Scherk in Clay


This innocent minimal surface, which can be obtained from Heinrich Scherk’s traditional surface by adding two wings and bending them towards each other, poses interesting challenges when printed (vertically, i.e. rotated by 90 degrees) in clay. First of all, there are three horizontal cross sections which look like branches of hyperbolas (but aren’t, not even for the original Scherk surface, in contrast what Wikipedia currently claims).

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When printing this layer by layer, the nozzle has to move from branch to branch, and as the printer can’t stop printing while it skips across, it leaves hairy artifacts.

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They clearly have their own charme.

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Another problem arises from the saddle points that are printed without support. This leads to other imperfections and sometimes structural complications that might take away from the elegance of the original surface but contribute to wild interior landscapes.

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Watching the printer work for two hours is dramatic, because failure in the form of collapsing walls can happen any minute.

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Deltoids in Clay

Clay printing currently works best for objects that change slowly from one horizontal layer to the next. This suggests to create 3-dimensional objects that realize a changing 2-dimensional configuration in one piece. An example of that is the rotating segment within the deltoid that at every stage foots on two sides of the deltoid and is tangent to the third.


As the deltoid itself doesn’t change shape, it will become a cylinder over the deltoid. On the other hand, the rotating segment will become a ruled, helocoid-like surface. If we printed the entire model like this, the interesting part, namely the rotating secant, would be mostly hidden. Therefore we will only use one edge of the deltoid, while the other two are implied only by the rotating endpoints of the line.

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Doing this in clay is not easy. First of all, we print it so that time is vertical. This allows to use the deltoid wall as a solid support. Each layer of the rotating secant then becomes a cantilever, supporting subsequent higher layers.

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The point when the secant turns into a tangent is particularly interesting. One can see the gravitational pull on the emerging new layer that bends towards us in the image above. The contrast between the static, cylindrical deltoid arc and the dynamic, rotating secant is compelling and hard to convey in a single image. But that’s a fair enough reason to make 3D sculptures.

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Quadrics in Clay

To get the orthogonal quadrics from Monday into clay using a clay printer, one needs to know about the limitations of Malcolm’s clay printer. It does nothing else but move a vertical tube full of clay horizontally around and vertically up, layer by layer. Simultaneously, it squeezes a continuous stream of clay, with no pause.

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The first few layers are pretty easy, clearly showing the elliptical and hyperbolic cross sections. We only print one half of the whole model, to have a solid foundation (the central cross section), and because it’s cool to be able to look inside.

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Things get interesting when the two branches of the hyperbola come together to connect to the single hyperboloid. We reach a critical point of the height function, and the clay printer clearly has problems with the Morse theory.

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Above you can see the nozzle in action, and more has happened: We have passed a second critical point when the two components of the hyperbola have separated from the ellipse. This is more complicated then the standard Morse theory of manifolds. The printer has do (quickly) move from one component to another at each layer, randomly dropping little chunks of clay on its way.

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This gets a bit messy when we reach the peak of the ellipsoid. Below is the completed print. It needs to dry and be fired. You will notice that we have only used two of the three surfaces. This is a pity, but the missing piece is one sheet of the double hyperboloid, and it is almost horizontal, and impossible to print.

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Arbeit und Struktur

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As hinted at in a previous post, I have been spending a fair amount of time this summer preparing 3D models for clay printing. I will talk about the models and the results at a later point. Today, we focus (or de-focus?) on watching the process. Printing a model takes time (say two hours for a model 20 cm in width) and requires almost permanent attention.

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So one naturally begins to pay attention to details. The shallow focus of a macro lens not only allows to pinpoint these details, it also blurs everything else into pleasant abstraction.

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Color is almost irrelevant, unless one wants to bring out the gradual change of clay type from layer to layer. Everything is reduced to utter simplicity, to the extent that the all too human question for meaning is becoming meaningless.

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What matters is structure, and the work to be done to maintain it.

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Arbeit und Struktur (Work and Structure) is the title of Wolfgang Herrndorf’s Blog-Diary that he wrote in the last three years of his life.
This diary distills much of what mattered to him while facing death, and the title is a further reduction of this to just two words.

Le Bateau Ivre (Loxodromes II)

A good way to embarrass oneself is to go to a book store in a foreign country whose language one is not fluent in, and buy a book. I did this multiple times, at least in France, Spain, and the UK.

I typically tried to get by without saying a single word as not to reveal my complete incompetence, but the punishment for that can be unexpected. During one of my first visits to Paris, I went and bought the Bibliothèque de la Pléiade edition of Arthur Rimbaud.

The catch was that the very pretty cashier tried to initiate a conversation by smiling at me and saying “Ah, J’aime Rimbaud”.
I blushed, payed, and made my way out. Embarrassing.

But it brings us to the topic, Rimbaud’s Drunken Boat.


The image is this, and it does not look like a drunken boat. What we start with are the loxodromes I have talked about before. They are the curves a sober boat would trace out on the sphere when heading in a fixed compass direction. Laying down one of these loxodromic double spirals as a base using Malcolm’s clay printer looks like this:

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Then, moving up, we deform the loxodrome that represents say North-North-West slowly into North-West and then West, which corresponds to a meridian, and therefore a straight line in suitable stereographic projection.

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Then, even higher up on the sculpture, we change course to South-West and thus reverse the direction of the spirals.

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This was our first rough prototype. The next step will be to make this larger, cleaner, and slightly drunken, so that the loxodromes swerve left and right.

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We’ll see shortly where we get…