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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Circles, Intersected

Lets look at circles with centers at points with integer coordinates and equal radii. When the radii are small, the circles will be disjoint. Something interesting first happens when the radius becomes 1/2, because then the circles touch.

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When the radii grow, the circles will intersect, and interesting patterns emerge. These patterns change continuously,
but when a special intersection occurs, the complexity of the intersection pattern increases. The next special intersection after r=0.5 occurs at r=0.7071, when circles that are diagonally across touch, and then again at r=1.

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Often, and due to the symmetry of things, whenever two of our circles touch, a second pair of circles must touch at the same point.
Then, at r=1.17851, we have true intersections of three circles at a single point (no touching!).

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Mathematicians find this interesting because the special intersections (touch or triple cross) mark singular points in the space of all such circle configurations. Understanding them means understanding the whole space.

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It is of course very satisfying that these singularities are also esthetically pleasing, as if they knew they are special and have dressed up for the occasion.

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