Between two maxima there is always a saddle point
No. 48, 1968
(Functions on manifolds)
India ink on paper, 21x32 cm.
Columns of men, lined up in parallel, stand on a plane that reaches far back onto a tumultuous landscape. A warped stone pillar soars into a stormy and energetic sky, where clouds flow like whirling cream and light rebounds off the folds in the distant hills. In general, this picture illustrates a theorem, the so-called saddle-point principle, which states that a smooth Morse function defined on a connected manifold, with at least two local maximum points, must have a saddle point somewhere in the middle. You can think of th.e proof intuitively. If you join the two maxima with a rubber band and let it go, it will settle at a comfortable place somewhere in the middle, designating a saddle point. In this image, the rocky landscape corresponds to a graph of a function with four maxima, three of which give rise to two saddle points. The third is not visible, hidden behind the central rock. The image also illustrates a type of topological restructuring in the theory of iso-energetic surfaces of integrable Hamiltonian differential equations.
