I started this little blog in late 2015 while being Director of Graduate studies in our department in order to keep sane. This proved (for me) to be very useful but a little irritating to the casual visitor who saw posts alternating between photography and a mild dose of mathematics.
This is about to change a little, because another project is taking my time. I call this the Minimal Surface Repository, a combination of blog and archive, that will vastly expand my minimal surface web pages. So my mathematical Monday posts here will become posts at the Repository, while the Friday posts will remain what this blog here was supposed to be: Thoughts about my inner state, accompanied by pictures, with an occasional game or puzzle thrown in.
So, please, head over, if you are interested in minimal surfaces and related topics. Otherwise, wait until Friday for more pictures from gloomy Indiana…
Below is something rare. You see two minimal surfaces in an (invisible) box that share many properties, but also couldn’t be more different.
Let’s first talk about what they have in common: They share lines at the top and bottom of the box, and they meet the vertical faces of the same box orthogonally. This means you can extend both surfaces indefinitely by translating the boxed surfaces around, in which they become triply periodic surface of genus 3.
How are the different? The red one is a little bit more symmetric and belongs to a 2-dimensional deformation family of the Diamond surface that has been known for about 150 years. You can see how these surfaces deform in an earlier post.
The other one belongs to a different deformation family that is only a few weeks old, discovered by Hao Chen, and of which you can see here some wide angle pictures, with clearly different behavior.
These surfaces existed right under our nose, but nobody expected them to exist, because minimal surfaces are usually content with a single symmetric solution. Chances are that these surface hold the key in understanding the entire 5-dimensional space of all triply periodic minimal surfaces of genus 3.
This story begins in 1988 with the first examples of doubly periodic embedded minimal surfaces where the top and bottom ends are parallel and asymptotic to vertical half planes. They were found by Karcher and Meeks-Rosenberg in two independent papers and look like these:
I like it how confusingly similar they are. The main distinction is that that the one on the left has horizontal straight lines on it about which you can rotate the surface unto itself, while the one on the right has a reflectional plane of symmetry instead.
These surfaces actually come in a 3-parameter family; what you see above are the most symmetric cases. The translational periods are horizontal, and there are vertical straight lines. If you divide by the translations, you get a torus as a quotient surface. Remarkably, this 3-dimensional family is all there is for genus 1, by work of Pérez, Rodriguez, and Traizet from 2005. In particular this means that the two surfaces above can be deformed into each other through minimal surfaces. This is not too hard to see.
Things get more interesting already at genus 2: At 1992, Wei found a 1-parameter family of examples of genus 2.
A specimen is below to the left.
A variation of it was constructed by Rossman, Thayer and Wohlgemuth in 2000 (above to he right). Again they look dazzlingly similar. I suspect that they cannot even be deformed into each other through embedded minimal surfaces, but I have no idea how to prove this.
Even better, Connor has numerically found another genus 2 example that looks significantly different from the ones above. One of the holes has is not symmetrically positioned anymore, and there is no way to get it back there…
Showing the existence of these surfaces would be good, and even better would be to find a way to distinguish it from the others.
But almost. It has a vertical axis, lots of horizontal lines, and it twists.
But it is part of something bigger, a triply periodic minimal surface. 32 copies of the above piece, replicated by rotations and reflections, look like this:
This surface sits in a rectangular box over a square. If you identify top and bottom edge of the original squarical helicoid, you get a doubly twisted annulus, which is intimately (confomally, that is) related to a hollow spiderweb:
When squeezing the height down, our non-helicoids become even more helicoidal. When pulling the height up, the helicoids disappear. What we have here is a deformation of the Diamond surface of Hermann Amandus Schwarz.
When he sees this, he will probably just nod.
What happens when we pull a little further? We see doubly periodic Scherk surfaces emerging, stacked on top of each other.
The first examples of periodic minimal surfaces with helicoidal ends (besides the helicoid itself) are Hermann Karcher’s twisted Scherk surfaces from 1988.
Here are a few of them, rendered with Bryce back in 1999.
As you can see, these can be twisted more and more so that they appear to become two helicoids glued together. In this case, the two helicoids turn the same way. A few years later, Martin and I were looking at more general ways of gluing helicoids together to obtain new minimal surfaces. The model case is what we called a parking garage structure: You can describe them mathematically as superpositions of complex argument functions, like so:
Here the numbers z(k) designate the location of the axis viewed from above, and the ε(k) can be +1 or -1, depending on the spin of the helicoid.
Note that the graph of the multivalued function arg(z) is half of a helicoid (that stays on one side of the vertical axis).
An example with three helicoidal columns of the same spin, placed at -1, 0, and 1, looks like this:
If you alternate the spin, you get surfaces that untwist to higher genus helicoids, we believe.
It is also possible to place the columns off a common line, like so:
Nobody knows what minimal surfaces these untwist to.
