Elusiveness (Fischer-Koch II)

By varying angles and edge lengths of the fundamental piece, one can repeat last week’s construction of the 6-ended Fischer-Koch surfaces and make surfaces with more ends. Above and below you see images of the 10-ended version, together with their twisted friends.

Again one obtains parking garage structures as limits, and the position of the helicoidal axes is indicated below: The case k=3 corresponds to last week’s surfaces, the case k=5 to the ones above. The colored disks represent helicoidal axes, with the color showing the different spins.

What about the case k=4, which should lead to 8-ended surfaces? Daniel Freese has shown that one can untwist the parking garage structures to screw motion invariant minimal surfaces:


But these surfaces can not be obtained using the Fischer-Koch construction. Below you see the completely untwisted version with annular ends.

One key difference to the Fischer-Koch surfaces is that opposing ends have opposite normals in Daniel’s surfaces (or differently colored sides, as visible above). If the vertical line was really a straight line, it would be a rotational symmetry line, and opposing ends had the same color.

So things are not always quite what they seem.

Stubbornness (Hemlock Cliffs VII)

About 9 months ago I went looking for the Arrowhead Arch near Hemlock Cliffs, and I thought it’s time to see how this place looks like in the autumn.

The stubborn spiders are out and about, and the stubborn leaves cling to the trees.

But, of course, what is really stubborn are the rocks of the Messmore cliffs.

This time I was stubborn enough to explore them all the way.

I like the complex and noisy landscape.

And there is the surprisingly harmonious contrast between the cold rock and the warm autumn colors.

Contrasts like these seem to need each other.

In places like this, sometimes, something magical can happen, pure serendipity.

Lost in Translation (Fischer-Koch I)

In 1990, Werner Fischer and Elke Koch classified embedded triply periodic minimal surfaces that can be obtained by extending Plateau solutions for Euclidean polygons.

Above to the left you see a minimal 8-gon, extended to a twisted minimal annulus to the right. The horizontal lines make a 60 degree angle, and if the height of the shorter vertical segments is one half of the gap size between two of these segments, further rotations will deliver an embedded surface.

It is hard to believe that something like this is possible, isn’t it?

Hermann Karcher describes a variation of this construction that creates 6-ended singly periodic minimal surfaces. He also mentions that these surfaces can be twisted, i.e. deformed into screw-motion invariant minimal surfaces with helicoidal ends.

Above you can see what happens when the surface is twisted clockwise. On the right, we approach a parking garage structure with five helicoidal columns, four of them with positive spin and axes at the corners of a square, and one with negative spin at the center of the square. Not getting lost in a parking garage like this would be very difficult…

As these surfaces are chiral, twisting them counterclockwise leads to essentially different surfaces. In this case, the limit parking garage structure consists of three helicoidal columns with axes placed on a straight line. So during this entire deformation, two of the helicoids have magically cancelled.

Next week we will see a very recent and surprising variation of this construction.

Aporetic Water (Ohio XI)

The second long hike this fall in Ohio took me to the waterfalls in Hocking Hills State Park.

Flowing water is infinitely attractive. It’s common to capture waterfalls through long exposures to get that seductive silkiness.

Waterfalls can be as beautiful as the human body.

One little project I have is to make a time lapse movie with long time exposures of waterfalls, ideally in Iceland, over 24 hours, taking one shot every minute, which would give a single one minute long clip.

Then one can also capture instants time with very short exposures, dissolving the flow of a water into isolated droplets.

Here the plan would be to create a slow-motion film, taking hundreds of shots a minute, and exposing each for 1/8000 of a second, so that we can follow each droplet for much longer that it takes to fall.

Scragglies (Ohio X)

When I visited state parks in Ohio at the beginning of the pandemic lockdown in March 2020, I didn’t expect that we would still be struggling with isolation.

One of the places I visited then was Conkles Hollow, whose rim trail offers fascinating views of scraggly trees on the steep slopes of the nature preserve.

I was curious how all this would look in the late fall, and the difference is not big.

If at all, the trees have become a bit more scraggly (and older, like myself), many individual lonelinesses in immutable clusters.

Maybe they are patiently waiting, too, for time to pass, wounds to heal, and spring to come.

Would they move elsewhere, if they could, and become tall and beautiful?


With cooler temperatures and less humidity, it’s time to say good-bye to the mushrooms, and maybe these close-ups will do.

The now decaying fruiting bodies of have done their work and put out spores for new mycelium to grow.

Underneath the mycelium will keep composting and cleaning up and waiting patiently for spring and warmth.

Wouldn’t it be wonderful if inside us could grow fungi, too, with moody tendrils of a soul-mycelium absorbing worries and fears, and strangely shaped fruiting thoughts sending out spores to grow elsewhere?

Spring and warmth will come.

Braun (Berlin XI)

Taste changes, as do necessities.

What you see here are impressions from the new Berlin-Brandenburg Airport, which has been controversial for two decades, mainly for financial reasons.

My own reservations are more of a more esthetic and, alas, political nature. At first, I was struck by its boredom.

We have low ceilings, steel, glass, and — brown decor.

The prevalence of the color brown (mostly in the form of dead wood and marble tiles) darkens the place, and, much worse, reminds me of an esthetic that I believed had been overcome long ago.

Curiously, the older airport Berlin-Tegel (which had replaced airport Berlin-Tempelhof, famous for the being used for the Luftbrücke during the Berlin Blockade) still feels much more modern, with its successful usage of simple geometric shapes like hexagons and triangles to create an efficient and stimulating atmosphere, was co-designed by the same architect, Meinhard von Gerkan.

Admittedly, here in Indiana I am used to the color brown, and I have always feared that it supports an unhealthy emotional state.

At least here, at BER, there is a solution built-in: Leave.

Autumn Tapestry

The autumn colors are finally here, too, and Shades State Park, about which I can’t possibly have written often enough, offers from it’s Inspiration Point an excellent view across Sugar Creek to the tree covered hills of the other side.

I could have spent hours there, letting the colors soothe mind and soul, like music for the eyes.

Does it ever get boring? The trees don’t think so.

Well, to have some variation, here is a wider view of the scene: