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

${\displaystyle z=x^{3}-3xy^{2}.\,}$

It belongs to the class of saddle surfaces, and its name derives from the observation that a saddle for a monkey would require 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 show that the monkey saddle has three depressions, one can write the equation for z using complex numbers as [1]

${\displaystyle z(x,y)=\operatorname {Re} [(x+iy)^{3}].}$

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

Another example of the monkey saddle is the Smelt petal defined by the equation ${\displaystyle x+y+z+xyz=0}$.[2][3]

Smelt petal: ${\displaystyle x+y+z+xyz=0}$