Rationality

Let’s do something simple today, something rational. We all know what a circle is, and most should know that there are points on the unit circle with rational coordinates, like (3/5, 4/5). This is because of the birational map called stereographic projection, known since antiquity:

Stereo

Take a point t on the real axis, connect it to the north pole (0,1) on the unit circle with a straight line, and find the second intersection of that line with the circle. This is a rational expression in t. So when t is rational, you get a point on the circle with rational coordinates. This is, of course, also a quick way to get (all) Pythagorean triples. But today we are going elsewhere. Now that we have many points with rational coordinates on a circle, we can make rational polygons, like the one below.

9 gon

This 9-gon is not only rational, but super-rational in the sense that all its edge lengths are rational numbers. Try it out. Even better: All the diagonals are rational as well. Is this a miracle? Are there others? No and yes, of course. Let’s get started:

Formulas

Using rational versions of the sine and cosine functions, we can write down rational rotation matrices. They will (for rational t) rotate any point with rational coordinates on a unit circle to another point with rational coordinates. What we are interested in are superrational rotations: Those that rotate a point to any other point. The example above suggests that there are many of those.

I will give the answer next time. For the moment, only a hint: The superrational rotations form a subgroup of the group of rotations. Which is it?

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2008 Recap

Yes, that’s right. Let’s begin the year with a recap of not last year, but of 2008, the year 10 years ago.

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This year brought photographically two significant changes into my life: My move to full frame digital (and the ability to use a handful of SLR lenses I still had from film days), and the adjustment to the Indiana landscape.

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It is not that the Indiana landscape is featureless. It is more a assembly of countless insignificant features that tire the eyes, with occasional exceptions.

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Some are less obvious then others, but the only chance finding them is to look.

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Sometimes I am being asked why I bother carrying a heavy camera when there is nothing worth to photograph.

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Visiting some of the state parks has helped to open the eyes, like McCormicks Creek, Turkey Run, Shades, or Falls of the Ohio. This had been a good year.

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Single Trees

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One of my enlightening tea experiences was a sample set that my little local tea shop in Bonn (now TeeGschwendtner and not so little anymore) offered many years ago: Four different early invoices from first flush Darjeeling teas. I had never had tea that expensive before, nor had I tasted tea like that. It changed my tea trinking habits dramatically.

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This year, Harney & Sons are offering small tins with labels saying distinctively Tree 1 through Tree 4. These are harvests from four individual Song Zhong tea trees, each prepared in a different way. Harney characterizes them as floral/fruity/body/creamy.

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These are all Oolong teas with a slightly peachy flavor, and they are really all different in taste. In found the second one most interesting with strong Maracuya notes, and the last one rather blunt.

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While not as mind-opening as my Darjeeling experience many years ago, this has still been a worthwhile experience.

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Reportedly, Chinese buyers can purchase an entire tea tree and receive the yearly harvest over their life time. I would like that.

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Grayness

Finally, the Grayness has arrived.

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Some places have a fifth season. California, for instance, has a few weeks of High Summer where the air seems fresher and the sky more lucid than usual. Indiana, on the other side, has Gray Winter.

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There are ways of resistance. One is through structure,

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another through subversive use of color.

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In Michael Ende’s masterpiece Momo, the men in gray talk people into depositing their free time (usually spent with relaxation or talking to other people) into retirements accounts, which is of course fraud, because the men in gray feed on other people’s time. Fortunately, there is Momo.

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Early morning frost and earlier flooding accentuate these little attempts of resistance that nature puts up.DSC 0245

Let’s not become the gray planet.

Teotihuacan (25 Years Ago)

The first three days of my Winter Break excursion to Mexico in 1993 I spent in Mexico City, and one of them in Teotihuacan.

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By bus it takes about an hour to get there. I should have arrived much earlier, to beat the crowds and have better light. One of the most fascinating aspects of this place is how little we know about it.

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It had been abandoned by about 700 CE, reaching a population well over 100,000 before. The reasons? We don’t know. Who lived there? We don’t know. Of course there are speculations and theories. 

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What fascinates me is the discrepancy between the longevity of what’s preserved and the fragility of what is gone. Did they care what would survive? If we knew we’d be gone in a century, would we care to leave something behind?

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Would it be art, pomp, or an attempt of a message?

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Perhaps it should just be a vision: This is how we liked it to be. This was us. 

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Laura Hare Nature Preserve at Downey Hill

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The long title of this post names one of the many places I have missed so far that are in 1 hour driving distance from Bloomington. If you are after spectacular rock formations or water falls, this isn’t the place. But it features a 4.5 mile loop through Southern Indiana landscape at its best. Sinkholes …

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… miniature canyons …

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… creeks …

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… and the trees, of course. All this makes up the landscape in the large, accented by the play of light or the lack of it. Then there are the small things: Leaves clinging on,

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moss providing unexpected greenery,

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hopeful trees sprouting,

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and older trees offering vistas in the past.

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This was a most enjoyable hike.