Happy Second Birthday!

This little blog is now two years old. As a birthday present, here is yet another visit to Strahl Lake in Brown County State Park, this time at a full moon.

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This turned out to be more difficult than I thought, and I tried it twice this year to capture the lake front in moonlight. The image up above shows how dark it really was. Below is a  picture an hour after sunset.

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Of course, the camera allows you to expose as long as you like, capturing more light than real. Below is what the camera thinks it should look like, assuming all images require the same amount of photons on the camera sensor.
If not for the stars, this could be a faded color print of a daylight photo. Eerie.

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Let’s end this year with a little more realism. We will need the light.

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Yoga for the Soul (Treescapes 2)

After a light freezing rain I went at sunrise to Brown County State Park to admire the trees. It was very cold, but, as I said before, I enjoy that these days.

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So yes, you can have views even in Indiana. Nothing man made visible here.

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The trees have been temporarily transformed. I am tempted to say that they appear to be frozen, but that is what they are.

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They seem to move even less than usual.

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I wish this would happen to us, too: For a couple of days being forced to stay put, to contemplate and reconsider, and then to be allowed to thaw.

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Double Parity (From the Pillowbook VII)

Here are the 36 pillowminoes introduced last time, arranged by their imbalance, i.e. according to how many more convex than concave edges they have. Isn’t that a pretty bell curve?

Pillowminoes weight

This time we will focus on the pillowminoes near the border of existence, namely the six ones that have all but one edge either bulging in or out. They have an imbalance of +4 or -4. Gathered and recolored, here are the marginal pillowminoes:

Margin

Let’s tile some curvy shapes with these. A curvy rectangle has odd dimensions, so cannot be entirely tiled by pillowminoes. If we decide to leave a round hole in order to fix that, the entire curvy shape will be balanced. This means that we will need the same number of brownish pillowminoes (with imbalance +4) as bluish ones (with imbalance -4). In particular, we will need an even number of these pillowminoes, so the total area of our shape needs to be divisible by 4. That’s our double parity argument.

The simplest example is that of a 5×5 square with a center hole (it’s easy to see that skinny rectangles with one edge of length 3 are not tilable with marginal pillowminoes).

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The example to the left is the only one I could find, up to the obvious symmetries. To the right you see how one can inflate it to make frames, proving:

Theorem: If you can tile an axb holy rectangle with marginal pillows, then you can also tile a holy (a+4)x(b+4) rectangle with marginal pillows.

We have seen this trick before, talking about ragged rectangles.

The next interesting case are 7×7 squares. Here is one example that also teaches us another trick:

Theorem: If you can tile an axb holy rectangle with marginal pillows, then you can also tile a holy ax(b+4) rectangle with marginal pillows.

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This second trick decenters the holes, however.

Finally, two examples that employ all six different marginals. First a 5×7 rectangle with center hole, then another 7×7 square that uses four marginals of each kind, nice and symmetrically.

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More Leaves …

Besides leaves from books, the other kind of leaves that are essential for my well being are tea leaves. This year was an interesting tea year for me. It began, as usual, with First Flush Darjeelings. One of my favorites this year was the the very mild First Flush Imperial from the garden Runglee Rungliot. Wonderful tippy leaves and a bright yellow cup.

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Considerably stronger in flavor (papaya) was the Okayti Wonder.

Wonder

A curiosity from the same garden is called Golden Treasure. It came to me from Harney, one of the few consistently good US based distributors. This tea tastes a bit like a good Chinese black tea, with pleasant cocoa flavors, but still a hint of second flush Darjeelings.

Golden treasure

Then, my afternoon favorite this year, the Himalaya Shangri-La Ruby from Nepal. Its golden leaves are curled, and the dark color of the cup reminds me of strong Assam teas. But the complex taste is milder, with hints of cocoa and cognac. I have never tasted a tea like that. Its sold out now, probably because of me. What I have left will get me through the long winter.

Shangri la

This tea came to me from Germany, via Tee Gschwendner: This a German company with local shops all over Germany that carry an amazing collection of loose tea. After graduating from grocery store loose tea tin boxes, I had been sampling their less expensive teas for several months. One day in Spring changed my life, when they had flown in small samples of the first invoices of that year’s First Flush Darjeeling from a handful of tea gardens. The price for the sample set was more than I would spend on tea in a year.

