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:



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:


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 crossDSC 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.


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.


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.


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.


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


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.


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:


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:


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:



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:


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:


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.


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.

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