Out of Focus

For several years, I have toyed with the idea to get one of Lensbaby’s odd experimental lenses. Against good advice, I have purchased the Velvet 56. This is a full frame 56mm lens, with maximal aperture 1.6. It is my most specialized lens by far. It excels at not being sharp.

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Even when stepped down, it is blurry near the edges of the frame. I decided to take it to its other extreme, and use it wide open. Then, the shallow depth of field and the radial decay of sharpness join forces. There are other artifacts, too. The glow around edges for instance is possibly caused by drastically exaggerated chromatic aberration. People have claimed that all this can easily be achieved in Photoshop.

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Maybe so. The images have a strange depth that might be hard to achieve. But even if somebody comes up with a Velvet 56 filter, this is not the point.

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For me, the most exciting aspect about photography is the moment when I take the picture. I transforms what I see and feel at this moment into a rather selective image that I hope will represent what I have seen and felt in some way. Improving the outcome in post processing is of secondary importance.

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The Velvet 56 is the most limiting lens I have used. No filter will make these images sharp. Some might view this as a fundamental flaw. I view this as a creative challenge. You have been warned.

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Domino meets Towers of Hanoi

When a neighbor and colleague of mine told me he has a blog about abstract comics, that concept fascinated me to the extent that I had to make one myself. Here it is:


This, by the way, makes a nice poster. I called it Migration, and didn’t give a clue where it came from. There are very smart people who have figured it out by just looking at it, but you can’t compete, because you have already read the title of this post.

Let’s begin with the Towers of Hanoi. This puzzle is so famous that I will not explain it here, mainly because I was traumatized as a high school student when I was forced to solve the puzzle with four disks on TV, in the German TV series Die sechs Siebeng’scheiten. I just pray that no recording has survived.


In any case, after a healthy dose of abstraction, let’s look at the Towers of Hanoi from above, and treat it as a card game.
The disks are replaced with cards that have a disk symbol on it. For the three disk game, there are three different cards, showing a small, medium, or large disk. To make everything visually more appealing, we color the disks, and to emphasize size, we show empty annuli around the smaller disks, as above. Then the solution of the three disk puzzle would look like this:


Because a card hides what is possibly underneath, a position requires context. This is one of the two ways the puzzle is mutating into a story. In the next step, we use domino shaped cards consisting of two squares instead of square cards. Here are the six hanoiomino cards:


The puzzle is played on a 2 x 3 rectangle, with all six cards stacked like this in the top row:


Note that we have modified the Hanoi-rule: In the original version, a card can only be placed on an empty field or on a card with a larger disk. A hanoiomino must be placed so that each of the squares either covers an empty square or a square with a disk of at larger or equal size. This allows for more choice, which causes the second mutation of puzzle into story.

The migration story now tells how to move all the hanoiominos to the bottom row, to the same position, albeit reversed. It is the shortest solution, and unique as such, unless you want to count the backwards migration as a second solution.

Balance (Zion National Park)

Every symmetry needs to be broken.

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The tree up above stands on a pass that separates the Upper East Canyon in Zion National Park from the area south of it that eventually drops into Parunuweap Canyon. The casual tourist driving on route 9 will not wonder what else there is beyond the magnificent scenery that is accessible from the road. We did. The symmetric tree on the pass is not an indication what to expect.

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The way up through the sometimes narrow Checkerboard Mesa canyon is not difficult, and the view back from the pass is already rewarding.

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Turning around, the landscape opens up. We are on top of an intermediate mesa, and can stroll around, even climb some minor peaks.

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Few people come here, we had all this for ourselves. Still, there are regions higher up, not (yet) revealing their secrets to us.

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Then, this rock, put by chance upon much smaller support that did not erode away like everything else, and kept it in balance.

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So this is what we seek: Broken symmetry, but still balance.

Not Tangram

As a kid, I did like puzzles, at least until I discovered Tangram. Few things can be enjoyed on so many levels: It’s simple, reusable, facilitates abstraction and meditation. I don’t know where to stop. I have printed photos onto a tangram square and used it as a miniature puzzle. I have written two Tangram stories, i.e. picture books with few words where each illustration consist of two or three tangram puzzles.

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I even saved money and bought Dumont’s New Tangram, which has eight new tangram shapes and comes with beautifully designed puzzles. It was not a success with me, maybe because the pieces were cheaply made, maybe because the design of the pieces felt too arbitrary.

Here now is my own take on a tangram variation. There are five different pieces, for a total of eight, that fit together as a circle.

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The curved pieces allow organic puzzles. Here is an easy one:

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Compact puzzles are a little harder, but nothing is really difficult:

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I think of these pieces more as of means for designs that for puzzles. Here is my favorite so far:

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So, go ahead, make your own designs, and write Not Tangram stories. I am done with that. For now.

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Red Planet (Kodachrome State Park III)

When using film, we always joked that Fuji’s films leaned towards intense greens, while Kodak favored strong reds. I wouldn’t call it a tint. I even heard the theory that Americans had a special gene that suppressed a sensitivity towards red colors.

