Verweile Doch (Nordhouse Dunes 2)

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A river with its strict sense of flow is a universally used symbol for the passage of time. Resistance against that is, in contrast, best contemplated by looking at the relentless forth and back of waves along a coast line.

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Time is reduced here to repetition, it seems.

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The weeds and twigs shown are gone. Their only action then was to write on the water, invisibly, and immediately forgotten.

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But there is more. They are defending a territory beyond the water.

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They are also witnesses for an esthetic of complexity beyond the untextured and timeless water.

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And finally, they did leave their traces, elsewhere, in memory.



War & War (Nordhouse Dunes I)

For many years we went camping to Nordhouse Dunes at Lake Michigan, and an episode of nostalgia made us revisit this place one last time before my daughter is off to college & life.

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In László Krasznahorkai deliberately cryptic book War & War, the hero György Korin is depersonalized: He just symbolizes a single function of our lives, namely delivering the past into the future, becoming the horizon between the below and the above.

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This happens concretely by carrying an old manuscript to New York, a place that the four inhabitants of that manuscript haven’t seen yet. These inhabitants are cryptic, too, bemoaning the loss of the noble, the great, and the transcendent, this causing also the loss of peace, so that the world now consist of only war & war. 

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Korin realizes however that delivering the manuscript is not enough, he feels the calling to complete it, to find an exit for its inhabitants. There are several attempts for this, one being by writing on water.

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This year, the dune grass that so gracefully used to create sand drawings, is now doing this in the water, thanks to water levels two feet above normal. The water itself leaves very temporary traces on the disappearing beach.

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Is it maybe the author’s dream that his protagonists keep writing the story?

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Noli turbare circulos meos! (Annuli VII)

In science, our goal should always be to present with clarity. Since the discovery of perspective drawings, a realistic representation of 3-dimensional objects has become almost mandatory. However, very often these objects have an appeal beyond their scientific truth which gets lost if its is shown in full clarity.



This blog has two series of posts titled “Spheres” and “Annuli” that both showcase images of simple 3-dimensional mathematical objects which deliberately forsake clarity in order to convey that other appeal. While accurate perspective renderings are used, the  perspective and textures are chosen as to emphasize the abstract aspect. 

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The example above shows a triply orthogonal system of surfaces. An easy way to create such a system is by taking a doubly orthogonal system of curves in the plane, revolve them about a common axis to obtain two families of surfaces of revolution that intersect orthogonally, and add all planes through the axis of revolution. For instance, we can choose two families of touching circles that pass through a common point, as above.


A single circle, rotated about the black axis, will revolve into a torus. To spice things up, let’s apply an inversion at a sphere centered at the intersection of the circles. This turns the tori into special cyclids like the one above, which all have the appearance of a plane with a handle. Using both a red and a green circle will invert-revolve in two such cyclids that intersect in a straight line and a circle:


These are still attempts of realistic drawings, but we already get the feeling that things aren’t completely evident anymore. For instance, the two cyclids above should be equals: but where did the corresponding red handle go?


Above is the same pair of objects from a different perspective. Now we can see the two handles and the intersection in a line, but where is the intersection circle? Also, where do we need to place the third surface family, which consists of inverted planes, i.e. spheres? The answer to that question is indicated below.


Other perspectives allow amusing variations:


For the top image, I have used several cyclids from each family, and several spheres, clipping them between two planes. To appreciate the image, all this knowledge might be irrelevant. To create it, it is essential.