Monkey Saddle Visualization | Three Hills, Three Valleys, a Flat Point

The idea: a saddle with room for a tail

The monkey saddle is the surface

z = x³ − 3xy²

It is the graph of a polynomial of degree 3, so it is a cubic surface, the simplest one after the quadrics. A horse saddle, z = x² − y², goes up in two directions and down in two: room for the rider's two legs. The monkey saddle goes up in three directions and down in three: two valleys for the legs and a third one for the tail.

Two discs seen from above. Left, the ordinary saddle z = x² − y²: the lines y = x and y = −x cut the disc into four 90° sectors, two hills (red, +) on the left and right and two valleys (blue, −) at the top and bottom, marked leg and leg. Right, the monkey saddle z = x³ − 3xy²: the lines x = 0 and y = ±x/√3 cut it into six 60° sectors, three hills at 0°, 120° and 240° and three valleys at 60°, 180° and 300°, marked leg, tail and leg. — Seen from above: the horse saddle has two valleys, the monkey saddle three.

On the canvas the surface is drawn over a disc around the origin, coloured by height: red is high, blue is low. The dropdown n switches to the ordinary saddle (n = 2) or to a saddle with four valleys (n = 4), and a stretches the height, z = a(x³ − 3xy²).

Polar form: r³ cos 3θ

Put x = r cos θ and y = r sin θ. Then

x³ − 3xy² = r³(cos³θ − 3 cos θ sin²θ) = r³ cos 3θ

by the triple-angle formula. (It is also the real part of (x + iy)³.) So the distance r only sets the size, and the angle θ decides up or down. Walk once around the origin on a circle: cos 3θ goes up and down three times. The tops are at θ = 0°, 120°, 240° (the hills) and the bottoms at 60°, 180°, 300° (the valleys). The horse saddle is x² − y² = r² cos 2θ: two of each. (Demo 1: walk around: three hills, three valleys)

The cut z = 0: three lines

The polynomial factors:

x³ − 3xy² = x(x² − 3y²) = x(x − √3y)(x + √3y)

So z = 0 on three lines through the origin: x = 0 and y = ±x/√3, at 90°, 30° and 150°. They are 60° apart and cut the plane into six sectors where z is positive and negative in turn. For the horse saddle, x² − y² = (x − y)(x + y) gives two lines and four sectors. In the panel "the cut z = h, seen from above" the signs + and − mark the sectors.

Cuts z = h: cubic curves with three branches

A cut z = h ≠ 0 is the curve x³ − 3xy² = h, a cubic curve. In polar form r³ cos 3θ = h, so r = (h / cos 3θ)^1/3 wherever cos 3θ has the sign of h. That gives three separate branches: for h > 0 one around each hill, for h < 0 one around each valley. Each branch runs off to infinity along two of the lines of z = 0. As h shrinks to 0, the branches squeeze onto the three lines; past 0 they come out in the other three sectors. (Demo 2: slide the cut z = h from 0.75 to −0.75)

Left, the contour map of z = x³ − 3xy² for |x|, |y| ≤ 1.4: the three black lines of z = 0 through the origin, three red branches for z = 0.25 and z = 1 in the sectors at 0°, 120° and 240°, three blue branches for z = −0.25 and z = −1 in the sectors at 60°, 180° and 300°, and a green circle r = 1. Right, the height along that circle, z = cos 3θ for θ from 0° to 360°: three peaks of height 1 at 0°, 120°, 240° and three dips to −1 at 60°, 180°, 300°. — Each level is three branches; around the circle r = 1 the height goes up and down three times.

The flat point: the second-derivative test says nothing

For f(x, y) = x³ − 3xy² the first derivatives are f_x = 3x² − 3y² and f_y = −6xy. Both are 0 at the origin, so it is a critical point and the tangent plane is z = 0. The second derivatives are f_xx = 6x, f_xy = −6y, f_yy = −6x, and all of them are 0 at the origin too. The Hessian is the zero matrix and D = f_xxf_yy − f_xy² = 0: the second-derivative test says nothing. Such a point is called degenerate. (Demo 3: the flat point: gradient and Hessian are 0)

	horse saddle x² − y²	monkey saddle x³ − 3xy²
gradient at 0	(0, 0)	(0, 0)
Hessian at 0	(2, 0; 0, −2)	all 0
D	−4 < 0: saddle point	0: the test says nothing
along the x-axis	z = x²: a bottom at 0	z = x³: flat at 0, no bottom, no top
near 0, \|z\| is at most	r²	r³: much flatter

So the answer has to come from looking at the function itself. Along y = 0 the surface is z = x³: it rises on one side and falls on the other, so the origin is no maximum and no minimum. It is a saddle point, just one that the test cannot see. The page Critical Points and Singular Points shows this same point as a contour map, next to ordinary maxima, minima and saddles.

Two graphs of the cut y = 0 near the origin with the tangent line z = 0 dashed. Left, the horse saddle: z = x², a parabola with its bottom at the origin, marked minimum along x. Right, the monkey saddle: z = x³, which lies flat on the tangent line at the origin and passes through it, below on the left and above on the right, marked no minimum, no maximum. — Along the x-axis the horse saddle has a bottom at 0; the monkey saddle passes straight through, flat.

The family rⁿ cos nθ

The same pattern works for any n: z = rⁿ cos nθ, the real part of (x + iy)ⁿ, has n hills and n valleys, and its cut z = 0 is n lines through the origin.

n	z in x, y	hills, valleys	cut z = 0	Hessian at 0
2	x² − y²	2, 2	y = ±x	not 0: saddle point
3	x³ − 3xy²	3, 3	x = 0, y = ±x/√3	0
4	x⁴ − 6x²y² + y⁴	4, 4	y = ±(√2 − 1)x, y = ±(√2 + 1)x	0

All of them are harmonic: f_xx + f_yy = 0 everywhere (for the monkey saddle, 6x − 6x = 0). A harmonic function can never have a top or a bottom in the middle of a region, so every one of these flat points is a kind of saddle. The stretch a, in z = a(x³ − 3xy²), makes the hills taller but does not move them. (Demo 4: ordinary saddle → monkey saddle, Demo 5: four valleys: r⁴ cos 4θ)

Facts

property	z = x³ − 3xy²
degree	3: a cubic surface, the graph of a polynomial
polar form	z = r³ cos 3θ = Re (x + iy)³
symmetry	turning by 120° about the z-axis; mirror in the plane y = 0; turning by 60° flips the sign of z
critical points	only the origin, degenerate (Hessian 0)
cut z = 0	three lines x = 0, y = ±x/√3
cut z = h ≠ 0	a cubic curve with three branches

See also Hyperbolic Paraboloid (the horse saddle, with its two families of straight lines), Level Curves (contour maps of other functions) and Quadric Surfaces (all surfaces of degree 2). Other surfaces of degree 3 and 4 are on Cubic Surface, Cayley's Nodal Cubic, Kummer Surface and Quartic Surface, and all of them side by side on Surfaces by Degree.

What the page leaves out

The curvature of the monkey saddle (negative everywhere except at the origin, where it is 0), higher-order tests for degenerate critical points, other cubic graphs such as z = x³ + y³, and the hidden-line drawing of the surface: the wireframe shows the back of the surface through the front.