Minimal Surfaces, At Their Limit

The minimal surfaces in the post about Möbius strips were made using a formula by Emmanuel Gabriel Björling, a Swedish Mathematician from the 19th century. For a given curve in space, this formula allows you to write down a parametrization for a minimal surface that not only contains the given curve, but is also tangent to the curve in any way you wish to prescribe. For instance, the multiple times twisted Möbius strips all contain a circle, and touch the circle by spinning around it more or less often.


This formula works not only for circles but also for other curves, like the helix above. The difficulty is that in most cases, the equations are so complicated that they become meaningless. There are some pretty exceptions, like this knotted minimal strip:


In the search for interesting and simple curves where Björling’s formula gives manageable results, the multiply twisted minimal strips are particularly useful. We saw that the surfaces in their associate family are also closed strips, but their core curves are not just circles anymore. Using these as new core curves can be used to compute surprisingly simple formulas for surfaces like this one:

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This can get quite complicated. Try to view this stereo pair cross-eyed or with a stereo viewer.

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These shapes appear to contradict what we think should be a minimal surfaces. But that’s what we do: Seek what goes against the convention.

La terre qui penche

In Carole Martinez’ extraordinary novel La terre qui penche we encounter the Middle Ages through the eyes of adolescent Blanche (and her mysterious and timeless alter ego, the Old Soul). Nature has not been conquered yet: Imagination and poetry instead of science are the primary means of comprehension.

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Southern Indiana’s landscape is hard to capture, because it is full of ruthless vegetation, and the harsh sun provides unwanted contrast. I usually resort to taking pictures of carefully selected views before or just at sunrise. This works well, but doesn’t capture how it really looks like.

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So I decided to go all the way to the other extreme, using a 15mm fisheye lens. We truly have the world tilting now, and this is how it feels like between the creek and the bluff at Cedar Bluffs Nature Preserve.

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Depending how one uses the fisheye, the effect can be more or less intrusive. One can have a peaceful valley that is a bit too curved, or a disorienting view down the bluff.

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I will need to revisit these images when I am less under the impression of La terre qui penche.

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Möbius and Friends (Scrolls V)

I am sure we all have cut a Möbius strip in half, and been irritated by not getting two pieces but instead a doubly twisted strip.


The quickest way to parametrize a Möbius strip is as a ruled surface, letting line segments rotate by 180 degrees while moving them around the same circle where we usually cut. This raises the tantalizing question whether we could possible make a book whose pages are Möbius strips. For the moment I don’t know, I haven’t been able to find many explicit bendings of the ruled Möbius strip, except for its Doppelgänger:


This, however, is not a closed band but instead continues on periodically. There is a version of the Möbius strip as a minimal surface that also has a circle as a core curve.


As with ruled surfaces, you can twist these minimal surfaces more or less often around the circle. Here for instance is the triply twisted version.


All these minimal surfaces can be bent in their associate family. They stay minimal, and, surprisingly, also closed bands (after two turns), except for the case of a doubly twisted band. The conjugate of the triply twisted band looks like this.


Of course three is never enough, so here is the 40 fold twisted version. Amazingly enough, all these surfaces have explicit formulas.


This would be another possibility for a book project: One long twisted sheet of paper, bent into disk like pages…

Methuselah Trail

The White Mountain area in the eastern Sierras is home to the Great Basin Bristlecone Pines. The arid climate and high altitude limits the growth periods of the pines to a mere two weeks per year.


So they grow slowly, and get very old. Some of them are over 4,000 years old, making them the oldest trees on the planet.


We humans don’t think in these time spans. We occasionally consider the next year, and rarely the next decade.


The age of the cathedral builders is gone, who knew they would not live to see their work finished.
What would we do with so much time? Would we plan ahead and mold the future, or would we just keep adapting and contorting?


Maybe we should develop a better sense of being content with what we have.


The Anti-Cone and its Doppelgänger (Scrolls IV)

The literature has not many interesting examples of ruled surfaces in Euclidean space besides cylinders, cones, hyperboloids, and the helicoid. Let’s fix that. A cone, or more precisely a frustum (latin for piece), can be described by following a horizontal circle (as a directrix) counterclockwise and rotating the generators also counterclockwise, horizontally pointing in the same direction as the point on the directrix, and with a fixed vertical component.


The result is a flat surface and thus not interesting for making a curved book. But we can also follow the circle and let the generators rotate the other way. I will call the resulting surface an anti-cone. It is certainly not flat anymore.


Take for a ruled surface all of its generators (the straight lines) and shift them so that they pass through the origin.
Their intersections with a sphere centered at the origin is called the spherical indicatrix of the ruled surface.

In this case, the spherical indicatrix is a pair of horizontal circles, both for cone and anti-cone, but differently orientated.

An old theorem about ruled surfaces states that you can deform any ruled surface by changing its spherical indicatrix pretty much arbitrarily. It turns out that there are typically two different solutions to do this, even if we trace the indicatrix in the same direction.

In other words, for a given ruled surface, there is a second ruled surface along a different directrix but with pairwise parallel generators. Let’s call this the doppelgänger of the ruled surface. For the anti-cone, it looks like this:


One can visualize the relationship between the two surfaces by putting them together and coloring corresponding (i.e. parallel) lines with the same color.


