Praising the Underrated (Enneper I)

In 1760 Joseph Lagrange writes, after establishing the minimal surface equation of a graph and observing that planar graphs do indeed satisfy his equation, that “la solution générale doit ètre telle, que le périmètre de la surface puisse ètre détermine a volonté” — the general solution ought to be such that the perimeter of the surface can be prescribed arbitrarily.

For a hundred years, little progress was made to support Lagrange’s optimism. Few examples of minimal surfaces were found, and most of them with considerable effort. Then it the second half of the 18th century, it took the combined efforts of Pierre Ossian Bonnet, Karl Theodor Wilhelm Weierstraß, Alfred Enneper, and Hermann Amandus Schwarz to unravel a connection between complex analysis and minimal surfaces that would become the Weierstrass representation and revolutionize the theory.

One piece in this story is Enneper’s minimal surface. Enneper was not so much after minimal surfaces but after examples of surfaces where all curvature lines are planar. This was immensely popular back then, and the long and technical papers are mostly forgotten.

Above is an attempt to visualize the planes that intersect the Enneper surface in its curvature lines.

Visually easier to digest are the ruled surfaces that are generated by the surface normals along the curvature lines, because here the ruled surfaces and the Enneper surface meet orthogonally. While not planar, they are still flat, and invite therefore a paper model construction (that one can do for the curvature liens of any surface):

Print and cut out the five snakes. The orange centers are the curvature lines. Also cut all segments that go half through a snake, and fold along all segments that go all the way through a snake, by about 90 degrees, always in the same way. Then assemble by sliding the snakes into each other along the cuts, like so:

The three long snakes close up in space and need some tape to help them with that. Here is a retraced version of the same model which might help.