Monkey Saddle
z = (x³ − 3x · y²)/20 · cos(t)
Open in the app The dials and keys named below are the app's.
What it draws
The height is the cubic (x³ − 3xy²)/20, scaled by cos t. In the polar pair the same cubic is r³·cos(3θ)/20. The height grows as the cube of the distance from the origin, so the middle is nearly flat and the edges do all the work, and the sign turns over three times in a circuit.
cos t multiplies the whole sheet at once. At Speed 1 it takes 2π ≈ 6.28 seconds to carry the shape through flat at t = π/2, right over at t = π, and back.
Three legs
cos(3θ) is 1 at θ = 0°, 120°, 240° and −1 at 60°, 180°, 300°. So three ridges rise and three valleys fall, alternating every 60°. An ordinary saddle drops away in two directions, which is enough for two legs. This one drops away in three, leaving a place for the tail, and that is where the name comes from.
The origin is the only point where the sheet is level, and it is level in an unusual way: the first derivatives vanish there and so do all three second derivatives. The quadratic that normally separates a summit from a hollow from a pass is simply absent, and the cubic decides instead. That is a degenerate critical point, the one classified as an elliptic umbilic. The cubic is also Re((x + iy)³), so it satisfies Laplace's equation. A harmonic function has no summit and no hollow anywhere inside, which is why every part of the sheet is a pass and the curvature stays negative away from the origin.
Try
- Top in the deck, looking straight down: three light lobes and three dark ones, turning colour every 60°.
- Write it in polar instead, r³·cos(3θ)/20 · cos t: the picture does not change.
- Then ask for four descents, r⁴·cos(4θ)/200 · cos t: four ridges and four valleys, and no tail.
- Drop the clock, (x³ − 3·x·y²)/20, to hold the sheet still and orbit it.
- Raise Span of x, y and watch the corners run away as the cube of the distance.