Studies in Black and White

Turkey Run State Park has maybe three locations that define the park for me. They are both intensely beautiful and unique.
To capture the essence of a place it is often necessary to reduce it, to strip it from some aspects of its appearance. For instance, to distill the structure of a place, it can help to view everything in black and white.

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The first of my three places is at the suspension bridge over Sugar Creek. At the right time just after sunrise when the low sun brings the shore to maximal contrast, the wooden structures, rocks, and vegetation become equal contributions to a dazzlingly complex whole.

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Next there is Wedge Rock. Many times I have tried to capture it in its entirety, but I found it more appropriate to only hint at its size by showing a small portion of it. The three trees cover about as much area in the picture as the rock, and this balance emphasizes the contrast between the two so different main structural elements. On the other hand, they both contribute diagonals to the geometric flavor of the place.

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Then, still in Rocky Hollow Nature Preserve, the two main structural elements are the horizontal segments of the steps in the from and the background canyon wall in the back, and the vertical opening between the canyon walls. The function of the steps is not clear from this image. In wetter conditions, the canyon floor will be impassable due to water torrents, and the trail bypasses it on the right side of the wall. In any case, the two paths both give choices without a clear hint where these choices might lead.
The perceived equilibrium between the two choices is a photographic choice: The “heavier” path through the canyon is closer to the center, while the “lighter” steps are further to the side, creating a balance by weight on an imaginary scale. Also the lighter color of the stairs and their unexpected appearance trick the eye into spending equal amounts of time with both elements.

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The last picture is from a location that I hadn’t visited until recently. I find this image quite successfully spooky. The two main structural elements, the elegantly layered rocks in the front and the tree that dares to grow inside them both frame a third structural element, the black void just above the rocks. The almost artificial arrangement of rock and tree suggests that there is more to the place, putting a growing question mark into what we might think of as a cave entrance.

The Obscure Object of Desire

The trails of the Pine Hills Nature Preserve are naturally bordered to the north by the Indian Creek, a tributary to the Sugar Creek. For most of the time, all one can see from here to he west is this triangle riddled view:

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This has become one of my many obscure objects of desire. Fortunately, I am mentally sane enough to have learned that you do not get all what you want in your life, so I have been happy keeping it this way.

Even more fortunately, this fall the water level in the Indian Creek was so low that one could easily get to that strangely suspended tree in the center triangle. So on we go…

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This tree, growing on a small patch of earth at a nearly vertical cliff is an easy metaphor for too many things. You pick.
For me, almost more surprisingly, the possibility to move forward also opened the possibility for a view back.

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So maybe, even if we don’t always get what we desire, sometimes we should get it, if only to be able to reflect about the change that just happened.

And on we go. Following an abandoned path along the Indian Creek, we meet another cliff, with Morse code writing on it that appears to tell a story for an audience long gone.

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And on we go, exploring the little piece of new territory. Finally, we arrive at a new border: The Sugar Creek, that connects Turkey Run State Park with Shades State Park.

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Also, unreachable from here, a covered bridge that would allow to cross the creek.

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Indiana doesn’t have a National Park. This whole area, including Shades State Park and Turkey Run State Park, is so full of quietly beautiful places, that it would make an ideal candidate. But maybe it is better to leave this area alone, and hidden, most of the time.

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The Pawpaw tree is one of the more interesting trees that are native to North America.
Pawpaws are small and like shade. In the spring they make small colorful flowers. I don’t know whether its common that differently colored flowers appear on the same tree.

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When the big leaves turn yellow, they produce potato sized fruits.
They will not stay long on the trees, as most animals (from squirrel to deer) seem to like them even when not yet ripe. You need to harvest them when they are getting soft and begin to smell.

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They will not keep fresh for very long, so go ahead and peel them. The easiest way to deal with the large seeds is to eat the fruit in chunks and to spit the seeds out. If you are more patient, you can also remove the seeds, put the fruits into a blender and make a very delicious pulp. The taste is banana like with an exotic touch that is hard to pin down.

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The Gyroids (Algorithmic Geometry III)

When we use squares bent by 90 degrees about one diagonal and extend by the rotate-about-edges rule, we get Petrie’s triply periodic skew polyhedron {4,6|4} which has six squares about each vertex. The two tunnel systems it divides space into are another crude approximation of the primitive surface of Schwarz.


Coxeter observed that this polyhedron can be used to construct Laves’ remarkable chiral triply periodic graph as follows. Choose any diagonal of any of the squares of {4,6|4}. Take an end point of the diagonal, adjacent to which are six squares. Look at the six diagonals of the squares that share the end point as a vertex, and take every other of them, starting with the already chosen diagonal. Keep extending the emerging graph like this.


You obtain the 3-valent Laves graph. At each vertex, the edges meet 120 degree angles. It turns out a mirror symmetric copy fits onto the {4,6|4} without intersections. These two graphs are the skeletons of the two components of the Gyroid, a triply periodic minimal surface discovered by Alan Schoen. You can read all about the discovery at his Geometry Garret.


The Laves graph also lies on the dual skeleton of the tiling of space of rhombic dodecahedra. That means that you can get a solid neighborhood of the Laves graph consisting of rhombic dodecahedra:


This can be done both for the Laves graph and its mirror still leaving a gap in which one can fit the gyroid. Alan Schoen also discovered a uniform polyhedral approximation of the gyroid, consisting of squares and star hexagons. To build it, take a star, attach a square to every other edge, bending the squares alternatingly up and down. Then attach six more stars to the free edges of the first star, fitting them to one free edge of one of the squares each:


Two copies of this piece (without the downward pointing stars and and squares) make a translational fundamental piece of the uniform gyroid.


