Simplicity

We have learned simplicity,
we sing in the choir of cicadas

 

In 2004, the midwest of the USA became the region of the largest outbreak of biomass on the planet. Brood X emerged, the largest of the periodical cicadas, with a life cycle of 17 years.

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They spend most of their life underneath as nymphs, going through several instars, and feeding from root juices of trees. Then, almost all on the same day, they emerge, and crawl up.

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They molt into adulthood, which takes up to an hour. All this takes place above an abyss. If they fall, the soft wings will not unfold properly.

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In the last stage, the wings unfold.

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They hide and rest for a few days. This is supposedly the time they are most delicious.

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Then, they tumble around, with their poor eyesight and clumsy bodies, causing harm to none, and begin their irresistible mating chant, droning sound patterns that move through the landscape like ambient music.

17 years underneath, for a few weeks of a single song. Who will question the meaning of life?

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Death of Proof (The Pleasures of Failure I)

Die Wahrheit ist dem Menschen zumutbar.

Occasionally, after confronting students with evidence of fact (Euler Polyhedron Theorem is a great example), I ask them whether they want to see a proof or prefer to accept the statement as a miracle. The overwhelming majority is always happy with the miracle. Such are the times. Below is such an evidence of fact: A minimal surface with 3 ends and of genus 2.

 

Almost

Should we doubt its existence? In 1993, John Horgan published an article in Scientific American questioning whether proofs were about to become obsolete, in times where shear length and difficulty made validation next to impossible, and numerical experiments supplied by computers could be an acceptable substitute. For many reasons, large parts of the mathematical community were outraged.

Notquite

Above is another example of that surface, for a different parameter value, but something seems off. There appears to be a little crack. Maybe I didn’t  compute accurately enough? Changing the parameter a bit more widens the gap.

Fail

The question whether this surface does actually exist hinges on the possibility to truly close that gap, for at least one parameter value. It appears that we have done so in the top image. But the parameter value there is 1.01, pretty close to 1, where the surface will clearly break down. A more accurate computation shows that there still is a gap at 1.01, which we can’t see, or don’t want to see. But maybe 1.001 will do?

Side

David Hoffman and Hermann Karcher analyzed this surface in 1993, the same year as Horgan’s article, and it became known as the Horgan surface. One can indeed prove that the gap cannot be closed, so, despite all the evidence, this minimal surface does not exist.

Top

Nebelland (Land of Fog)

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Fog is an overused trope. It serves of course as a backdrop for everything spooky, because fog makes us afraid of not knowing. There is also the personal, psychological fog that makes us forget the past or prevents us from seeing clearly into the future, like in Kazuo Ishiguro’s The Buried Giant or Miguel de Unamuno’s Niebla.

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More dramatically, there is the governmentally imposed fog, as in Alfred Kubin’s visionary Die Andere Seite.

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Today’s pictures are from a very recent trip to the Indiana Dunes. A bit of fog helps to hide the view of the industrial port areas north of Gary. This is scenic, too, but I can’t post pictures here, because, alas, the Indiana Administrative Code explicitly forbids to take pictures of the port, even when standing outside the area.

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But spreading fog, for whatever purpose, has also the side effect to make the things that remain visible to appear more true to themselves, like the trees here.

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Ingeborg Bachmann, whose thin oeuvre has many references to fog, insisted, while charging her co-writers to become eye-openers for others, that we humans are capable of bearing the truth.