If we had paper ribbons shaped this way, we could bend one into the other.

Here is another image of the generators of the anti-cone doppelgänger. Giving up on clarity can increase the esthetical appeal when we add ambiguity.


Ahr Valley (Wine Biking II)

The Ahr valley is about 90 minutes away from Bonn by bike. This valley marks the northern end of the Eifel, a volcanic low mountain range in northern Germany.


The microclimate and terroir (slate) makes it suitable for growing wine. For many years, tradition and local demand resulted in largely unremarkable sweet Pinot Noirs.


When I came back from California and felt I needed to cultivate my acquired habits also at home, I went on bike rides to the few wine makers who dared to go against the tradition. My favorite was Weingut Kreuzberg, run by a friendly family. The two sons were happy to sell me one or two bottles of their Dernauer Pfarrwingert Auslese, to be carried home by bike.


One day there was excitement: They just had won a price in Berlin for their illegally planted and harvested Cabernet Sauvignon. German wine laws are quite German, indeed. Each region is supposed to grow only their allowed varieties of grapes, and exceptions need a special permission. The Kreuzbergs didn’t get permission, because the wine association didn’t believe that Cabernet would grow in this climate.


The punishment was cynical: The Kreuzbergs had to rip out their Cabernet plants, because they were planted without permission, but were allowed to replant them, because they had proven that Cabernet could actually grow there.

Things have changed since. More quality wines are being produced at the Ahr (among them Cabernet Sauvignon), and the little known wines made by the Kreuzbergs are now sold out faster than it takes to get from Bonn to the Ahr (by bike).

What am I doing here? Applauding the break with traditions and simultaneously lamenting their loss? Silly me.

Lawson’s Klein Bottle (Annuli IV)

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This is what it might look like if you got stuck inside a highly reflective Klein Bottle. Wait – Klein Bottles don’t have an inside.
From the outside they can look like this:


Crawling through into the pipe at the bottom gets you from outside to inside. This is spooky, and responsible for this odd behavior is a Möbius strip. There are other versions of the Klein Bottle. Here is the figure eight version, obtained by rotating a figure 8 bout the vertical axis, and giving thereby the 8 a half twist.


Note that in both versions, the bottle cuts itself along a circular closed curve. That the two versions above are really quite different can be seen by looking at the bottles near these circles. In the first case, it looks like an annulus intersected with a cylinder, while in the second case we see two intersecting Möbius strips. The latter description helps to understand the geometry of the next version better.

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As we know, a Möbius strip has just one boundary curve. The image above shows a Möbius strip where the boundary curve is a perfectly round circle. Taking a second copy of this Möbius strip and attaching it to the first along the boundary circles produces the stereographic projection of Lawson’s Klein Bottle, a minimal surface in the 3-dimensional sphere.

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This is really complicated, so let’s look at the anatomy of this beast. The top translucent part, when turned around and after a paint job, reveals himself as a doubly twisted cylinder.

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The other (bottom) part is still rather complicated. It consists of two pieces of the Klein Bottle that intersect along the orange circle.

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One of them without its distracting sibling is once again a Möbius strip.

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So Lawson’s Klein bottle is anatomically just a union of two intersecting Möbius strips and a doubly twisted cylinder.

Patterned Ice

A side effect of having the temperatures vary between 10 and 60 degrees Fahrenheit within days is that one can admire interesting ice formations while hiking in pleasantly warm weather.

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What we see here are thin layers of ice over shallow running water, in some of the little creeks that flow through Yellowwood State Forest.

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Rocks make for quicker thawing, and the hickory leaves from last fall adapt by becoming translucent.

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Near hexagonal patterns like this one occur often in nature when homogeneous material breaks under uniform pressure. It’s the first time I have seen this with ice, and I would love to see how it forms in a time lapse movie.

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Another one, at a different location.

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The Hyperbolic Paraboloid (Scrolls III)

If you build a wire frame into the shape of four consecutive edges of a regular tetrahedron, dip it into soap water, and carefully pull it out again, you get a piece of the Diamond surface. If you cheat and just span wires between corresponding points of opposite edges, you get a doubly ruled surface, the hyperbolic paraboloid. Here is one such surface, together with a mirror image. The eight corners coincide now with the eight corners of a cube.

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As a digression, we can fit a total of six such paraboloids into the same cube, creating a curved version of Kepler’s Stella Octangula.


But let’s return to paper making. The home recipes include the usage of a mold, which is a wireframe that is used to get the right amount of paper pulp into shape and, most importantly, dry. For flat paper one can just use a flat wire frame, like a window mesh screen, which is purchasable. The hope is that, using modest force, such a screen can be stretched into tetrahedral shape. We’ll work on that later.

For the moment let’s delight in previewing how the paper would bend.

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Up above you can see three sheets. The darker bottom one is the actual hyperbolic paraboloid, while the two lighter and greener ones are bent versions that are still attached to each other along the middle straight line that is pointing towards us. This will be our spine. Here is a top view:

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