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.

Limit1

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

Limit2

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…

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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

Wp1After 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

 

Hanging Lake (Colorado II)

Last week’s post was a bit of a cliff hanger, and so is Hanging Lake.

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It is precariously sitting on top of a cliff, with waterfalls in the back as a bonus.

 

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The emerald green water creates an eery play between underwater world and the reflections of the upper world behind.

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What more could one wish for? Well, there is more. A very short hike up above is Spouting Rock, a single, taller waterfall that by itself is worth a visit.

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Long time exposure doesn’t do it much good.

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In this case, I like the dramatic spattering or the quiet drip-dropping much better.

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It is a wondrous place. Remember, come early.

The Translation Invariant Costa Surface

Out of the flurry of minimal surfaces that was inspired by the Costa surface, a particularly fundamental new surface is the Translation Invariant Costa Surface, discovered by Michael Callahan, David Hoffman, and Bill Meeks around 1989.Chm11

Like Riemann’s minimal surface, its ends are asymptotic to horizontal planes, but it is invariant under a purely vertical translation, and the connections between consecutive planes are  borrowed from the Costa surface. Surprisingly, in a few ways this surface is even simpler than Costa’s surface. To see this, let’s look at a quarter of a translational fundamental piece from the top:

Quarter

It is bounded by curves that lie in reflectional symmetry planes, and cut off with an almost perfect quarter circular arc. Hence the conjugate minimal surface will have an infinite polygonal contour, like so:

Conjugate

It is not too hard to solve the Plateau problem for such contours, and adjust the edge length parameter so that the conjugate piece is the one used for the Translation Invariant Costa Surface. It is also possible to argue that the Plateau solution is embedded, and conclude the same for the Translation Invariant Costa Surface. All this is not so easy for the Costa surface itself.

Chm12

Above is a variation with one handle added at each layer. Surprisingly, the corresponding finite surface does not exist. One can add deliberately more handles. Below is a rather complicated version that I called CHM(2,3), with a wood texture rendered in PoVRay in 1999, when I had figured out how to export Mathematica generated surface data to PoVRay.

Chm23 wood

Hanging Lake Trail (Colorado I)

Hanging Lake is one of the most popular hikes in Colorado. In the summer, the daily 1000+ visitors don’t hike the trail anymore, but stand in line all the way up and down.DSC 8523

I avoided all this by getting there at 7am, which gave me time to enjoy the trail itself.

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It climbs up steeply among trees and rock cliffs. On a crowded day, it would be impossible not to overlook perfect sceneries like this one.

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The semi-vandalized hut below hints at how much work it must be to maintain the trail and keep it in its almost pristine condition.

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What to do with an incessant stream of visitors? Let it grow, or cut?

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It all depends, of course, what is at the end of the hike. We’ll learn that next week…

Boundary Considerations, Part II

As promised, today we will look at a close cousin of last week’s surface. A good starting point is the CLP surface of Hermann Amandus Schwarz, about which I have written before.

 

Clp

Up above are four copies of a translational fundamental piece. There are horizontal straight lines meeting in a square pattern, vertical symmetry planes intersecting the squares diagonally, vertical lines through edge midpoints of the squares and horizontal symmetry planes half way between squares at different heights. What more could one want? Well, CLP has genus 3, and we wouldn’t mind another handle.

 

 

Nearsingly

 

There are various ways of doing that, and one of them leads to today’s surface, shown above. For adding a handle we had to sacrifice the vertical straight lines, but all other symmetries are retained. These are, in fact,  essentially the same symmetries we had in last week’s surface, except that there, the squares in consecutive layers were shifted against each other. The similarities go further.

Neardoubly

Again we can ask how things look at the boundary. Pushing the one free parameter the the other limit, gives us again doubly  periodic Scherk surfaces and Karcher-Scherk surfaces. There is a subtle difference (called a Dehn twist), however, how the two types of Scherk surfaces are attached to each other in both cases.

Flat

Finally, as usual, the cryptic rainbow polygons that encode everything. Today, the two fit together along their fractured edges, which has to do with the period condition these surfaces have to satisfy.

 

Trillium Grandiflorum

Spring this year was short, and so was the wildflower season.  

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This post is dedicated to the largest trillium in the state, the trillium grandiflorum. It is not particularly rare, but the large white flower petals whither quickly, and is a favorite food of the abundant deer population.  DSC 1292

Both the petals and leaves are deeply veined. The flower sits on a stem above the leaves, in contrast to the drooping trillium where it drops down below the leaves.

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One of the woodland trails in Turkey Run State Park has them usually in abundance, but only this year I saw them in their prime time.

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How Does This Look Like At The Boundary?

A common recommendation to the layperson who is stranded among a group of mathematicians and doesn’t know what to say is to ask the question above. It will almost always trigger a lengthy and incomprehensible response.

 

Single

 

For example, let’s look at the surface below. It constitutes a building block that can be translated around to make larger pieces of the surface. That this works has to do with the small and large horizontal squares. It is similar to Alan Schoen’s Figure 8 surface, but a bit simpler (it only has genus 4)

 

Pyramid

This surface belongs to a 5-dimensional family about which little is known. The only simple thing I can do with it is to move the squares closer or farther apart. So, how does this look at the boundary? On one hand, when the squares get close, we see little Costa surfaces emerging, as one might expect:

Nearcosta

At the other end of infinity, things look complicated, but depending what we focus on, there is a doubly periodic Scherk surface or a doubly periodic Karcher-Scherk surface:

Nearscherk

Below are, for the sake of their beauty, the two translation structures associated to two of the Weierstrass 1-forms defining this surface. Next week we will study a close cousin of this surface.

 

Flat

 

Little Things in Yellow and White and Green

The Turkey Run State Park has not only some of the most interesting rock formations in Indiana, but also an exciting vegetation.  Today we focus on the little things. Let begin with the liverworts (marchantiophyta), belonging to the bryophytes family. These bryophytes neither use roots nor make flowers, but interesting leave patterns. This stuff is what covers the rock formations, unless the rocks have been abused as slides. People should (i) look and (ii) think before they do their thing.DSC 1231

Here is another very little one I don’t know the name of. 

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Below, I think, is a buttercup flower, getting ready.DSC 1257

Unfortunately, searching for yellow with white hair is not very helpful. So I also don’t know what this one is called. But it does know about right angles and 5-fold symmetry.DSC 1258

Finally, another flowerless plant with pretty hair:DSC 1286