Minimal Graphics (Spheres IX)

This post in the Sphere Series is motivated by the recent Circles post. It’s easy enough to conceive a generalization where we place spheres with centers at the points with integer coordinates in space, and set the radius so that something interesting happens.

There is a problem, though. We could visualize the 2-dimensional circle case because we could look onto the plane from our privileged position in 3-space. To do the same with spheres, we would need to step outside 3-space into 4-space. Let’s not do that.

Instead, let’s look at the simplest case of circle intersections. We can think of the quarter arcs as deformed straight edges of squares.

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To make things visible, we have to remove some of them, and a natural choice is to remove every other arc, like so:

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A similar approach works in three dimensions. Here, the spheres are arranged in a cubical lattice, and we can think of this as tiling by cubes where each cube has been replaced by an inflated sphere.


This would still be too busy, so I have removed some of the spherical shards. The choice for that is suggested by a minimal surface, the P-surface of Hermann Amandus Schwarz.


You can think of it as consisting of plumbing pieces that have connectors in six directions: up, down, left, right, front, back. There is a coarse polygonal approximation by it, using squares. Both the original minimal surface and its polygonal approximation divide space into two identical parts. A rat could not tell whether it lived on the insid or outside of the plumbing system.


If we push the squares a little as to create four-sided pyramids, alternating the direction, we get the prototype of the model of sphere shards. In the spherical version, the shards meet just at the corners, leaving enough space so that light can get through.


To make the sculpture more interesting, I have varied the colors, and moved it (sort of) off center. I feel it is a a visual representation of minimal music. Granted, there are many kinds of minimal music, and I do like many of them, but not all. The one I have in mind here would have to be composed by Steve Reich.


This would make a nice pendant sculpture. As material, I would prefer ceramics, not glass.


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After scanning some 400 negatives from pictures I took in the summer of 1990 (on my first hiking trip to the Pyrenees), the selection process feels difficult. I could go about it chronologically and tell about all the little mishaps, like the inept preparation (who would pack a full tracking backpack and in addition wrap a large bag to hold camera and multiple lenses around the neck?), or the virus infested water at Gavarnie we learned about too late.

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All this is silly, of course. Why should one go hiking in the Pyrenees to begin with? One reason to hike the Haute Randonnée Pyrénéenne we had not in mind was that this trail is transversal to the famous Camino de Santiago, used by pilgrims even today for personal enlightenment. Which brings us to a theme.

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The light in the Pyrenees is special. It combines the mediterranean softness with the clarity of high altitude.

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And there are special places, too, that deserve clarity. Like the Brèche de Roland, where Roland, after losing the battle against the Basques in 788, destroyed his sword Durandal. leaving a 40 meters wide gap in the mountains, part of which can be admired above.

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They all should have done the same with their weapons before the battle.

Circles, Intersected

Lets look at circles with centers at points with integer coordinates and equal radii. When the radii are small, the circles will be disjoint. Something interesting first happens when the radius becomes 1/2, because then the circles touch.


When the radii grow, the circles will intersect, and interesting patterns emerge. These patterns change continuously,
but when a special intersection occurs, the complexity of the intersection pattern increases. The next special intersection after r=0.5 occurs at r=0.7071, when circles that are diagonally across touch, and then again at r=1.

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Often, and due to the symmetry of things, whenever two of our circles touch, a second pair of circles must touch at the same point.
Then, at r=1.17851, we have true intersections of three circles at a single point (no touching!).

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Mathematicians find this interesting because the special intersections (touch or triple cross) mark singular points in the space of all such circle configurations. Understanding them means understanding the whole space.

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It is of course very satisfying that these singularities are also esthetically pleasing, as if they knew they are special and have dressed up for the occasion.

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Sugar Creek

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Sugar Creek is a tributary of Wabash River (which continues into the Ohio River and the Mississippi).
It connects Shades State Park with Turkey Run State Park, and is a highlight of both parks.

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At Shades State Park, most trails touch the creek at some point, or at least provide an unobstructed view across onto a vast wooded slope.


