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.
Can we do this, too — transform our transient stream of thoughts, worries, and hopes into something else, like frost transforms water into ice?
How do we begin? How do we get ready for it like water always is?
In Adalbert Stifter’s Rock Crystal (quotes below in the translation by Marianne Moore) the two siblings Sanna and Conrad get lost on a vast glacier in a snow storm on their way home in the Alps. Frost has transformed the landscape, and is transforming the children, too, to the absolute essential.
The boy maintains hope, despite evident hopelessness, and his little sister maintains trust.
At last they came to a tract with not a tree on it. “I don’t see any trees,” said Sanna. “Perhaps the road is so wide we can’t see them because of the snow,” said the lad. “Yes, Conrad,” said the little one.
“Sanna, we cannot go over there,” said the lad. “No,” said the little one. “We shall just turn around and get down somewhere else.” “Yes, Conrad.”
These dialogues continue like this, while the children spend the night on the mountain in the ice. Conrad and Sanna are becoming ice, too, Conrad refracting reality and Sanna reflecting it back to him.
Frost is usually an unpleasant experience for us, but I take the arrival of a much needed cold front here as an opportunity to look at its benefits.
There is for instance hibernation — and I don’t mean some form of revocable death, but a state of reduced activity of body and soul. Reportedly, bacteria in Antarctica have not only survived hundred of thousands of years in the ice, but stayed active repairing their DNA during the entire time.
100 Years of Solitude are nothing compared to this, and one year of covid should also be survivable, in proper hibernation.
The contemplation of beauty in the frozen landscape has become my personal way of hibernation these days.
Two Lakes Trail is a 13 mile loop in Southern Indiana that took me 6 hours to hike today.
I met not a single soul on the entire trail, leaving plenty of time to think about place, time, and consistency. Now that winter is almost here, colors are not as vibrant anymore, and I like the distance this creates. It allows a strong focus.
I also like the concept of a loop, as a design pattern, a way to return to oneself, remaining consistent, but still transformed.
We are familiar with transformation through time (I can still feel my sore muscles). The transformation through space is more subtle, it requires to inhabit a place, and the only way to do this is through consistency.
While horizontally distanced, let’s use the freed time for a contemplation of the vertical.
The first two sets from today are, as so often, taken at the DePauw Nature Park.
I have arranged the 1×3 pictures in triptychs to further emphasize their narrowness by framing the depth of the center piece with contrasting walls. I find the effect disconcerting in a positive way, maybe because the obvious path is not necessarily the only path.
This last one is from my other favorite place here, New Harmony.