Monkey's saddle

An ordinary saddle has two pommels and two flaps. The front and the rear parts of the saddle are directed upwards, the sides fall downwards. Moreover, the saddle is everywhere curved, i.e. non-flat. More precisely, the plane tangent to the saddle cuts the saddle in every, even the tiniest, environment of the point of osculation and therefore has points on both sides of the saddle.

If a similar saddle were to be designed for a flat-nosed (New World) monkey, i.e. for a monkey with a non-negligible tail, it would require three flaps and three pommels. It turns out, unfortunately, that such a good saddle cannot exist. There will always be at least one point at which it is locally flat.

The reason is that the extreme curvatures of the cuts of a smooth surface that are perpendicular to the tangent plane and cross the surface at the point of osculation lie on orthogonal planes. Under the threefold symmetry of the surface - and this is the case with monkey's saddle - this can only be achieved when the extreme values are equal to zero, i.e. when zero is the curvature of each such cut.