Paris at Night

Paris has many things to offer, and not few of them are best savored at night. One popular option is to take the RER to La Défense, and take a look at La Grande Arche.

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This monumental building was designed by Johann Otto von Spreckelsen and Erik Reitzel, and is one of several Grands Projets by France’s former president François Mitterrand.


Its shape is inspired by a common projection of the hypercube into Euclidean space.

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Like every good piece of art, it is worth looking at from different angles.

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I took these pictures in the summer of 1991, just before a backpacking trip to the French Alps.

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The platform under the Grande Arche is typically so bright and the area behind so dark that
the casual visitor will not notice what the long time exposure reveals.

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Gestundete Zeit (Sand Art III)

My first digital camera was a Fuji Finepix 1400. Yes, the 1400 means that it had a staggering 1.4 Megapixels. That pretty much filled the screens these days, and also the memory cards. The first trip I took the camera to was to Amrum, an island in the North Sea. As I only had a really small memory card at first, I had to reduce the image size. Back then this was fine. Today they look puny.


Visiting Amrum in winter seems like a dumb idea: No rolling in the dunes, no swimming in the sea during long summer nights. Instead, hikes along the frozen beach during brief days, and sauna in the evening.


The tides flood the extended beaches and leave behind compelling patterns, which are brought out to perfection by the low sun.


Freezing and thawing helps to make patterns that the waves alone don’t accomplish.


The coastline looks like the alien landscape of a cratered moon.


I wish I could come up with sculptures like these: Simple, but utterly compelling.

Trillium Sessile

Now that June has arrived, it is time to say goodbye to the trillium sessile, the most common trillium in Indiana.

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It is beautiful already in its budding state, where the characteristic 3-fold symmetry is broken.


Like the four leaf clover, there is the rare exception of a quadrillium.

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Here is the same plant in full bloom.

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I have revisited the same spot in subsequent years, but haven’t seen it again. Some things are not meant for repetition.

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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