In any case, this is about Kodachrome State Park (again), and its glorious reddishness.

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This is of course a joke, I could have tinted all the images green and called moved everything to Fujichrome State Park. What is important, though, is the overwhelmingly monochrome landscape. While painters always have complete freedom over their color palette, the (nature) photographer can exert control only within limits. What do you do when a nice rocky landscape is ruined with green weeds? This does not happen on Kodachrome Planet, so almost any view allows undistracted contemplation. Be it the sun scorched earth above, or the enormous canyons below:

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Clay sculptures grow on the cliffs, unsure about wha shape they want to take,
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and rocks in intimate embrace wait for us to leave. Was this once just one rock that split, or are these two rocks that time has shaped like this?

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Oh yes, there is some greenery. It reminds us that we are only tolerated, too.

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Polysticks (Polyforms II)

One of my favorite polyforms are polysticks on a hexagonal grid. These critters consist of connected collections of grid edges.
I stipulate that whenever two edges of a polystick meet, we add a a joint to the figure. This is in order to avoid indecent intermingling of legs as shown by the two polysticks in the figure below. Blush. The properly decorated green polystick can only watch in dismay.

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We want to use the polysticks as puzzle pieces, and we want to keep things simple. So here are all four hexagonal polysticks with three legs and just one joint. I like to call them triffids.

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Two of them are symmetric by reflection, so I leave it up to you to count them as one or two. We can us three of them to tile a small triangle easily like so:

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By tiling I mean that we want to cover all the edges of the given shape, do not allow that two polysticks share a leg or joint (what a thought!), and do not require all vertices to be covered. We could do so, limiting the possibilities dramatically.

Below are two more examples. First a larger triangle, tiled using three kinds of triffids.

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I have not found a way to tile this triangle (or a larger one) with just one kind of triffid. And here is a hexagon that uese all four triffids to be tiled:

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Now go and make your own. If you want to use triffids, make sure that the number of edges of your shape is divisible by 3.

Secret Passage (Utah)

Continuing the exploration of special places in Kodachrome State Park, here is the Secret Passage, on a optional side loop of the Panorama Trail.

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It is bordered by a tall vertical wall on one side, a sloping climbable rock on the other, and leads nowhere, symbolized by the two meaningless boulders.

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So what is special about this place? The texture of the vertical wall is so rich of detail and variation that I just stood there for a while, staring.

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Of course everything is mindnumpingly red.

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This reminded me of an exhibition of large format abstract paintings by Emil Schumacher that left me unimpressed until I discovered their textural richness.

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In both cases, the fractal-like richness of detail seems to provide a non-spatial third dimension to the otherwise mostly flat wall.

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The Projective Plane

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This image (a variation of which I used for many years as a desktop background) is a close-up of the large sculpture below that can be seen at the Mathematical Research Institute in Oberwolfach.

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It is a model of the projective plane, a construct that simultaneously extends the Euclidean plane and describes the set of lines through a fixed point in space.
The simplest way to make your own model is via the tetrahemihexahedron,


a polyhedron that seems to take every other triangle from the octahedron and twelve right isosceles triangles to close the gaps left by the removed four equilateral triangles. That, however, is not the only way to look at it. These right isosceles triangles fit together to form three squares that intersect at the center of the former octahedron, in what is called a triple point.

So we truly have a polyhedron with four equilateral triangles and three squares as faces which can be unfolded like so


where arrows and equal letters indicate to glue. From this flattened version we recognize a (topological disk) with opposite points identified, which is yet another abstract model of the projective plane. The tetrahemihexahedron suffers not only under the triple point at the center, but also under six pinch point singularities at the vertices. Maybe it was this model that made Hilbert think that an immersion of the projective plane into Euclidean space was impossible, and having his student Werner Boy work on a proof. Instead, Boy came up in 1901 with an ingenious construction of such an immersion, which has an elegant connection to minimal surfaces.


Robert Kusner constructed a minimal immersion of the thrice punctured projective plane into space, with three planar ends, that you can see above. Applying an inversion, as suggested by Robert Bryant, produces images that are very close to what Boy had in mind.

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This explicit parametrization served as the basis for the model in Oberwolfach.

Cool Cave (Utah)

This year was the fourth time that I spent Spring Break in Utah, and it has become a mixture of revisiting familiar places and exploring new ones. One of the new discoveries is the Kodachrome State Park, a detour for people traveling Highway 12, much less overwhelming than nearby Bryce National Park, but in a very positive way. I met just two other hikers on the 10 mile Panorama Trail. The landscape is serene and has many spots that feel special. Let’s begin with the most remote of them, the Cool Cave.

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The pine trees guard the narrow entrance and the color palette suddenly becomes monochrome.

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Inside, there is just one open space. One hears the wind and clicks from small rocks falling down. Apparently, sometimes the rocks can be larger, too.

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The simplicity of this description is deceiving.

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The view back to the cave entrance, for instance, could be the work of an artist. The tonality is miraculously supporting the depth of the image, and the interplay between light and rock offers ample material for contemplation.

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