In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the monkey saddle is the surface

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

 defined by the equation


It belongs to the class of saddle surface

Saddle surface

A saddle surface is a smooth surface containing one or more saddle points.The term derives from the peculiar shape of historical horse saddles, which curve both up and down....

s and its name derives from the observation that a saddle

Saddle

A saddle is a supportive structure for a rider or other load, fastened to an animal's back by a girth. The most common type is the equestrian saddle designed for a horse, but specialized saddles have been created for camels and other creatures...

 for a monkey

Monkey

A monkey is a primate, either an Old World monkey or a New World monkey. There are about 260 known living species of monkey. Many are arboreal, although there are species that live primarily on the ground, such as baboons. Monkeys are generally considered to be intelligent. Unlike apes, monkeys...

 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

Critical point (mathematics)

In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...

 of the function z(x,y) at (0, 0). The monkey saddle has an isolated umbilic point with zero Gaussian curvature

Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...

 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 number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s as
It follows that z(tx,ty) = t³ z(x,y) for t ≥ 0, so the surface is determined by z on the unit circle

Unit circle

In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

. 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.

Horse saddle

The term horse saddle is used, in contrast to monkey saddle, to designate a saddle point

Saddle point

In mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...

 that is a minimax

Minimax

Minimax is a decision rule used in decision theory, game theory, statistics and philosophy for minimizing the possible loss for a worst case scenario. Alternatively, it can be thought of as maximizing the minimum gain...

, that is to say a local minimum or maximum depending on the intersecting plane used. The monkey saddle has just a point of inflection. To see this, suppose y = kx: then z = (1 − 3k²)x³.