Pflaumenmus

While many fruits and berries are being cultivated or stored so that you can buy them “fresh” year round, some are either too delicate or not popular enough for this treatment. So you have to get them when they are ripe, and find your own means of preservation.

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Plums are one such example. My own plum trees lose what little they produce to the greediness of the birds well before they are ready for human consumption, so I have to resort to local stores.

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Plums are also interesting, because the American style plum butter is a far cry from what this fruit deserves. Plums are too juicy for the standard ways of jam making. To produce a real mus, they need to be stoned, mixed with sugar (1 cup for 3 pounds), and spices (try cardamom and clover!).

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Let this sit for at least two ours and discard the juice (or dink it, if you like sweet treats).
Then put this into a baking dish and bake for at least two hours at 350 degrees Fahrenheit, stirring occasionally. Leave the oven door open for the first 30 minutes to get rid of even more liquid. You want the result to be really gooey. Be warned: 3 pounds of plums make less than a cup of mus.

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Now all that is needed is good bread.

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Clouds (Iceland XII)

Most places have their own very distinctive appearance of clouds. Landscape painters know this and therefore prefer to live close to the ocean or the mountains. Needless to say, clouds in the midwest are either dull or very dangerous.

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Iceland has both ocean and mountains so that one can expect the best of the best.

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When I was little they told us in school that life forms can be distinguished from lifeless matter by a few criteria: Ability to move, react to the environment, and reproduce. Clouds can do all that.

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So I started thinking that being a cloud might be an interesting way to live.

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Unfortunately, the only cloud based life forms in the near future will most likely be rather virtual.

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Primitivity? (Algorithmic Geometry II)

The construction of the polygonal diamond surface via bent rhombi can be varied. If we take as the bent rhombus two adjacent faces of the regular octahedron instead of the tetrahedron and follow the same rule to extend the surface by 180 degree rotations of rhombi about edges, we first get a less crooked hexagon,

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four of which can be assembled to a translational fundamental piece

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of a triply periodic polyhedral surface

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that approximates Schwarz’ so-called primitive surface.

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In this case, the ribbon representation has a much simpler appearance than the rhombic image.

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After all, apparent complexity is often only a matter of the presentation.

If you want to make quick paper models of either the diamond or the primitive surface, cut out lots of equilateral triangles, divide the edges into thirds, and bend the three triangles at the corners upwards. These smaller triangles serve as flaps that you glue to the front sides of the central hexagons inside the original triangles.

Model

Music for the Eyes (Iceland XI)

Waves are endlessly fascinating. Iceland, being surrounded by water, has plenty of them. The image below might appear quite ordinary, but for the rather irregular ripples at the bottom right. They were caused by the high frequency vibrations of the motor on the boat from which I took the photo.

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Thanks to a large amount of inland water, you can find more waves virtually anywhere, like here at Geyser, with colorful deposits.

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Just a few feet away, the landscape at your feet changes dramatically, but still offers waves.

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And even without water, you will see waves. After staring at rocky sand

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and lava beds in the large

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or more up close,

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when you finally have enough and look up at the sky…

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There is no escape.

Diamonds are Forever (Algorithmic Geometry I)

When you dip a closed wire frame into soapy water and pull it out, the soap film you see is a minimal surface. Finding a formula for this surface is generally a very difficult problem, and one of the earliest successes was achieved by Hermann Amandus Schwarz, using four edges of a regular tetrahedron as the contour.

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One feature of minimal surfaces is that they can be extended across a straight line by means of a 180 degree rotation about that line. Doing so for the tetrahedral patch above generates the diamond surface, named so because it has the symmetries of the diamond cubic crystal structure.

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One can explore this crystal structure together with the diamond surface at a much more elementary level. Start with two sides of a regular tetrahedron. They constitute a blue rhombus that has been folded along the shorter diagonal. Rotate this blue rhombus by 180 degrees about any of its four edges to obtain a second (red) rhombus that is attached to it, like in the image above.

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Keep extending this surface, always by rotating a bent rhombus about a boundary edge by 180 degrees. Above you see how you can obtain a hexagonal shape, and below an annulus.

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Again the surface will extend indefinitely. The following piece is a fundamental piece in the sense that mere translations of it will produce the whole infinite surface.

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There is another way to render the bent rhombi by replacing each rhombus by a circular ribbon which ends at the far corners of the rhombus. I learned about this from Alison Martin at the Shape-Up conference in Berlin, 2015.

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So what we are trying here is to visualize an abstract algorithm (extend by rotating about an edge) that can reapplied to varying geometric contexts.

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Avoiding Collisions (Helices I)

One of the simplest line configurations in space just utilizes the parallels to the coordinate axes that pass through the (red) points with integer coordinates.

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If we want to avoid the triple collisions at all these points, we can shift the lines one half unit each, like so:

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This results in a dense packing of cylinders. Another possibility to avoid the collision is to let the lines spiral around the red points. I haven’t found a nice way to do this because the three helices would need to pass through the eight cubes surrounding a red point, meaning this is impossible in a symmetric way.

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However, there is another line configuration where the lines pass through all the main diagonals. This is more complicated, because we have now four sets of parallel lines. Again we can shift the lines to avoid collisions.

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Now, with four lines through each intersection, we can replace them by helices in a pretty symmetric fashion.

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