Wrapped Packages

Another fascinating minimal surface from Alan Schoen’s NASA 1970 report is his I-WP surface.

The name I-WP indicates the two skeletal graphs of the complement: The I-graph and the WP-graph. WP stands for wrapped package. You can see 8 copies of the surface below.

Because it is cut by symmetry planes into simply connected pieces, the conjugate surface is tiles with minimal polygons. This is Steßmann’s surface, discovered 1931, about 40 years earlier.

Berthold Steßman’s thesis determined the Enneper-Weierstrass data of all minimal quadrilaterals such that rotations about the edges generate a discrete group, completing work begun by Riemann and Enneper another 70 years earlier. One might wonder why Schoen’s I-WP surface was not discovered much earlier. Likewise, one might wonder why Steßmann (and Carl Ludwig Siegel, his advisor in Frankfurt), was interested in the Enneper-Weierstrass representation when Jesse Douglas and Tibor Radó had established the existence of much more general Plateau solutions by 1931.

The reasons for progress or the lack of it often lie in human fate. I could find only little about Berthold Steßmann. A short biographical note by the German Mathematical Society mentions 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 note also mentions that Steßmann was Jewish. I have some hope that his doctoral degree helped him to emigrate in time. Another historical note mentions that he received the Golden Doctoral Certificate at the 50th anniversary of his doctorate in Frankfurt.

Such were the times: such are the times.

The story of I-WP continues a little further. Hermann Karcher found a tetragonal cousin which he called T-WP:

A final riddle: Sven Lidin, Stephen Hyde and Barry Ninham discovered that the associate family of the I-WP surface contains several embedded triply periodic minimal surfaces at angles that are multiples of 60º. These are, however, all isometric to the I-WP surface. It is conceivable however, that they possess different kinds of deformations.

Make Your Own Gyroid

It’s that time of the year when I like to evade all the hectic and retreat to a little craft. One of my favorites to make is Alan Schoen’s uniform gyroid, which he discovered on his way from the Laves Graph to the minimal gyroid.

This is a triply periodic uniform polyhedron of type 3-3-6-3-4, meaning that all vertices are equivalent by a motion of space, and at each vertex meet triangle, triangle, hexagon, triangle, square, always in this order.

The triangles and hexagons group together to planar six pointed stars, which I prefer for model making, because it makes the model more sturdy and is less work… If you want to make your own, you need to print the templates on colored card stock. Make sure not to scale the pdfs below so that squares and stars have the same edge lengths. I use two different colors for the stars so that touching stars get different colors.

Take a star, attach a square to every other edge, bending the squares alternatingly up and down. Then attach six more stars to the free edges of the first star, fitting them to one free edge of one of the squares each, as shown above.

Two copies of this piece (without the downward pointing stars and and squares) make a translational fundamental piece of the uniform gyroid. Then you keep going until you get tired or run out of paper. If you have access to a a hyperbolic universe, you can also make a hyperbolic version (which would make a wonderful carpet design):


Printable PDF templates of stars and squares

Alan Schoen’s Nasa Report 1970

Alan Schoen celebrated his 94th birthday earlier this month, so it is only fitting to write a little about his NASA report in my series of blog posts of highly influential papers about the construction of minimal surfaces.

This task is not easy, because there is too much worth discussing, so I decided to split this into multiple posts, beginning today with his simplest surfaces. These are H’-T, H”-R, S’-S”, and T’-R’. I add to these P and H, found much earlier by Hermann Amandus Schwarz, because these six surfaces share enough properties so that a single piece of code can be used to compute all of them.

They all have a reflectional fundamental cell consisting of a right prism over a triangle, which must be of one of the types (3,3,3), (2,4,4), or (2,3,6), the numbers representing the triangle angles as fractions of 180º. Below is a piece of the T’-R’ surface in a (6,2,3) prism, and next to it how this prism fits into a translational fundamental piece.

If we remove from the skeleton of the prism the edges that intersect the surface, we obtain the two skeletal graphs of the surface, shown below in extreme wide angle perspective from above. That one of the graph has triangle layers and the other rhombic layers is the reason for Schoen’s naming convention: The letters T and R stand for triangle and rhombus.

The piece in a prism can also be used to effectively parametrize these surfaces. To do this, note that the vertical faces of the prism meet the surfaces in two arcs: One is a segment without corners, the other has two 30º corners where the arc switches from one face of the prism to another. These corners are also the only points with vertical normal. We can therefore conformally parametrize a surface piece in a prism by the shaded rectangle below. 

