Enneper Surface
Shape · (x, y, z) = f(u, v, t)
(v · (3 + sin(t/2))/5 · cos(u) −(v · (3 + sin(t/2))/5)³/3 · cos(3u), (v · (3 + sin(t/2))/5)² · cos(2u), v · (3 + sin(t/2))/5 · sin(u) +(v · (3 + sin(t/2))/5)³/3 · sin(3u))
Open in the app The dials and keys named below are the app's.
What it draws
A disc of Enneper's minimal surface whose edge grows and shrinks. In each expression s = v · (3 + sin(t/2))/5 is the distance from the centre and u the angle round it. The sheet is x = s cos u − s³/3 cos 3u, height y = s² cos 2u and z = s sin u + s³/3 sin 3u.
With a = s cos u and b = s sin u these are the textbook equations: a − a³/3 + ab², b − b³/3 + ba² and the height a² − b². Wavelace is y-up, so the height sits in the middle expression. Written by angle and radius, the disc ends in a round edge rather than a square one.
As v runs from 0 to π, the radius reaches π · (3 + sin(t/2))/5. That edge breathes between 2π/5 ≈ 1.26 and 4π/5 ≈ 2.51, once every 4π ≈ 12.6 seconds at Speed 1.
Where it meets itself
Near the centre the surface is a saddle, rising along the x axis and falling along the z axis. Further out the cubic terms fold the rim back over the middle.
At radius √3 ≈ 1.73 the folds first touch. The points at angles 0 and π both land on (0, 3, 0), and those at ±π/2 both on (0, −3, 0). Past that radius the sheet passes through itself along two curves, one above the centre and one below.
The edge is below √3 only while sin(t/2) is under −0.243. That is about 5.3 seconds of each 12.6, from t ≈ 6.8 to 12.1, and the rest of the time the surface crosses itself.
Why it is minimal
The mean curvature is zero at every point: the sheet bends up in one direction exactly as much as it bends down in the one across. That is the property of a soap film, which is why such surfaces are called minimal.
History
Alfred Enneper introduced the surface in 1864, in his work on minimal surfaces. It comes from the simplest choice in the Weierstrass–Enneper formula, which builds a minimal surface out of two complex functions. Robert Osserman later proved that a complete minimal surface with total curvature −4π is either a catenoid or this one.
Try
- In all three expressions replace 3 + sin(t/2) with 2: the edge stays at radius 1.26, and the disc never touches itself.
- Replace it with 4 instead: the edge stays at 2.51, and the two curves where the sheet crosses itself hold still.
- Front: the sheet seen side on, with the two curves where it crosses itself one above the centre and one below.
- Drag Mesh to its top stop: the solid shows where the sheet passes through itself.