Monkey saddle

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The monkey saddle

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

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

It belongs to the class of saddle surfaces and its name derives from the observation that a saddle for a monkey 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 umbilic 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

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

It follows that z(tx,ty) = t³ z(x,y) for t ≥ 0, so the surface is determined by z on the unit circle. Parametrizing this by e^iφ, 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.

[edit] Horse saddle

The term horse saddle is used, in contrast to monkey saddle, to designate a saddle point that is a minimax, that is to say a local minimum or maximum depending on the intersecting plane used. The monkey saddle will have a local maximum along certain planes, but it won't be a local minimum along others — just a point of inflection.