There are sights that stun instantly, and others that require some time.


In Turkey Run State Park, (almost) every visitor will cross the suspension bridge and enjoy a view like this:

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Paths (Corsican Landscapes, Part III)

Another landscape paradigm to explore is that of the path. Usually, we plainly think of a path as something that helps us to get from here to there. Sometimes, circumstances can hide the paths, and the signs disappear in the landscape.


Or the path disappears, because it really does not matter where you walk.


In other cases, the ever changing surroundings make it necessary to find a new path each time.


Often, the path is obvious, but there arise doubts about where it leads.


And finally, Freudian stairs can lead to hidden desires of the mind.


Corsican Rock Faces

One of my manic disorders forces me to see faces in all kinds of rock formations.


The varied geological nature of Corsica, together with heavy erosion due to wind and water, provides ample material.


Sometimes, sunlight helps, too.


I am sure that today, after more than 20 years, these rocks are better preserved than the negatives I had to deal with. Besides the usual dust and scratches, some of them have deteriorated beyond help. I have stored all of them in proper sleeves and under dry conditions. Still, I noticed large speckled areas on some of them that clearly were not present 20 years ago.

Horizons and Limits

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How does the landscape we live in influence our concept of the beyond? For instance, if our daily view consists of a seemingly endless chain of mountain ridges, do we expect that these mountains will continue indefinitely? Or is there a last one, after which the earth just falls off?

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Islanders are obviously in a special situation. The island Corsica is even more special, because besides the unlimited view into the mediterranean see, it also offers serious mountains that help obstructing the view.

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I visited Corsica in the summer 1992 for two weeks. This was a difficult year of changes and decisions. In retrospect, meditating about the limits imposed by landscapes provides adequate means to examine one’s own limitations.

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À la recherche du temps perdu

The summer and fall 1992 I spent my free time reading Marcel Proust’s À la recherche du temps perdu. That winter, I took the photos from this page, and revisiting them now is just one of several connections to Proust’s recherche.

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Of course these were shot with film, and in black and white, appropriate for season and theme. The location is the Sieg valley near Bonn in Germany, where I happened to come across a temporarily abandoned construction site.

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The time was literally frozen. Everything had been more or less orderly put ready to use.

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This was a curious sight. Unless our daily business is construction, we usually do not see these things, because they are buried or covered up, in the hope that they will function even in hiding.

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Not esthetics, but pure purpose is the reason for these designs. And because I did and do not know the actual purpose of them, they became for me the abstraction, the idea of purpose itself.

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This tilting away from reality towards abstraction has always fascinated me, already (at least) 23 years ago.

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Landscape Without Sky

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Of course this landscape has a sky. But everything in Zion National Park is so big that our human field of vision is somehow inappropriate.

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It’s like the romantic landscapes of Caspar David Friedrich about which Heinrich von Kleist wrote that when looking at them, he felt like his eyelids had been cut away.

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After a while, the desire to grab the widest lens in the bag and to take it all in fades. We become aware of a landscapes full of still lives.

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This is in particular true for the eastern part of the park, where most hikes are off trail (and which is much less overcrowded).

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Navigating this terrain is fun, but one needs to be careful. What appears easily accessible can well end in sheer cliffs.

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Also, be sure to pack plenty of water. The trees will thank you for it.

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Keep Going (Spheres VIII)

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This, like most of my images from the Sphere series, has its origin in a 2-dimensional picture. Below you see (parts of) the Apollonian Gasket. Descartes and Princess Elisabeth of the Palatinate discussed formulas for a fourth circle tangent to four other circles, more than 300 years circle packings became fashionable.


Now, in three dimensions, begin by placing four equally sized spheres into a larger sphere, like so:

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Start filling the empty space with more spheres, each as large as possible to touch four other spheres.

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Keep going.

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and going, and going.

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Finally, when you are tired and done, remove the first four largest spheres to create an empty space, and have a peek inside. What you see might look like the image at the top.