Monkey Saddle -- from Wolfram MathWorld

Monkey Saddle

A monkey is a surface which a monkey can straddle with both legs and his tail.

A simple Cartesian equation for such a surface is

(1)

which can also be given by the parametric equations

			(2)
			(3)
			(4)

or in cylindrical coordinates as

(5)

The monkey saddle has a single stationary point as summarized in the table below. While the second derivative test is not sufficient to classify this stationary point, it turns out to be a saddle point.

			point
	2	0	saddle point

The coefficients of the first fundamental form of the monkey saddle are

			(6)
			(7)
			(8)

and the second fundamental form coefficients are

			(9)
			(10)
			(11)

giving Riemannian metric

(12)

area element

(13)

and Gaussian and mean curvatures

			(14)
			(15)

(Gray 1997). The Gaussian curvature can be written implicitly as

(16)

so every point of the monkey saddle except the origin has negative Gaussian curvature.

Peckham (2011) asked about the existence of monkey saddles in natural landscapes, and several were subsequently identified by Coté et al. (2020).

References

Coté, J.-J.; Elgersma, M.; Kline, J. S.; and Wagon, S. "Monkey Business." Math Horizons 28, 16-17, Sep. 2020.

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 365, 1969.

Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, p. 202, 1999.

Peckham, S. "Monkey, Starfish, and Octopus Saddles." Proc. Geomorphometry 2011: Five days of Digital Terrain Analysis. Redlands, CA, pp. 31-34, 2011.

Monkey Saddle

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