Images of larger portions are hard to parse, but it makes a wonderful model.



When discussing the options for traveling with a three weeks old baby from California to Indiana, friend Bryce reminded me that while today we view traveling as the unavoidable side effect when to get from A to B, there used to be a more conscious form of travel that one can metaphorize as a journey. Thrilled, we decided to take this trip by train. The idea was to spend two nights in a sleeper car, and the days sightseeing.


The comfort is minimal, but so are the demands of a three week old.


California becomes Nevada. Notice the difference in architecture and functionality (railway station vs. correctional facility).


Nevada becomes Utah and Colorado.


Then, in Iowa, when we start feeling the heat and humidity of summer in the midwest, the power of all passenger cars fail. For hours, the Amtrak personal shuffles the cars in order to put the one with the faulty cable at the end. In vain.


When we arrive in Chicago 8 hours late in the third night, Amtrak pays for a hotel with view.


We have arrived! Moral: Each journey should result in a story.

Vegetation (Iceland XIII)

This post is about ignorance. While I like plants, I know next to nothing about them, with the possible exception of cacti.
Moreover, I wan not prepared to encounter any interesting plants in Iceland at all. If I get the chance for a second visit, I’ll pack a macro lens. Let’s begin with Pinguicula vulgaris, the common butterwort. This is the second carnivorous plant appearing on this blog.

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The next one is the minuartia arctica, or the arctic sandwort. The german derivative of the old english wyrte is -wurz, which also appears and connotes with Gewürz, meaning spice. I haven’t tasted any of these.

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The previous ignorances could be covered up thanks to Google image search. The next one, which I find particularly pretty, I am clueless about. The blossoms were not more then 3-5mm in diameter. Help!
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Things get more complicated. Of course, it is not the actual plant is a soulful being that interests me, but rather its idea as a shape forming entity. Like so:

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These grow on the stunning black sand beaches. Because of the harshness of the environment, I suppose, the plants in Iceland are more exposed. While in lusher zones, the abundance of growth (and decay) is also camouflage, here, where there is nowhere to hide, everything becomes subject.

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World Cat Show in Novi, Michigan, 2013

In November 2013, my cat addicted daughter and I went on a trip to Novi, Michigan, to the World Championship Cat Show.

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What you see there are not the typical neighborhood alley cats, nor your (my) regular neighbors.

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The bond that apparently forms between two and four legged animals reminds me of what Albert Camus says in L’Étranger about the old neighbor and his dog: “À force de vivre avec lui, seuls tous les deux dans une petite chambre, le vieux Salamano a fini par lui ressembler.”

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What might I look like in a decade or so after living with that lovely Persian cat up above…
I have the feeling that the Birman down below would allow me an easier transition.

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It seems that all the efforts we humans have made to create special cat (or dog) breeds are only a feeble attempt to help us with our own transformations.


While many fruits and berries are being cultivated or stored so that you can buy them “fresh” year round, some are either too delicate or not popular enough for this treatment. So you have to get them when they are ripe, and find your own means of preservation.

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Plums are one such example. My own plum trees lose what little they produce to the greediness of the birds well before they are ready for human consumption, so I have to resort to local stores.

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Plums are also interesting, because the American style plum butter is a far cry from what this fruit deserves. Plums are too juicy for the standard ways of jam making. To produce a real mus, they need to be stoned, mixed with sugar (1 cup for 3 pounds), and spices (try cardamom and clover!).

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Let this sit for at least two ours and discard the juice (or dink it, if you like sweet treats).
Then put this into a baking dish and bake for at least two hours at 350 degrees Fahrenheit, stirring occasionally. Leave the oven door open for the first 30 minutes to get rid of even more liquid. You want the result to be really gooey. Be warned: 3 pounds of plums make less than a cup of mus.

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Now all that is needed is good bread.

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Clouds (Iceland XII)

Most places have their own very distinctive appearance of clouds. Landscape painters know this and therefore prefer to live close to the ocean or the mountains. Needless to say, clouds in the midwest are either dull or very dangerous.

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Iceland has both ocean and mountains so that one can expect the best of the best.

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When I was little they told us in school that life forms can be distinguished from lifeless matter by a few criteria: Ability to move, react to the environment, and reproduce. Clouds can do all that.

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So I started thinking that being a cloud might be an interesting way to live.

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Unfortunately, the only cloud based life forms in the near future will most likely be rather virtual.

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Primitivity? (Algorithmic Geometry II)

The construction of the polygonal diamond surface via bent rhombi can be varied. If we take as the bent rhombus two adjacent faces of the regular octahedron instead of the tetrahedron and follow the same rule to extend the surface by 180 degree rotations of rhombi about edges, we first get a less crooked hexagon,


four of which can be assembled to a translational fundamental piece


of a triply periodic polyhedral surface


that approximates Schwarz’ so-called primitive surface.


In this case, the ribbon representation has a much simpler appearance than the rhombic image.


After all, apparent complexity is often only a matter of the presentation.

If you want to make quick paper models of either the diamond or the primitive surface, cut out lots of equilateral triangles, divide the edges into thirds, and bend the three triangles at the corners upwards. These smaller triangles serve as flaps that you glue to the front sides of the central hexagons inside the original triangles.