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.

Moon 1

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.

Moon 4

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

The Real Helicoid (Scrolls II)

After talking about the other helicoid first, it would be impolite to ignore the real helicoid, which is of course much more famous,
mainly because it also is a minimal surface.


A such, it possesses a deformation into the catenoid

which every textbook on the geometry of surfaces mentions at least as an exercise. Half way the helicoid will look like this:


I have chosen the size of the helicoidal paper so that the two spiral edges almost touch. This deformation is, however, problematic for book making, because no curve could serve as a spine.


But there are other ways to bend the helicoid, that are not anymore discussed in text books. We can keep the horizontal generators of the helicoid as straight lines, but let them slope upwards a little, like above,
or like below, with steeper lines.


Russian River Valley (Wine Biking I)


One of the must-sees for tourists in California is Napa. To be honest, the wines are overpriced, and the landscape is underwhelming. Go a bit further north, to the Russian River Valley, and enjoy the scenery by bike.


You will know that your are doing this with the right sort of people when they cross the river like this


even though there is a bridge and they didn’t have any wine (yet).


The winemakers were friendly and let us taste for free even though they knew we would not buy much wine.



I like games or puzzles that create something while being played. Here is a simple example which I call Unbalance. The single player version is played on a rectangular grid, like this one:

Board 01
A move consists of drawing a horizontal or vertical line segment of length 5 on the grid and within the box.

The first line segment can be placed arbitarily. All subsequent segments must cross exactly one already
drawn segment. Only two types of intersections between two segments allowed: They either
both divide each other both in the proportion 1:4 or both in the proportion 2:3. Contacts at an end point are
not allowed.

Legal 01

The last intersection not allowed because the vertical segment divides the horizontal segment in the proportion 2:3, while the horizontal segment divides the vertical segment in the proportion 1:4.

The goal is to place as many segments as possible without violating the rules. Here is an example with 11 segments.

Attempt 01

The are many variations. For instance, you can play with red and blue segments. Here it is required that segments of the same color divide each other in the same proportion, while segments of different color divide each other in different proportions.

Twocolor 01

For several players, you can start on a larger board, and each player uses their own color. The last player who can still make a move wins the game, following the rules with different colors otherwise.

The New Petzval Lens

Let me introduce you to one of my Christmas presents:

DSC 0983

This is a remake of a lens design from 1840 by Joseph Petzval, then one of the leading physicists working in optics. For its time, this lens was very fast and very sharp at the center. Today, we have faster glass, and good lenses maintain sharpness across the entire frame, so why would one bother? One reason is that the progressive vignetting that occurs towards the boundary of the frame creates a radial blur unlike anything else.

DSC 0372

This effect can be controlled by choosing appropriately sized aperture blades. Yes, this lens is so not automatic that you have to manually insert blades with aperture holes. You see them in the top picture scattered around near the lens. The lens comes with a set that have circular holes as one would expect, but nothing prevents you (or me) to use plates with holes in different shapes.

The effect is simple: A small point-like object (like a light source) that is out of focus is usually rendered as a slightly blurred small disk. This is what makes up the bokeh of the lens, and it is one of the most important characteristics of fast lenses (where you will have a lot of the frame out of focus, usually). If the hole in the aperture blade is not a disk but (say) a square, then the small dot that is out of focus will become a small square. Likewise, you can have star shaped blurs or even multiple blurs if the aperture plates has several holes.

Aperture Plates

Above is my first set of self-designed blades for the Petzval lens. I created this by first scanning in the actual plates for size and shape, vectorizing them in Adobe Illustrator, adding my own design, exporting them as an AutoCAD DXF file, and importing them into the software that drives my Cameo Silhouette die cutter. The result are little pieces of card stock paper.

DSC 0414

Here, for instance, is a neocubistic sculpture (from the Sculpture Trails Outdoor Museum near Solsberry, like all portraits in this post), using an aperture plate with several square shaped holes. Below is an image using a plate with a fractal cross.

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This opens up many possibilities. One can design aperture plates to complement the motive by enhancing the background, or one can distort an otherwise distractive background beyond recognition.

DSC 0378

Imagine a technology where a modern lens contains instead of regular aperture blades an electronically controlled screen that, somewhat like liquid ink, can be used to create aperture holes of any shape. In a film camera this would make it possible to continuously modify the out of focus area. Alfred Hitchcock would have used this to make the famous tower scene in Vertigo even more vertiginous.

The Other Helicoid (Scrolls I)

I have been thinking for a while to make a book out of curved paper, and my new year resolution for 2016 is to make this happen.

Usually, a book consists of a few rectangular pieces of paper that are attached to each other along one side of the rectangles to form the spine of the book. The fact that we can turn a page nicely uses the fact that flat sheets of paper can be bent into cylindrical or conical shapes without the need to bend the spine as well. A good choice of a shape for curved paper that behaves similarly is that of a ruled surface or scroll. The latter name is not in common use anymore, but I like it better.


For instance, we could take paper in the shape of a hyperboloid of revolution. This consists of a family of generators (the orange straight lines) that are attached to a directrix (the waist circle, for instance). We will now cut open this hyperboloid along one of the generators and bend it a little along all generators simultaneously, thus making them more horizontal.


We can bend further, making the generators truly horizontal. This gets us to the other helicoid:


That it is not the standard helicoid that you get by lifting and rotating a horizontal straight line along a vertical axis becomes evident in the top view.

HyperboloidalScroll top

Cross sections of this helicoid with vertical planes are graphs of the reciprocal of the sine function, in case you have wondered.
We can deform further, arriving at more scroll like images.


Here the idealized paper is slicing through itself, which looks interesting, but will, like most ideals, require some trimming in reality.

Wood and Water

Usually the McCormick’s Creek flows gently into the White River, which itself is during dry summer months reduced to a muddy mess.
After winter rain storms, the White River floods into the plains, and pushes the McCormick’s creek back, forcing the foam caused by the recent storm to spiral in waiting — for what?

DSC 0963

Time like the water has come to an unnatural standstill.

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The trees that will teem with life in a few months look tired beyond hope.

DSC 0275

But ripples in the water shake us back to life. We have paused only for a brief moment.

DSC 0272

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.

Knoid 5 b

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.

4Noid Big

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.



The Konica 3200 was one of the fastest color negative films ever made. ISO 3200 might sound lame these days when sensors are advertised with speeds reaching into the millions, but back in the 1990s, this was a revelation and gave ample opportunity to experiment.


Of course, we were talking about grain, while today it is called noise.


One of the unconventional uses of these high speed films was to take portraits at parties when it was getting dark and the usually shy victims were getting relaxed and more tolerant towards photographic intrusion.


I am normally hesitant to post pictures of friends because these are, well, private in the sense that they are not of general interest.


But friendships become memories


and posts like these messages in a bottle, to be lost or to be found.