The Inner Frame

Steßmann’s Surface (Wrapped Packages II)

In the paper Periodische Minimalflächen, published by the Mathematische Zeitschrift in 1934, Berthold Steßmann discusses the minimal surfaces that solve the Plateau problem for those spatial quadrilaterals for which rotations about the edges generate a discrete group.

Contour

Arthur Moritz Schoenflies had classified these quadrilaterals, there are precisely six of them, up to similarity. For the three most symmetric cases, Hermann Amandus Schwarz had found the solutions to the Plateau problem in terms of elliptic integrals, and Steßmann treats the remaining cases. One of them is shown above. It is easier to describe the contour for three copies: Take a cubical box. Then the contour above consists of two (non-parallel) diagonals of top and bottom face, to vertical edges of the box, and two horizontal edges that lie diametrically across.

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Extending the surface further produces the appealing triply periodic surface above. Below is a top view. This would make a nice design for a jungle gym. Unfortunately, this surface will not stay embedded; you see this at the corners where three pairwise orthogonal edges meet.

Top

However, the conjugate surface is embedded, and concludes the story from a few weeks back. The surface introduced there is the I-WP surface of Alan Schoen, and he mentions in the appendix of his NASA report on triply periodic minimal surfaces, that the conjugate of his I-WP surface had been discussed by Steßmann. Below is a more traditional view of the I-WP surface.

I WP cube

Its name (explains Schoen), stands for Wrapped Package, because a translational fundamental piece of its skeletal graph looks like four sticks wrapped together into a package:

Wrappedpackage

The internet knows little about Berthold Steßmann. There is a short biographical note by the German Mathematical Society, telling that he was born on August 4, 1906 in Hüllenberg, Germany, studied in Göttingen and Frankfurt to become a high school teacher, which he completed in 1933. Then, a year later, he received his PhD about periodic minimal surfaces, with Carl Ludwig Siegel as advisor. The same year, the Mathematische Zeitschrift published a paper of Steßmann, covering the same topic. The note also mentions that Steßmann was Jewish. This leaves little hope.

Puttabong Organic Moondrops First Flush 2017 vs 2018

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This year I had a little of last year’s first flush Organic Moondrops tea harvest from the Puttabong garden in Darjeeling left, so when the new harvest arrived I decided to compare the two.

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Both harvests show exceptional leaves (samples of 2017 above, 2018 below). Reportedly, this tea is harvested in the early morning hours when there are still dew drops on the leaves.

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The overall appearance is that the 2017 harvest is more yellow, while the 2018 more green.

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This becomes most evident in the pictures of the steeping leaves, and is clearly an effect of the leaves maturing over time.

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In the cup, there is no visual difference. The taste, however, is miles apart. Not only is the 2018 fresher with notes of green grass, it also has the slightly liquorish aroma of an execeptional first flush Darjeeling.

I think the 2017 harvest was generally rather problematic, so that the difference in taste is less a sign of aging but rather of a difference in quality of the harvest.

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I was curious to learn what the effect of sweeteners on the taste would be, so I also tried both teas with a little Stevia added. While I found that this can occasionally enhance the flavor of teas (for example strong Assam teas), here it completely leveled out the differences between the two harvests. More precisely, the sweetened 2018 tasted almost exactly as the 2017 harvest. So, do not add Stevia to prime teas, you might loose the nuances.

One, Two, Four

At the MSRI in Berkeley, there is a marble sculpture by Helaman Ferguson showing Klein’s quartic surface. Kleinquartic1

This is a Riemann surface of genus 3 with 168 automorphisms. Our Euclidean brains have a hard time seeing all these. Let’s start with an automorphism of order 7, and a tiling of the plane by π/7 triangles:

Heptatile

Fourteen of them fit around a common vertex (at the center of our hyperbolic universe), and the black geodesic indicates how to identify edges of the green-yellow 14-gon (repeat the pattern by 2π/7 rotations). Euler will tell you that the identification space has genus 3. A little miracle is that these π/3 triangles fit nicely into a tiling by π/3 heptagons. This becomes evident like so:

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The geodesic we used to indicate the 14-gon identification pattern becomes a geodesic in the heptagon tiling that passes through edge midpoints of eight consecutive heptagons, and all such geodesics will be closed on the identification space. This allows to define this surface also as an identification space of 24 heptagons (using the same geodesics). As this description is intrinsic to the heptagon tiling, it is invariant under all symmetries of that tiling, which include rotations of order 2 and order 3, in addition to the order 7 rotation.

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Why is this surface called a quartic? Replacing the hyperbolic π/7 triangles with Euclidean (1,2,4)π/7 triangles in three different ways and keeping the identifications, we obtain three different translation structures on the Klein quartic, which define a basis of holomorphic 1-forms. Playing with their divisors show that these 1-forms satisfy the equation x³y+y³z+z³x, showing that the canonical curve of Klein’s surface is a quartic curve in the complex projective plane.

More Smallness

I have written before about the perspective vertically down, and complained that in Indiana, you only see mud or decaying leaves. So, let’s have a look. DSC 1030

What is this stuff? I have only seen it at the DePauw Nature Park, near water. It is likely organic, but never green. Is there a zombie-plant whose natural state of existence is that of decay?

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But not everything is decaying. Roots are feeling their way, and algae cover everything in wondrous patterns.

