Squaring the Circle

Squaring the circle is easy, you just need to know what you want to do. My personal favorite method is to use elliptic functions defined on rectangular tori to map rectangles to disks, as shown below for a square. These maps don’t preserve area (which is what the Greeks had wanted), but they preserve angles.

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I had some leftover architecture images from Columbus and wanted to see how they look when made circular. Here, for instance, is the AT&T building

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and this is a circular version:

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There are three degrees of freedom one can play with (the dimension of the automorphism group of the hyperbolic plane), which means that one can squeeze parts of the image towards the boundary cirle. Here are two other versions of the same image.

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Another favorite of mine is the atrium of the Cummins office building with its wonderfully intricate play with straight lines and black and white.

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Now we only have to find architects and builders who create buildings that have these curves in reality.

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Bucket of Blood Street (Arizona II)

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The little town Holbrook in Arizona offers convenient accommodation after visiting the Petrified Forest National Park. This is not a wealthy town, but  the downtown area has its own nostalgic charm. You wonder what life was like here a hundred years ago.

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Then you come across this street sign. Choosing a name is a delicate thing. Apparently, in the good old times a saloon shooting ended in such a way that the establishment was renamed the Bucket of Blood Saloon. In the long run, this didn’t help much, and after the building fell apart, the name survived as the street name, to this day.

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Other local attractions allude to that bit of the town’s history in appropriate color.

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The moral? Appearances change, names stay. But it seems the town hasn’t quite figured out whether that name is a curse or an opportunity.

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Tulip Trestle

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This is the Richland Creek in Green County, Indiana. Nothing spectacular about it, but this is how much of southern Indiana looks like in winter. This creek is bridged by an enormous railroad trestle, whose insisting presence almost justifies is existence.

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Near the ground, the steel posts are hidden in the brush. Then they emerge, undeniably.

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Made, not grown, and grabbing all the space there is.

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Is dialogue possible?

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The letter T, with which both Tree and Trestle begin, (and Tower, Tall, Top, and others, worse) seems to indicate in its negative space the lack of anything above.

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Will Time say nothing but I told you so?

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Bridges, Rivers, and Walls

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Locals rarely go on sightseeing tours for tourists, which contributes to their different perception of things. For instance, living behind a wall is nothing strange when you grow up with it. When I (re)visited Berlin in 2015 for a conference, the conference excursion was a boat trip on the Spree.

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While I knew that Berlin has that river, it had never become part of my perception of Berlin as a city. The Spree does nothing for Berlin like the Seine does for Paris, the Thames for London, or the Danube for Budapest or Vienna, the Rhine for Cologne (to name a few). It is small, largely canalized, and so much covered with bridges that one can easily overlook it.

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These bridges connect the northern with the southern part of the city. Some of them are old, other modern. In contrast, the Spree connects the eastern with the western part, and it did so even when there was a wall. Rivers are hard to stop, like time. Crossing from west to east on this little river was a special moment for me.

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Above is the cathedral, to its left the (re-)construction of the Berlin City Palace, replacing the East German Palast der Republic, that it turn had replaced the original Baroque City Palace in 1950.

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Buildings serve many purposes, but they also mark and preserve time — for a while. The architects should never forget that time will flow on, inevitably.

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Stellated Triacontahedron

If you have mastered the Slidables from last year and had enough of the past gloomy posts, you are ready for this one.

Let’s begin with the rhombic triacontahedron, a zonohedron with 30 golden rhombi as faces. There are two types of vertices, 12 with valency 5, and 20 with valency 3. In the image below, the faces are colored with five colors, one of which is transparent.

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The coloring is made a bit more explicit in the map of this polyhedron below.

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We are going to make a paper model of one of the 358,833,072 stellations of it. This number comes from George Hart’s highly inspiring Virtual Polyhedra.

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In a stellation, one replaces each face of the original polyhedron by another polygon in the same plane, making sure that the result is still a polyhedron, possibly with self intersections.

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In our case, each golden (or rather, gray) rhombus becomes a non convex 8-gon. The picture above serves as a template. You will need 30 of them, cut along the dark black edges. The slits will allow you to assemble the stellation without glue. Print 6 of each of the five colors:

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Now assemble five of them, one of each color, around a vertex. Note that there are different ways to put two together, make sure that the original golden rhombi always have acute vertices meeting acute vertices. This produces the first layer.

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The next layer of five templates takes care of the 3-valent vertices of the first layer. Here the coloring starts to play a role.

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The third layer is the trickiest, because you have to add 10 templates, making vertices of valency 5 again. The next image shows how to pick the colors to maintain consistency.

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Below is the inside of the completed third layer.

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Two more to go. Layer 4 is easy:

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The last layer is again a bit tricky again, but just because it gets tight. Here is my finished model. It is quite stable.

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Points of Support

Three points make a triangle.

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After witnessing the remnants of the American democracy in free fall during this year’s election, I found some peace in contemplating the stability that is achieved by three points
on a long walk through Turkey Run State Park’s maze of canyons.

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Fall is almost over here, and the dead leaves and trees are awaiting the mercy of frost and snow.

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There is beauty still in all of this, maybe because it cannot be made a target of hatred.

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Let’s call this a prayer, if such thing still exists.

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Columbus out of Focus

A while back I confessed that I had acquired a Velvet 56 from Lensbaby. Yesterday I decided to try this lens with street photography and architecture, abusing charming Columbus (Indiana) for that purpose. Of course, all images were taken wide open. Brace yourself.

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Above is the entrance of the Cummins Headquarter building. This lens has clearly difficulties here. The overall softness distracts from the graphical elements. If you don’t know what Cummins is making, you can see one of their products below. It is not a space ship, nor a gun.

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While Cummins would probably not use this image for marketing, the Velvet 56 does a much better job when there is an obvious foreground. With the rental bikes lined up below, one can see nicely how the lens progresses into unsharpness and how it deals with highlights.

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I still like that image, and even more so the image below. Beautiful couple on beautiful bikes.

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My favorite is the last one, however. The unreal mini-halos about all the highlights on the chairs complement the mural as if the lens just came out of the bar…

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