Such surfaces are called "anticlastic" surfaces. A "synclastic" surface has the center of curvature of any plane intersection always on the same side of the surface.

It can be proved, that for any continuous anticlastic surface, at any point on the surface, there is at least one plane intersection direction where the curvature is zero.

Well it has to go to zero to get from positive curvature to negative curvature, if it is continuous.

As a result, anticlastic surfaces are susceptible to buckling, along a direction of near zero curvature. That's why egg shells, are built synclastic, and not anticlastic.

This post imported from StackExchange Mathematics at 2014-06-16 11:19 (UCT), posted by SE-user George E. Smith