What to Keep

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I have often been trying to capture personal time in this blog through old and new photographs. What you see above, is a recent (like 20 minutes ago) photo of an old toy of mine. I received this early edition of Spirograph when I was maybe 9 years old. You can now purchase a 50th anniversary edition (without the tasty pins …).

This is something that I (and my daughter) have used intermittently over all these years, and it has acquired a meaning for me way beyond its mere presence. Already back then I cherished it so much that I kept the products in the box. So this is, well, an ancient artifact:

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The lavishly illustrated instruction manual promised perfection that I never achieved. Too often one of the wheels started sliding instead of just rolling, or the pins didn’t quite hold. What counts, however, is the process. We are, truly, not interested in the ideal, the mathematical perfect curve, but in the process of getting there.

The curves that one can make with Spirograph are called Cycloids. You can get them abstractly by tracing a point on a wheel that is rolling along a curve. In its simplest form, you roll a circle along a line,

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and you learn that these curves can be found on the icy Saturn moon Europa, or as geodesics in the upper half plane when using the Riemannian metric 1/y ds (which is not quite the hyperbolic metric, of course). The ancient ones used them to model planetary orbits when popular belief pinned man into the center of the universe.

As my early Spirograph experiments show, the results make nice designs. Using contemporary software like Mathematica allows you to create these to perfection, you think? Unfortunately, plotting the true cycloids will result in images that are either inaccurate (not enough anchor points) or difficult to manipulate in Adobe Illustrator (too many anchor points). So, to make this:-:,

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I replaced the cycloidal arcs between intersections by cubic Bezier splines that have the same curvature as the cycloids at their end points. Again, this was just to find satisfaction in the process to approximate the ideal.

Choice and Fate

Jigsaw puzzles are terrible. They tap into our subconscious desire to complete tasks even if they are pointless, the reusability is minimal, and they offer next to nothing for the creative or just inquisitive mind.


When in one of my former lives I needed a creative activity for young children that would encourage them to view themselves as part of a group, I designed the anti-puzzle above. It consists of only one piece, which is blank. Each child would receive a large printout of the tile, cut it out, decorate it with something personal, and put it on the wall.


There is no difficulty putting the pieces together. Everything fits, and you cannot make mistakes. The only choices that are left are the designs of the individual pieces, and the place on the wall. I can imagine grownups could use these for brainstorming, post-it style.


In a later part of that life, I recycled the idea for older children. Here, there are three different snowflake shaped jigsaw pieces (which I had cut out in large numbers and many colors using a die cutter). This turned out to be surprisingly difficult, because the pieces appear to allow you some choices. However, if you want to fill larger regions without gaps, it will always look like this:


The hexagonal lattice is something our squared brains have a hard time to adjust to, apparently. Still, choices can be made by selecting the colors and shape of the design.


Methuselah Trail

The White Mountain area in the eastern Sierras is home to the Great Basin Bristlecone Pines. The arid climate and high altitude limits the growth periods of the pines to a mere two weeks per year.


So they grow slowly, and get very old. Some of them are over 4,000 years old, making them the oldest trees on the planet.


We humans don’t think in these time spans. We occasionally consider the next year, and rarely the next decade.


The age of the cathedral builders is gone, who knew they would not live to see their work finished.
What would we do with so much time? Would we plan ahead and mold the future, or would we just keep adapting and contorting?


Maybe we should develop a better sense of being content with what we have.



When I was a graduate student I lived next to the Kottenforst, a decent forest in the south-west of Bonn. While largely unremarkable by itself, it offered me the opportunity to try things, like getting pictures of the reflection of a full moon on a frozen pond.

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The way to get there was partially paved and good enough for a safe bike ride at night. I had left my bike at the bridge that leads over the pond, and didn’t expect to see anybody.

Moon 2

While I was busy with tripod and macro lens, I heard a car approaching. This was not only unusual for the time of day (midnight), but also because the narrow road was not permitted for cars.

Moon 3

The car stopped, and police lights went on. They must have had their suspicions. I shouted at them from the far shore of the pond that I was just taking pictures of the moon. They left me alone.

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Whenever I am stopped by the police these days (it doesn’t happen that often), I am tempted to use this excuse again.

Balance (k-Noids II)

The Catenoid is one of the prototypical minimal surfaces, a building block for more complicated objects. The two openings (ends we call them) spread out to fill almost half of Euclidean space. If we want to have more such ends, we have to chop them off early enough.

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This, for instance, is a 5-Noid, because it has five such catenoidal ends. They are quite symmetrically placed, which is not necessary, at lest not to this extent.

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Here is a 4-Noid. The two little catenoids poking out (like eyes??) at the front push their bigger brother and sister backwards, suggesting a rule of balance that must be followed. This is indeed the case: the direction vectors of the ends (the way they poke), scaled to take their size into account, must be in balance. This is one of the many reasons why minimal surfaces are so esthetically pleasing: They keep a sense of equilibrium.

This is convenient for the mathematician, who knows that whatever minimal surface we discover, it will be pretty, but disappointing for the artist, who can’t claim credit for its pre-established harmony.

