In the Beginning, Leonhard Euler created the Calculus of Variations and gave many examples. In one of them he proved that if you want to minimize the area of a surface of revolution (among surfaces of revolution), you will get a piece of a catenoid or plane.
The story is not as simple as it seems, because catenoids stop being minimal at a certain size. Above, for instance, two catenoids that have the same two circles as boundaries. Clearly only one of them has minimal area.
One of the early stories of failure was the attempt to add a handle to the catenoid. Maybe one could even save area this way?
Not so, as the handle doesn’t even extend far enough to close up. Rick Schoen proved more generally that catenoids don’t come with handles of any sort and number. But one can try other things, like making the catenoid more symmetric. This is of course silly, but Luquesio Jorge and Bill Meeks did just that by turning the reflectional symmetry at a horizontal plane into a dihedral symmetry, thus creating the k-Noids.
This works for any order and gets a little boring after a while.
But, somewhat surprisingly, one can add a handle to these k-Noids when k is at least 3:
Like last week, k does not need to be an integer, and one can see clearly what goes wrong when one pushes k below 3: Here we have a broken catenoid with a handle.
Even though Berkeley is lush enough by itself (in 1993 at least), it has its little parks like People’s Park or the Rose Garden. I happened to live right below the Rose Garden and could, like the occasional deer or burglar, just hop over the little wall.
One of my (these days somewhat neglected) hobbies is to take stereo images, and the gorgeous roses were patient victims. All these pictures were taken hand-held, and they are supposed to be viewed cross-eyed, i.e. the image for the left eye is on the right.
The rule of thumb for taking stereo images handheld was to choose a landmark point at the center, take a picture, take a step to the left, recenter, and take a second picture. This works very well for average street scenes without moving objects, like MacArthur Station below.
Our brain is more than happy to ignore little inconsistencies. If you do that with flowers, you will of course end up taking an entirely different picture. So you have to scale down and move just a centimeter to the left. This is still pretty wide and gives these flowers the appearance of rather large objects. You can do the opposite by taking images from a plane and wait 10 seconds between the shots. This will give the stereo image the appearance of a toy landscape. I’ll dig out an example when we come back from Mexico (early 1994, that was).
Finally for today, a stereo image of Hermann Karcher, also from 1993, thinking about Helicoids.
Up above is Meinhard Wohlgemuth’s first surface. It is very similar to Costa’s surface, but has one more end and genus 2. In Mathematics, like in every experimental science, you gain intuition about truth through observation. For instance, every minimal surface person I know will probably believe in dihedralization.
Whenever you have a minimal surface that has two vertical symmetry planes making an angle π/2, there should be more symmetric version where the planes make the angle π/n. Like up above for n=5, observed for Wohlgemuth’s first surface.
Then there is Wohlgemuth’s second surface. It has genus 3, squeezing in a little suspicious handle between the two planar ends, so that it looks a little bit like two copies of the ill-fated Horgan surface stacked together. But this time it works, one (Meinhard) can prove this surface does indeed exist. So, let’s make it more symmetric:
This is what I get when I run the experiment. The parameter values are so extreme that I don’t think this actually exists. So, do we trust our intuition, or the experiment? Neither. The only way out is to dig deeper. Let’s cut Wohlgemuth’s surface open (in half, to be acurate).
We can now apply the dihedralization continously, slowly increasing n=2 to n=3, thereby tracking the numerical solutions. They show a clear deformation,up to approximately n=2.9 (below). After that, the periods don’t close accurately enough anymore to make me feel confident about the surface.
So I tend to think that this is the first case where dihedralization fails.
In 1993, I went to Berkeley for a year. Among many other things, I went backpacking quite a bit, and I will share some of the images over the following months, celebrating the 25 year anniversary.
The second hike I went on (the first I already wrote about) was in Yosemite, to the Ten Lakes basin. To get there, you have to cross a plateau with gorgeous views.
The scenery is serene, and there is almost no way to get lost. Just don’t make the mistake I made, trying to get from the slower group to the faster group by following the trail. The faster group had stepped off the trail for a minute to enjoy the view, so that I rushed past them, getting more and more nervous towards the evening because I couldn’t find anybody.
I was only mildly relieved when a few campers told me I had indeed reached the Ten Lakes area, where I waited nervously for two hours until the rest of the group finally arrived.
There are two clear indications that you hike with people from Berkeley: They bring text books and actually read them, and they go skinny-dipping in every little pond. Well. Above is partial proof.
On the way back you have to get up to that Plateau again, and then it gets interesting when you entire the granite fields transformed by the afternoon light.
The rocks that are scattered around there cause suspicion that they have been purposefully placed,
and it is up to us to decipher the message.
For me, this was easy. It meant Come Back.
Die Wahrheit ist dem Menschen zumutbar.
Occasionally, after confronting students with evidence of fact (Euler Polyhedron Theorem is a great example), I ask them whether they want to see a proof or prefer to accept the statement as a miracle. The overwhelming majority is always happy with the miracle. Such are the times. Below is such an evidence of fact: A minimal surface with 3 ends and of genus 2.