You know what happened.

Sofia

Sofia

I like local produce. And I like cheese. One of the warnings I got from my European friends was that the people in America would spray their cheese on crackers. While it is true that one can buy something named cheese in spray cans and do whatever with it, this is still a free country, so nobody forces anybody to do so.

Moreover, there is no such thing as the people, and there is tolerance for perfectionist cheese makers. One of them are the family from Capriole Farm in Southern Indiana who make amazing goat cheeses. Above is Sofia, allegedly named after one of their daughters.

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Of course this cheese goes well with good bread (more about that at a later point). I also like minimalist recipes, with a maximum of three ingredients. For instance:

Red beets, goat cheese, sage:
Slice and sliver everything. Bake 15 minutes
at 400 degrees. Serve warm.

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Enjoy.

Berlin 12-19-2016

Berlin has changed a lot since I grew up there, in the western part of the then divided city. Here are some pictures I took in 1991, already only a visitor, after the Wall.

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The first one is a view from the Teufelsberg, the Devil’s Mountain, an artificial hill made from the rubble after the Second World War. This is one way to get rid of ruins.

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I like to revisit places, and therefore I like it when ruins are being kept. This one, the ruin of the the Anhalter Bahnhof, close to the Wall, used to be surrounded by unused land. It is one of the key places in Wim Wender’s film Wings of Desire. Another most important place in the film is yet another ruin, the Gedächtniskirche below. I took the picture from the top of the Europa Center, closed to visitors now, maybe because of the film.

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These are all summer pictures. In December 2008, 8 years ago almost to the day, I tried to capture this view again. You can see the traditional Christmas Market at the foot of the two churches.

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It is a good thing that we cannot see into the future.

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In Memoriam, 12-19-2016.

Durissima est hodie conditio scribendi

The regular pentagon is a curious thing. It doesn’t tile the plane, but we can use twelve of them, three around each vertex, to tile the sphere, obtaining the dodecahedron.

Dodecahedron

This is one of the five Platonic solids. Their symmetries have intrigued mankind back way before Plato and any written history, but today’s story is contemporary. Johannes Kepler needed more than five regular shapes, because he had set his mind to explain the universe. In his Harmonice Mundi, he analyzed regular polygons, star polygons, polyhedra, and (re-)discovered star polyhedra, two of which I will look at today.

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The Small Stellated Dodecahedron as conceived by Kepler does not have 60 triangles but rather 12 star pentagons as faces. It also has only 12 vertices and 30 edges. This leads to the annoying observation that this polyhedron has Euler characteristic -6, meaning it is topologically not a sphere, but a surface of genus 4. Similarly, his Great Stellated Dodecahedron

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has 12 usual regular pentagons as faces, but is only immersed. To unwrap these, we need the hyperbolic plane, tiles by regular hyperbolic pentagons whose interior angle is 72 degrees so that five of them fit around a corner.

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That is not part of Kepler’s story but that of William Richard Maximilian Hugo Threlfall, who was probably the first who understood the hyperbolic nature of Kepler’s polyhedra, and their group theoretic implications. So we can tile the hyperbolic plane with regular pentagons, five around each vertex. One of the surprising features of the hyperbolic plane is that shapes do not scale as in the Euclidean plane. Pentagons half the area have actually right angles, so that four of them fit around a vertex, as indicated by the reddish grid in the picture above.

Curiously, there also is a uniform polyhedron where four pentagons fit around each vertex, the so-called Dodecadodecahedron (yes, these names are odd).

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It has as faces both pentagons and star pentagons.

There is another connection between Kepler and Threfall. Kepler begins the introduction of his Astronomia Nova from 1609 with the sentence Durissima est hodie conditio scribendi libros Mathematicos, praecipue Astronomicos. In 1938, Seiffert and Threlfall published a book (Variationsrechnung im Großen) that has as its motto the shortened quote Durissima est hodie conditio scribendi libros Mathematicos.

That was a risky thing to do back then.

Kepler was an interesting personality. It must have been maddening for him to believe himself on the verge of unraveling the universe and be constrained by earthly powers that threatened to burn his mother as a witch. There is a biographical novel about him by John Banville (whom I generally like for his affinity to bizarre characters). In this case, I am afraid, he falls short. Maybe only a scientist can truly understand scientific obsession.