June 2018 – The Inner Frame

Mimosa Tea

We have four Mimosa trees in the garden. They have been busy blooming for a while, and the fragrant flowers are a wonder to look at.

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I couldn’t resist to harvest some of them and turn them into tea. DSC 9685 Half a dozen in a cup turn instantly pale green when infused with boiling water. DSC 9699

The cup is still bright green, and the typical mimosa scent is overpowered by a herbal note that is not really unpleasant but distracting. What did turn out more spectacular was to use some of the flowers for pressing and scanning. Mimosa003 2

The colors became stronger after a day under heavy books.

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The Naming of Things

Studying minimal surfaces has become a bit like hunting for rare new species of plants or animals. Having a newly discovered specimen named after a person might be considered an honor to some, but a confusion to others, because it typically says more about the name giver than the actual specimen.

Asingle

Let’s, for instance, consider the minimal surface above, which is part of a triply periodic surface. It has genus 5 (after identifying opposite sides by translations), and no name yet. In my book it comes with the code name (1,0|1,1). To explain this code, look at how the symmetry planes cut the piece above into eight pieces, and look just at the fron-top-left. There are, on the piece, four points where the surface normal is vertical: Two of them are on the left side, with normal pointing up and down, and two more in the middle of the picture above with normal pointing up at both points. So the 1 stands for up, the 0 for down, and the vertical bar separates the two boundary components. A phi1 01

You can see this also in the frieze pattern associated to this surface via the Weierstrass representation. The uppr contour repeats left-left-right-right turns, while the bottom just alternates left-right-left-right. Replace left by 1, right by 0, to get 110011001100… and 1010101010…, respectively, and take only the first two digits of each sequence, to get the name code. Below is a larger copy of a deformed version with the same code.

Acopies

That seemed a clever thing to do, because there are at least four more such codes for genus 5 surfaces (one of them has been named Schoen’s Unnamed Surface 12 before…). Unfortunately, codes can be deceiving, because there is also this surface:

Bsingle

It follows the same up/down pattern of the surface normal and hence gets the same code, but the top edge bends differently. Neither the frieze pattern

B phi1 01

nor larger chunks of deformed versions Bcopies

allow to codify the difference. The mystery can be resolved by looking at what are called the divisors of the Gauss map, and use Abel’s theorem to distinguish them. But not today anymore.

Leadville (Colorado V)

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That little town you can maybe make out to the left and behind the lake up above is Leadville. At just over 10,000 feet, it is the highest incorporated city in the US, and used to be a bustling mining town, as the name hints at.

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There are attempts to cash in on the town’s history, like in charming Jerome.

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But while the drive-through city center is well kept eye-candy, the mining area tells another story.

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It doesn’t take long until things fall apart beyond repair…, but fortunately, sometimes they do this in style.

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The aesthetic appeal is enormous, maybe because a true relic conveys a stronger message than a fake facade.

The Angel Surfaces

One of the toy examples that illustrates how easy it is to make minimal surfaces defined on punctured spheres is the wavy catenoid. In its simplest form it fuses a catenoid and an Enneper end together, like so:

WavyCatenoid

I learned from Shoichi Fujimori that one can add a handle to these: G=1

This would make a beautiful mincing knife… Numerically, it was easy to add more handles:

G=4

I dubbed them angel surfaces, partially because of their appearance, partially because while we think they exist, we don’t have a proof.

They are interesting for two reasons: First, they are extreme cases of two-ended finite total curvature surfaces: The degree of the Gauss map of such surfaces must be at least g+2, where g is the genus of the surface. Here, we have equality.

Secondly, they come in 1-parameter families, providing us with an interesting deformation between Enneper surfaces of higher genus. G=2

Above is a genus 2 example close to the Chen-Gackstatter surface. Below is a genus 2 example close to a genus 2 Enneper surface, first described by Nedir do Espírito-Santo. Espiro Santo

In other words, we get a deformation from a genus 1 to a genus 2 surface.

Green is a Difficult Color (Treescapes III / Colorado IV)

Getting to higher elevation in late spring is a problem in Colorado, not so much because of snow, but because of the streams with hip deep and ice cold water that one has to cross. DSC 8735

After a while one resigns into not reaching that peak or lake, and finds consolation in the contemplation of the trees on the other side of the stream.