Should we doubt its existence? In 1993, John Horgan published an article in Scientific American questioning whether proofs were about to become obsolete, in times where shear length and difficulty made validation next to impossible, and numerical experiments supplied by computers could be an acceptable substitute. For many reasons, large parts of the mathematical community were outraged.
Above is another example of that surface, for a different parameter value, but something seems off. There appears to be a little crack. Maybe I didn’t compute accurately enough? Changing the parameter a bit more widens the gap.
The question whether this surface does actually exist hinges on the possibility to truly close that gap, for at least one parameter value. It appears that we have done so in the top image. But the parameter value there is 1.01, pretty close to 1, where the surface will clearly break down. A more accurate computation shows that there still is a gap at 1.01, which we can’t see, or don’t want to see. But maybe 1.001 will do?
David Hoffman and Hermann Karcher analyzed this surface in 1993, the same year as Horgan’s article, and it became known as the Horgan surface. One can indeed prove that the gap cannot be closed, so, despite all the evidence, this minimal surface does not exist.
This is a tomato plant. You can call it misfocus of the cheap digital camera I grabbed quickly, or well-staged dramatic suspense, because, as you will have noticed, there is something in the background.
A few minutes earlier my daughter had spotted an egg case out of which cute little monsters were emerging.
Praying mantises are fascinating. It seems so easy to say: Oh, that’s not one of our species. But then, why do we project aggressiveness into everything they do – hunting posture, sex life, the way they look?
One thing clearly distinguishes them from us: They are ready for life seconds after birth.
Whatever you think, that tomato plant was free of all other insects a few minutes later.
When you take two non-parallel planes, they will intersect in a line. The singly periodic Scherk surfaces are the only minimal way to “desingularize” this, in the sense that they are the only known minimal surfaces asymptotic to these two planes. To show this is one of the many famous open problems about minimal surfaces.
The situation gets vastly more complicated with three planes. Nobody has yet succeeded in constructing a minimal surface that is asymptotic to the three coordinate planes. That is another open problem. A case where we do know something is that of three (or more) vertical planes. Martin Traizet has shown in 1994 that in case the planes are reasonably general one can wiggle them a little bit and desingularize them by gluing in singly periodic Scherk surfaces. The concrete and very symmetric example above was known before that.
The only requirement on the Scherk surfaces is that they have the same translational period and share a horizontal reflectional symmetry plane to ground them. But nothing prevents us from shifting one of the Scherk surfaces by a half-period, like up above. To make the image, I assumed another reflectional symmetry at a vertical plane (roughly parallel to the screen). This still left me with a 1-parameter family, whose existence is truly only guaranteed near the limit that looks like three Scherk surfaces (with one of them shifted). But nothing keeps us from looking at the other surfaces in this family.
Above I have turned it around so that one can appreciate the handles better. What emerges becomes clear when one pushed the parameter further:
A singly periodic Costa surface! There is a similar one constructed by Bastista and Martín where the Costa-necks are rotated by 45 degrees. It then loses its reflectional symmetries but gains straight lines.
Among the more pleasant creatures who thrive in the current heat and humidity are the butterflies. My daughter’s obsession with them started when she was five, with a little moth.
It then took off in Michigan that summer, where the rustic campground offered plenty opportunity to look for cute little critters in all the unspeakable places.
It takes no time to figure out that the thing to do is to collect the caterpillars, feed them and watch them molt.
The tricky part is to find the right host plants for your caterpillar collection. So we focussed on monarchs and de-leaved all the milkweed plants in the neighborhood.
You put them in a box, and a stick on top of that box instead of a lid when the caterpillars get fat and restless.
If you are lucky, you get several of them lining up on the stick, and then you can see them molting one by one. The act of getting out of the cocoon is pretty dramatic.
When the wings are pumped up, be sure to have the food ready.
In 1982, Chi Cheng Chen and Fritz Gackstatter published a paper that described the surface below.
Like some of the classical examples of minimal surfaces, this surface is complete and has finite total curvature. A famous theorem of Osserman from 1964 asserts that any such surface can be defined on a punctured Riemann surface. In the classical examples, this had always been a sphere, but here we have a torus with one puncture. There were some earlier examples, but this one, while not embedded, was surprisingly simple. From far away, it looks just like the Enneper surface.
How does one make such an example? One problem is illustrated above: While Osserman’s theorem also guarantees that the derivative of a conformal parametrization has a meromorphic extension to the compact surface, the integration of these so-called Weierstrass data might leave gaps.
To close the gap, we use the help of symmetries: Two vertical planes cut the surface into four congruent pieces, each represented by the upper half plane. The Weierstrass forms and then turn out to be Schwarz-Christoffel integrands. The corresponding integrals map the upper half plane to (infinite) Euclidean polygons, shown above. The left extends to cover a bit more than a quarter plane, the right a bit less than a three quarter plane.
Incidentally, we can see the torus by fitting four copies of the right polygon together. We obtain the plane with a square missing. Identifying opposite edges of the missing square creates a torus with one puncture.
Now the condition that makes the gaps disappear is just that the two polygons fit together, which can be achieved by scaling. It’s really that simple. Similarly one can have more symmetric versions by just changing the angles in the polygons. Below is an example with sevenfold symmetry.