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Tiniest plants remind us that we are little, too. DSC 1062

Hence let us rest…

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Hyperbolic Architecture

In high school I usually loathed art class. But once we started an architecture project that got me excited: We were tasked to design our own house, with all bells and whistles. I decided on the rooms being regular hexagons, arranged in an annulus of six, with a center hexagon without windows. Hexa annulus

This is the motivation behind today’s surface. Six hexagons, arranged as above, with the additional stipulation that when you exit a hexagon you re-enter another hexagon as indicated by the arrows (and extended by rotational symmetry). You can easily check that when you travers the rooms by always leaving through opposite walls, you will pass alternatingly through two rooms and return. Moreover, there are a total of six rooms around each corner, which suggest that Euclidean geometry is not suitable for this architecture.

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In the hyperbolic plane, one can arrange six 60 degree hexagons around a vertex as above. The geodesics indicate which edges are to be identified, again implying silently that everything is rotationally symmetric.

Now in the above Euclidean model the identifications are done by Euclidean translations, defining what is called a translation structure. One can accomplish the same with other Euclidean polygons (that are still conformal images of a regular hexagon) like so:

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or so:

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In the above image we have a very short inner edge connecting the 240 degree vertices. There are a few more one can use, but they are not quite as pretty. In any case, they provide us with plenty of holomorphic 1-forms on the surface of genus 4 given by the algebraic equation w⁶=z⁶-1: This is, after all, a sixfold cover over the sphere, branched over the sixth roots of unity. The first model realized this geometrically by replacing the sphere by the double of a Euclidean hexagon.

The Little Ones

I am not good with names. I recognize maybe a handful of Indiana wild flowers, but that’s it. In particular the small ones I tend to ignore. So, please take the names in this post with a grain of salt. DSC 0914

This one above, for instance, I believe is called Salt and Pepper (Erigenia bulbosa), and it is tiny (the petals are just 2mm long). So you see, I have been exercising my macro photography skills.

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A little larger is the Bloodroot (Sanguinaria canadensis). Its white petals are extremely delicate.

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The one above was again very small, and I have no clue what it is. DSC 0995 The last one I also don’t know. The little buds or whatever they are were maybe 2mm in diameter. Very cute.

A Double Figure 8

Recently, a local artist had an intriguing question. Suppose you have a hook in the ceiling (who hasn’t?), and two spot lights in front of the hook, slightly to the left and to the right. Suppose also that you have drawn two curves on the back wall. Can you bend a wire and suspend it from the hook so that the two projections match the drawings?

Sketch

I first thought: Yes, this means we just have to determine the intersection of two cones, so this is possible but maybe tricky.

After playing around with it a little I realized that this is simpler than I thought: Of both curves have the same height, this is essentially always possible, and even completely explicit. In fact, this is almost as simple as using two perpendicular parallel projections.

For instance, below you see a single red wire that has two figure 8 curves as projections.

Then of course one wants to play with it and rotate the wire.

Clearly, there are two more rotational positions where one of the projections is again a figure 8, the one above and the one below.

Now we need to find somebody who can accurately bend wires for us.

Namring Upper (Darjeeling 2018 I)

This year everything seems to be late. DSC 8497

This is ok. Somewhat worse is that the prices for Darjeeling have gone up again. I can only hope that the workers benefit from it, too. DSC 8499

My favorite this year so far is the Upper Namring “Premium”. I don’t know whether these little epithets like “Premium”, “Wonder”, “Exotic” or “Supreme” have a qualifying meaning; I liked it better when they would just call it “Invoice 12”, counting the harvests. But clearly that requires explaining, while everybody seems to understand “Wonder”. DSC 8504

This “Premium” harvest ic clearly not completely uniform, but I like the mix of bright green leaves with the rolled darker ones, this gives the tea a slightly grassy note in addition to the floral character of a powerful Darjeeling.

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Next time I will try to visually convey the taste differences between 2017 and 2018 Puttabong Moondrops.

Mouthwatering.

Fake Diamonds

Below is something rare. You see two minimal surfaces in an (invisible) box that share many properties, but also couldn’t be more different. Dd4

Let’s first talk about what they have in common: They share lines at the top and bottom of the box, and they meet the vertical faces of the same box orthogonally. This means you can extend both surfaces indefinitely by translating the boxed surfaces around, in which they become triply periodic surface of genus 3. Dd1

How are the different? The red one is a little bit more symmetric and belongs to a 2-dimensional deformation family of the Diamond surface that has been known for about 150 years. You can see how these surfaces deform in an earlier post.

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The other one belongs to a different deformation family that is only a few weeks old, discovered by Hao Chen, and of which you can see here some wide angle pictures, with clearly different behavior.

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These surfaces existed right under our nose, but nobody expected them to exist, because minimal surfaces are usually content with a single symmetric solution. Chances are that these surface hold the key in understanding the entire 5-dimensional space of all triply periodic minimal surfaces of genus 3.

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Short Lived Magic

Snow in April is a rare thing in southern Indiana. DSC 8396

The snow cover was very light and didn’t stay long on the warm ground.

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But the trees were decorated with lots of tiny white accents from snow flakes and water droplets, creating an unusual winter landscape.

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Instead of the typically harsh winter sun, everything was bathed in ambient light.

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The pictures were all taken with the Lensbaby Velvet 85, fully open at 1.8. The stopped down images I took also look gorgeous, but have an entirely different, less surreal character.

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More snow is predicted for Friday night. I can’t wait.

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