# 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](https://www.wavelace.com/app#p=179) · [This page](https://www.wavelace.com/presets/enneper-surface)

### 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.

### Read more

- [Enneper surface](https://en.wikipedia.org/wiki/Enneper_surface)
- [Minimal surface](https://en.wikipedia.org/wiki/Minimal_surface)
- [Weierstrass–Enneper parameterization](https://en.wikipedia.org/wiki/Weierstrass%E2%80%93Enneper_parameterization)
- [Mean curvature](https://en.wikipedia.org/wiki/Mean_curvature)