Monkey saddle

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The Monkey saddle

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

z = x³ − 3xy².

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 origin of the monkey saddle is an example of a degenerate critical point.

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) = t^3 z(x,y) for $t\ge 0$ , so the surface is detetermined by z on the unit circle. Parametrizing this by $e^{i\phi}, \phi\in[0,2\pi)$ , we see that on the unit circle, $z (φ) = cos3φ$ , and z has three depressions. Replacing 3 with any integer $k\ge 1$ we can create a saddle with k depressions.

The term horse saddle is used, by contrast with 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.