Curved is also Beautiful

Among the many helicoids with handles, the translation invariant genus one helicoid is by far the simplest. It was first constructed by David Hoffman, Hermann Karcher, and Fusheng Wei. You can learn almost everything about it from a single image.

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The right hand side is a portion of the actual minimal surface, which extends by rotations about its horizontal and vertical lines to the complete surface.

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The quotient of this surface by its vertical translation is a torus, and the presence of the two straight symmetry lines hint that this is a rhombic torus, which you see outlined black in the left top left image. Its two diagonals become the two straight lines of the surface. The trick is to see the surface patch to the top right as the image of the colorful rectangle on the top left. The top left and bottom right corners of that rectangle are bent together so that they touch, the horizontal edges align as the horizontal line, and the vertical edges align as the vertical line of the surface.

The two semicircular arcs become the half turns of the two helicoidal arcs, this allows to truncate the surface image nicely. The mesh lines of the colored rectangle are, incidentally, obtained by conformal mapping a rectangle to itself in a quirky way:

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Here, the vertical edges of the left rectangle are mapped to the two semicircles, and the horizontal edges to everything else. The “extra” vertical lines are included so that we hit all vertices of the right rectangle by a parameter line.

So that is all very easy. The tricky part is make the right choices in order that the the two opposite corners of our parameter rectangle really meet. The horizontal alignment is achieved by using as a rhombic torus the funny 70.7083 degree rhombus we discussed last time. If you choose another rhombus, the two verical line segments will not match up.

To guarantee also a vertical alignment of the two corners, one needs to choose the location of the two points E1 and E2. To do this, one constructs a meromorphic 1-form on the torus which has simple poles at E1 and E2 and two zeroes at V1 and V2 (whose location depend on E1 and E2 by Abel’s theorem). The integral will map our colored rectangle to a slit domain consisting of two merged half strips. The ends of the half strip correspond to the two helicoidal ends of the surface.

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That the two slits line up in this picture is no coincidence. E1 and E2 have been chosen so that this happens (thanks to the intermediate value theorem). Tt is exactly what is needed to achieve the vertical alignment of the two corners.

Sacred Grounds Café (Southern Illinois I)

One of my little hobbies is to visit places that play a role in a book (or film) I like. This year, one of my favorite books so far has been John Burnside’s novel Ashland and Vine. Even though the author is Scottish, the book takes place in a generic Midwestern college town called Scarsville.

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The book is about a College student and an elderly Lady. I don’t like to write or read reviews, and in this case the title already is hint enough. It is a good book. We don’t learn much about the place, until a coffee place is mentioned with the uncommon (for a coffee place) name Sacred Grounds Café. Thinking about it, this is an excellent name for a coffee place, because it tells the customer that one is welcome to sit down for a while instead of just grabbing a quick coffee. In any case, I found the name odd. There are indeed real coffee places in the US with that name, and one of them is in Edwardsville, Illinois. Even better, it is also on Main Street, as in the book.

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I decided to pay the place a visit. Unfortunately, the staff didn’t know author or book, and the owner (who plays a role in the book, too), was not present.

The quiche with sweet potatoes and spinach was excellent.

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Jerome, oh why you treat me so cold? (Arizona IV)

Another example how to deal with days past can be admired in Jerome, an old miner’s town. Today’s population is a mere 5% of what it used to be 100 years ago when copper was plenty. So, where did all the people go, and what happened to the houses?

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The ones that appear not to be abandoned cultivate lucrative traditions

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or celebrate an unprofitable business by placing it at a location that makes it particularly difficult. How would you arrange the book shelves?

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For the curious tourist, accommodations are plenty,

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and transportation is traditional.

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Then there are the true and timeless inhabitants, always slightly annoyed.

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Recovery (DePauw Nature Park I)

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The main attraction of the DePauw Nature Park is its limestone quarry. It is a vast and eery place. After DePauw University took possession of the area, they removed buildings and other environmental damage, and allowed nature to recover.

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So one can see the effect of two different forces at once: The enormous momentous force of machines that cut through all that limestone, and the much slower force of plants that take roots again.

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This appears to be simple. We know the constituents: water, rock, grass, tree. But then there is weird stuff I don’t know. I will pretend it is harmless.

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Maybe Mars looked like this, too, millions of years ago.

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Sources of Healing

Last October I went on an early morning to McCormick’s State park, not expecting to see anybody.

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But there was this guy, sitting next to the little waterfall in the dark. We started chatting. He was from Florida, evacuating for the week because of Hurricane Matthew. Friends had told him to check out this place, and he was quite impressed.

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They had also told him of a Spring of Healing that could be found here, and he wanted to know about it.

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I couldn’t help.

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But it is true that some places are special. My virtual substitute here can only be a reminder that they still exist.

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They need our protection.

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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.

The Zone

In his film Stalker, Andrei Tarkovsky transforms the mysterious zones from Arkady and Boris Strugatzky’s book Roadside Picnic into a spiritual personal experience for the visitor.

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I like that concept of a place that is off the map where we can go and dream or contemplate.

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Fortunately, it is possible in Indiana’s slightly boring landscape to just step off the path and end up in one’s own little zone.

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Granted, there will be no alien artifacts to collect, and no wishes fulfilled. But that was never the true purpose of Tarkovsky’s zone. Instead, a zone allows undisturbed introspection.

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Sadly, these undisturbed space become harder to find, and maybe we have to move the zones into a virtual space. When the space limit on this blog runs out, I might call my next blog The Zone.

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