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.
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.
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.
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?
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.