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I have written twice about treescapes: First about the fall at Red River Gorge State Park in Kentucky, and then about the winter in Brown County State Park. So now it is time for a spring version.

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Green is a difficult color. When I make 2-colored surface images, I usually have a hard time picking a second color that complements any sort of green nicely. On the other hand, I find the natural shades of green in these landscapes positively overwhelming. My theory is that green goes well only with more green, or shades of gray.

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These images are from an attempt to reach the Flat Tops Wilderness. There will be another time.

Inside or Outside?

The last minimal surface that made it into Alan Schoen’s NASA report is the F-RD surface. It has genus 6 and looks fairly simple.

Tetra

A fundamental decision one has to make these days is to choose the side one wants to live on. If, for instance, we decide on the orange side, we will have the impression to live in a network of tetrahedrally or cubically shaped rooms with connecting tunnels at the vertices of each. Not too bad, but, as things stand, we will never know what life on the other side looks like.

Cubical

Luckily, our imagination is still free, and we can think about the other, green side. What we can hopefully see from the pictures above and below is that the rooms of the green world are all cubical, with tunnels towards the edges of each cube. Alternatively, we can also think of the rooms as rhombic dodecahedra, with tunnels towards the faces. That’s where F-RD got its name from: Faces – Rhombic Dodecahedron.

Double

Incidentally, the conjugate of the F-RD surface is again one of those discussed by Berthold Steßmann, with the polygonal contours having been classified by Arthur Moritz Schoenfließ.

A simple deformation of F-RD maintains the reflectional symmetries of a box over a square, but allows to change the height of the box. It turns out that there are two ways to squeeze the box together.

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In both cases we get horizontal planes joined by catenoidal necks, but differently placed in each case.

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Crossings (Colorado III)

You need to cross the stream three times until you reach Bridge One…

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Crossing a stream is a well-worn pattern, at least in Western culture: we think of Hades, Lethe, and all that. This post is about the pattern of multiple crossings.

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I was hiking No Name Trail (near Hanging Lake), when I met the hiker who informed me as above. She continued:

Bridge One is awesome. You should go there.

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And so I went, crossing the stream three times. A single crossing is like a terminal step, irreversible. Multiple crossings are like a dialogue: Hey, here we meet. We both have changed. Let’s meet again.

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When switching from one side to the other, we accept a change. On No Name Trail, this might be perceived as a change from pine and oak to birch.

…From Bridge One you can go on to Bridge Two…

At Bridge Two, there is a violent waterfall. Bridge Two itself, broken.

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…You can go even further, to a place I call The Top of the World…

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Will I ever get there?

Alan Schoen’s I6-Surface

Wp1 After Alan Schoen was fired from NASA at the end of 1969, he moved back to California and continued to experiment with soap film. In October 1970, he used two identical wireframes bent into figure 8 curves consisting of two squares meeting at a vertex. When he dipped them into soapy water at a small distance from each other and pulled them out, he could poke the flat disks between the two figure 8s and create a minimal surface that looks like the top half in the picture above. It extends triply periodically to a surface of genus 5.

Wp2

Several pages of notes with descriptions of successful experiments made it to Ken Brakke, who used his marvelous Surface Evolver to make 3D models of the surface. It was named I6, because it happened to be the 6th surface on page I of the notes. Hermann Karcher later called it Figure 8 surface. When you move the two figure 8s close to each other, you will get a surface that looks like a periodic arrangement of single periodic Scherk surfaces:

Wp3

Note that these Scherk surfaces are vertically shifted in a subtle pattern. More interestingly, there is a second, unstable surface you won’t get as a soap film:

Wp5

What you see here are Translation Invariant Costa Surfaces (or Callahan-Hoffman-Meeks surfaces) we looked at last time. So Alan Schoen’s I6 surface can be considered as a triply periodic version of the Costa surface, which Celso José da Costa discovered about 10 years later.

Of course you can poke more handles into I6, as you can with the translation invariant Costa surface. Below is an example of genus 7:

Wp6