My little excursions into the history of minimal surfaces continues with a contribution of Heinrich Scherk from 1835. Making assumptions that allowed him to separate variables in the so far intractable minimal surface equation, he was able to come up with several quite explicit solutions, two of which are still of relevance today.
In its simplest version, the singly periodic Scherk surface looks from far away like two perpendicular planes whose line of intersection has been replaced by tunnels that alternate in direction.
The next milestone concerning these surfaces took place 1988, over 150 years later, when Hermann Karcher constructed astonishing variations. Among others, he showed they can be had with (many) more wings
and even twisted:
Now, can they also be wiggled? The prototype here is the translation invariant Enneper surface. It has the feature that it can be wrapped onto itself after sliding it any distance.
In other words, it is continuously intrinsically translation invariant.
Hmm. I should patent this.
So we can switch out the boring flat Scherk wings with the wiggly Enneper wings, like so, still keeping everything minimal, pushing the notion to its limits.
Here is a more radical version. You don’t want to run into this in the wild.
A while back I posted a few images from a nearby industrial ruin. Here are some more.
This time, the theme are walls. They are back in fashion these days, and it might help to see what inevitably happens to them.
The function of a wall is rarely only to separate two sides,
but also to be a platform for comment,
a target for subversion,
or sometimes a symbol for future obsolescence.
We just need to make sure that the open space is significantly better.
Sometimes, the Enneper surface will just show up. For instance, when classifying complete minimal surfaces of small total Gauss curvature, it is unavoidable. Together with the catenoid it hold the record of having only total curvature -4𝜋. Next comes -8𝜋, and for this you will encounter critters like these that have look like an Enneper surface with two catenoids poking out.
There are many others, and I view them not so much as objects to be classified and put away but rather as play grounds where one can learn what design goals are compatible with the constraint of being a minimal surface.
For instance, adding a base to the surface above is possible but pulls the two top “lobes” of Enneper and with them the two inward pointing catenoids apart:
But still, the Enneper surface comes in handy. The k-Noids, which traditionally are minimal surfaces just with catenoidal ends, have to be well balanced: The catenoids pull and push in the direction of their axes, and get boring after a while. The Enneper surface is much stronger then any number of catenoids and will win any tug-of-war.
My last pre-digital visit to the Point Reyes National Seashore was in late fall of 2000.
I find it amusing to see how the way we view things can change in mere seven years.
There is a first image of which has become a leitmotiv since:
And of course, the accidental color among all the gray.
Another special feature of Enneper’s surface is that it is intrinsically rotationally symmetric. This means that if you had a marble version of it, and a paper copy (made of curved paper, that is) sitting on top of it, you could rotate the paper copy smoothly by 360 degrees just by bending the paper, but without tearing or stretching. Amusingly, there is no truly rotational symmetric in Euclidean space that is isometric to Enneper’s surface.
Enneper’s surface shares this surprising feature with a few other minimal surfaces, like the one with five ear lobes instead of just two above. By the way, that the lobes touch is an artistic choice. The surface extends indefinitely, intersecting itself, which has led to its partial demise. There are also intrinsically rotationally symmetric minimal surfaces with two ends, like the plane and catenoid, or the more amusing one below with a planar end and an Enneper style end at the center.
This rotational symmetry gets lost when you stack two equal Enneper surface on top of each other, like so:
In mathematics, when you give up something, you typically can gain something else. In this case, you gain flexibility. You can change the distance between the two wiggly Enneper ends and bring them so close together that cleaning in between becomes impossible. The version below would make an interesting wheel. Use at your own risk.
When I first saw this nicely bent tree in McCormick’s Creek State Park in the fall of 2008, I did not expect to see it again.
Arch like trees have become something like an archetype for me, or rather, as I am not so fond of C.G. Jung, a pattern, as in pattern language. They serve the (purely symbolic, of course) dual purpose of creating a connection between two sides and signaling a passage through, and all this under the apparent duress of being bent to the verge of breaking. In any case, this arch was still there in winter, the next summer,
and the following years.
Is it still there? I leave it to you to decide whether this year’s image shows he same tree again.
It does not matter. Thomas Mann explains in his tetralogy Joseph und seine Brüder his concept of time: Events, or motifs for stories, or patterns, reoccur or are at least thought to reappear over and over, with no hope to trace their origin or future repurposing.
There will always be trees ready to bend, even after countless others have been broken.
In Memoriam, Orlando 6-12-2016
My second visit to Point Reyes National Seashore was later in 1993, when the weather forecast promised high coastal winds, and Bryce suggested to go storm watching.
Above we are on our way to the Lighthouse, and below are the first storm clouds.
It got a little bit more dramatic,
but we stayed dry and took pretty silhouette pictures.
At the end, the colors returned.
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.
My first visit to Point Reyes National Seashore was on the occasion of the CHAOS Fall Gourmet Trip 1993. The rules for these trips are simple: Dress up and bring good food.
On the way to the camp site you were also supposed to help carry supplementary items like pieces of a portable hot tub.
After pitching the tents and admiring each other’s costumes, more serious activities would commence.
There were also opportunities to hunt for more food.
Which was obviously rather tasty.