almost periodic deformations whose local wavevectors add to zero. Based on this observation, we reproduce a wide spectrum of plant patterns and show how the occurances of the Fibonacci sequence and golden angle are natural consequences.