The images on this page were rendered with Bryce3D. In my first experiments with Bryce3D, I was captivated by the possibility to put alien looking abstract mathematical sculptures into more or less realistic landscapes.


However, while real landscapes have automatically meaning for us just because they exist, it is much harder for imaginary landscapes to acquire an equivalent meaning (maybe with the exceptions of the landscapes we dream about). So I abandoned the capabilities of Bryce3D as a landscape renderer but instead started to explore its immensely complex texture editor. The last image of today is an attempt of a reconstruction. I have lost the Bryce3D scene file, and only a very small version of the rendered scene has survived. So here is the new version, rendered using an old Mac laptop that still can run OS 9 and my old version of Bryce3D.


More Praise of the Mirror

So I could not resist to try out how my ancient Sigma 600mm/f8 mirror lens does on a Nikon D800. So out I went into this year’s mediocre winter.

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The static motives one can find in the woods are tree stumps, rocks, and dead leaves. Cheers.

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All pictures here are reduced in file size by about the factor 20, thereby sharpening quite a bit. None of the some 50 pictures I took were sharp at full resolution, but the scaled images look decent.

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So this is not a miracle lens. A solid tripod and precise focussing are a must, at least when it is as gloomy outside as it is here now.

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But this was not the point of the experiment. I wanted to play with the background distortions caused by the mirror design of the lens. The ideal targets were either small or far enough away, with some sort of non-uniform background at about twice the distance to the target.

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I usually use only telephoto lenses up to 200mm, and finding small targets in the distance is a great visual training.

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The driving force behind 3D computer graphics in the 1990s was the gaming industry, and the standard software library was OpenGL. Most attempts to bring 3D to a larger audience via web based 3D formats was largely a failure, very much like the reoccurring attempts to create a stable market for 3D movies. However, OpenGL had curious spin-offs. For instance, on the Mac, there was no OpenGL implementation until Conix3D Enterprise made one in late 1997. To advertise it, they also produced an Add-On for Mathematica that allowed access to use the complete OpenGL libraries through equivalent Mathematica functions.


I had just learned how to make (static) images of minimal surfaces straight in Mathematica, but the new technology allowed to produce vastly superior results. Smooth shading using surface normals, two-sided rendering of surfaces, and sophisticated lighting were a few of the immediate benefits.


In 2001, Wolfram Inc started to tease the users of Mathematica by showing off a beta version of Mathematica that supported OpenGL rendering. It took them until 2007 to ship Mathematica 6 that finally supported this.

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In the meantime, Apple had bought Conix3D, and the shift to OS X and Intel together with a new format for Mathematica add-ons completely doomed the OpenGL explorer.


The worst, however, was a limitation of the OpenGL explorer: It could only render images at screen size. And of course the big screens from back then are no match for today’s retina displays…

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Es reiche aber des dunkeln Lichtes voll, mir einer den duftenden Becher (Pyrenees 1996 – II)

Spending the night at a mountain lake is without comparison. Here we are at the Estany dels Monges at 2422m altitude, which was very cold, but we needed it. Walking around the lake in the evening and morning


The next two days brough disappointment: The area around Salardú had been heavily developed, to the extent that the GRP follows asphalted roads. Thankfully, heavy fog started to cover up all the ugliness.


Assuming that touristic development implies well marked paths was not a good idea. The plan was to reach the Col de Curios by day 7, which managed a day late, after losing the trail a couple of times and scrambling off trail whenever we felt like it.


A valley further we reached the Estany de Colberante at 2490m, which the HRP guide book praised as an ideal camping spot.
Unfortunately, the weather has deteriorated, and we were desperately looking for shelter, to no avail. So we pitched the tent and spent the night pretty much without sleep through two heavy thunderstorms with rain and hail.


I still do not know what the best survival strategy is in a thunderstorm at high altitude without any protection nearby. My guess is that the narrow valley was our savior, because the lightning strikes would rarely find their way all the way down to the valley floor. It was scary enough, though.

(to be concluded)

Geh aber nun und grüße die schöne Garonne (Pyrennees 1996 – I)

My second backpacking vacation in the Pyrenees was better prepared than the first. We had a tent, and we both had a fair amount of experience. The plan was to start start in Luchon, on the french side, and the hike the HRP until Andorra. We only made two mistakes: We started late in the Summer (end of July counts as late), which means hot weather in the valleys accompanied by thunderstorms, and our tour guide was from the previous year, i.e. too old. What saved us was the communication with the locals, who were enormously helpful.


The clouds on the picture above confirm what we had heard in Luchon: Heavy rain would come over night.
Fortunately, a friendly couple invited us to spend the night with them in a little hut they had the key for.


One of the highlights of the second day was to see the Garonne, that originates on the Spanish side, disappear in a sink hole, sneak its way under the mountains to reappear on the French side. The following day we had to cross the Port de la Picarde, which was slightly problematic, because it was still heavily snowed in:


(The other side is much steeper). The landscape changes rapidly between very very rugged to lush.


Similarly, the weather changes rapidly from sunny and hot to foggy and cold.


To be continued.