Monkey saddle z = x³ − 3xy²
Equation: z = x^3 - 3 * x * y^2
In polar coordinates, x³ − 3xy² = r³ cos(3θ). Three downhill and three uphill directions meet at the origin — room for two legs and a tail.
The Hessian vanishes there, so classifying the critical point needs third-order derivatives.