Monkey Saddle

In mathematics, the monkey saddle is the surface defined by the equation

It belongs to the class of saddle surfaces and its name derives from the observation that a saddle for a monkey requires three depressions: two for the legs, and one for the tail. The point (0,0,0) on the monkey saddle corresponds to a degenerate critical point of the function z(x,y) at (0, 0). The monkey saddle has an isolated umbilical point with zero Gaussian curvature at the origin, while the curvature is strictly negative at all other points.

To see that the monkey saddle has three depressions, let us write the equation for z using complex numbers as

It follows that z(tx,ty) = t3 z(x,y) for t ≥ 0, so the surface is determined by z on the unit circle. Parametrizing this by eiφ, with φ ∈ [0, 2π), we see that on the unit circle, z(φ) = cos 3φ, so z has three depressions. Replacing 3 with any integer k ≥ 1 we can create a saddle with k depressions.