Enneper Surface

Rotating Enneper Surface – Greg Egan

This is the Enneper surface, as drawn by Greg Egan using Mathematica. It’s a minimal surface, meaning one that necessarily gets more area if you warp any small patch of it. A soap film will make a minimal surface if it doesn’t enclose any air. But the Enneper surface intersects itself: it’s immersed in 3d space, but not embedded. So, you can’t make it with soapy water!

You can describe the Enneper surface using these equations:

$$x = u – u^3/3 + uv^2$$

$$y = -v + v^3/3 – vu^2$$

$$z = u^2 – v^2$$

As $u$ and $v$ range over all real numbers, the point $(x,y,z)$ hits every point of the Enneper surface.

Alfred Enneper and Karl Weierstrass were thinking about minimal surfaces back around 1863, and they discovered every such surface that’s simply a disk mapped into $\mathbb{R}^3$ could be described in a clever way using complex analytic functions. This is called the Enneper–Weierstrass parametrization. The Enneper surface is distinguished by the fact that it has an extremely simple parametrization of this sort.

The Enneper surface is also special because it is a complete minimal surface in $\mathbb{R}^3$ for which the integral of the Gaussian curvature over the whole surface is $-4\pi$. The only other surface with this property is the catenoid, obtained by rotating a catenary. The catenoid was the first minimal surface to be found after the plane: it was proved to be minimal by Leonhard Euler in 1744.

Can you figure out the symmetry group of the Enneper surface?

For more, read:

• Enneper surface, Wikipedia.

• Enneper–Weierstrass parametrization, Wikipedia.

• Myla Kilchrist and Dave Packard, The Weierstrass–Enneper representations, Dynamics at the Horsetooth 4 (2012).