The images above were made with Mathematics in 2001. Later I found a simple way to do this in PoVRay, which I might explain next time. Here an image from 2002:
Most people get easily lost in parking garages that have only two columns. It would be cool to have a computer game where one can walk around these more complicated structures, with the location of the columns moving in time …
I like it when apparently simple things evolve all by themselves into complex objects. Like watching cactus seeds grow into cacti. That was a distraction, but I do like it. Below is a left over piece of mathematics that would have fit nicely into a paper I wrote with Shoichi about triply periodic minimal surfaces.
It is, evidently, quite complicated. To unravel it, here is a smaller portion of it, its seed, so to speek.
That is a minimal surface inside a prism over a 2-3-6 triangle (which has a right angle, a 30 degree angle, and a 60 degree angle).
The curves in the vertical faces of the prism are symmetry curves of the surface, and reflecting at these faces of the prism extends the surface. The two curves in the bottom and top face of the prism are not symmetry curves, but when you place two prisms on top of each other (by translation), the curves will fit. The pattern the curves make on the prism determines the surface almost completely, there is just one degree of freedom. Here is another, equally pretty, version, using a different parameter.
Another way to seed these surfaces is through conformal geometry. Below is the conformal image of a circular annulus onto a polygonal annulus bounded by two nested 2-3-6 triangles. The parameter lines are images of radii and concentric circles, respectively. This map is the main ingredient in the Weierstrass representation of all these surfaces. Simple, isn’t it?
Computer scientists, dog owners, parents, and most other generic humans are happy when their trained subjects behave as expected. Mathematicians are happy when things develop other then expected.
For instance, Rafael and I have built a machine that takes an explicit planar curve, lifts it to a space curve, and twirls an explicit minimal surface around it. The emphasis here is on explicit, because that allows to do all kinds of things to the minimal surface that would be hard to do otherwise.
So we started feeding curves to the machine that we hadn’t built it for.
As a first example, the logarithmic spiral is lifted to a space curve such that both ends of the spiral move up, and the speed with which the surface twists is much faster at the inner piece. We call this the cobra surface.
A few years after David, Mike, and I had shown that the genus one helicoid is embedded, I was contacted by a science freelance writer. She said that this helicoid with the handle had been pretty cool, whether we had maybe some new examples that looked very different and cool, too. We hadn’t. But here they come. The Archimedean spiral is next. Again, the surface spirals faster when the curve is more strongly curved.
If you liked the trefoil surfaces, you will like the next one, too: Here we start with a common cycloid, and the lifted curve becomes another trefoil knot.
Finally, the pentagram cycloid lifts to a knotted curve without cusps, and we can make another prettily knotted minimal surface.
In mathematics, even the simplest things can have an astounding depth. Let’s for instance take the trefoil knot, the simplest knot there is:
One can replace the tube by a ribbon, like so:
This could be done with a simple ruled surface, but I like a challenge. To make this a minimal surface, one can use Björling’s formula. The game becomes tricky if one wants the surface to be of finite total curvature, but this can be done as well. Then it is not difficult to let the normal of the surface rotate once to get a knotted minimal Möbius strip.
Faster spinning normals create knotted helicoids.
Extending the surface beyond a small neighborhood of the trefoil knot makes things appear really complicated.
Of course the same can be done with more complicated knots.
Just three months before his death on July 20, 1866 (150 years ago), Bernhard Riemann handed a few sheets of paper with formulas to Karl Hattendorff, one of his colleagues in Göttingen.
Hattendorff did better than Riemann’s house keeper who discarded the papers and notes she found.
He instead worked out the details, and published this as a posthumous paper of Riemann. It contains his work on minimal surfaces. Riemann was possibly the first person who realized that the Gauss map of a minimal surface is conformal, and that its inverse is well suited to find explicit parametrizations. He used this insight to construct the minimal surface family that bears his name, as well as a few others that were later rediscovered by Hermann Schwarz.
Above is one of Riemann’s minimal surfaces, parametrized by the inverse of the Gauss map. This means in particular that the surface normal along the parameter lines traces out great circles on the sphere. Riemann discovered these surfaces by classifying all minimal surfaces whose intersections with horizontal planes are lines or circles. These are the catenoid, the helicoid, or Riemann’s new 1-parameter family.
The proof utilizes elliptic functions, which is not surprising: Riemann’s minimal surfaces are translation invariant, and their quotient by this translation is a torus, on which the Gauss map is a meromorphic function of degree 2. It is in fact one of the simplest elliptic functions, and one can use it to parametrize Riemann’s surfaces quite elegantly. What is not simple is the proof that these surfaces have indeed circles as horizontal slices. All arguments I know involve some more or less heavy computation. We are clearly lacking some insight here.
The longer one studies these surfaces, the more perplexing they become. There is, for instance, Max Shiffman’s theorem from 1956. It states that if a minimal cylinder has just two horizontal circular slices, all its horizontal slices are circles. The proof is elegant, magical, and still mysterious, just like Riemann’s minimal surfaces.
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.