The vertical edges correspond to horizontal symmetry lines, and the horizontal edges to symmetry lines in the vertical prism faces. In this way, the shaded rectangle corresponds to the flat structure of the height differential. The points marked a and -a correspond to the two corners, where the Gauss map as a pole and zero, respectively. The value of a for an (r,s,t)-prism is determined (a consequence of Abel’s theorem) as
a = \frac12\frac{s}{s+t}.
This has the curious consequence that the height of the corners of the surface piece are determined relative to the height of the prism. For instance, for T’-R’, the value of a is 1/5, meaning that if the prism has top and bottom face at height +1/2 and -1/2, then the two corners are at height +1/5 and -1/5.

The integrals of the Enneper-Weierstrass forms G dh and 1/G dh become Schwarz-Christoffel maps that map the horizontal gray strip to a periodic polygon or freeze pattern. For T’-R’ this looks like

For other surfaces in this group, the angles will change. The Schwarz P-surface corresponds to the (2,4,4) prism, a will be 1/4, and the frieze pattern has an additional symmetry:

The red lines correspond the the horizontal straight lines on the P-surface. We will see in a later post that the same method can be used to generate many more surfaces.

Finally, all the surfaces in today’s group come in a 1-parameter family and have similar limits. The catenoidal neck pattern and the singly periodic Scherk surface arrangement is encoded by the skeletal graphs:


Universal Mathematica Notebook

The Costa Surface

In his 1982 PhD thesis, Celso José da Costa wrote down the Enneper-Weierstraß representation of a complete minimal torus with two catenoidal and one planar ends, all with limiting vertical normals. 

I do not know whether Costa had any hope or even opinion that his surface might be embedded, but this is what David Hoffman and William Meeks realized and proved in 1985. It was the first complete, embedded minimal surface of finite topology after 1776 when Meusnier had proved that the helicoid is minimal. This breakthrough has spawned a vast number of new examples and triggered ongoing research. 

David Hoffman and William Meeks found more symmetric examples of higher genus and were also able to deform the middle planar end into a catenoidal end.

Putting the new surfaces under some regime of classification has proven difficult. Costa’s proof that 3-ended embedded minimal tori belong to the Costa-Hoffman-Meeks family is all but transparent, and the question whether there are other embedded minimal tori of finite total curvature is still open. Examples with more ends seem to require also more handles, like Meinhard Wohlgemuth’s examples.

Then there are periodic examples that utilizes Costa saddles as building blocks, like the singly periodic Callahan-Hoffman-Meeks surface and the singly periodic Costa-Scherk surface below to the right that is different but possibly related to the Batista-Martín surface (of which I haven’t made a picture yet).

Below are two doubly periodic surfaces where the Costa saddles are rotated by 45º. The left one is the Lübeck-Batista surface, the right one a doubly periodic Callahan-Hoffman-Meeks surface with reflectional symmetries and without straight lines.  Can one rotate a Costa saddle continuously by 360º in any such configuration?

Finally, there are several triply periodic Costa surfaces. The left is Alan Schoen’s I6 surface from around 1970, found through soap film experiments, and predating the Costa surface by over 10 years. The middle one is Batista’s surface, and the right one a new example of genus 4 that actually has the Costa surface as a limit, and not the Callahan-Hoffman-Meeks surfaces.

All this is only a beginning. Laurent Hauswirth and Frank Pacard have smuggled a Costa saddle into Riemann’s minimal surface, making it a genus one surface with infinitely many ends. Laurent Hauswirth has also used Costa saddles to construct families of singly periodic surfaces with annular ends.


Mathematicians like to classify things. Among the complete, embedded minimal surfaces of finite total curvature in Euclidean space or space forms, this has been accomplished in most of the simplest possible cases. Let’s summarize:

In Euclidean space there are only two surfaces of genus 0 in this class: The plane and the catenoid. This is a consequence of the López-Ros theorem, proven in 1991 by Francisco López and Antonio Ros.

For translation invariant surfaces (or equivalently, minimal surfaces in ℝ³ divided by a translation, the surfaces of genus 0 (in the quotient) are the general Karcher-Scherk saddle towers. These surfaces have an even number 2n of annular ends and 2n-3 free parameters with which they can flap their ends. This has been proven by Joaquín Pérez and Martin Traizet in 2007.

If you want planar ends, the lowest possible genus is 1, and Bill Meeks, Joaquín Pérez and Antonio Ros have shown in 1998 that the Riemann minimal surfaces are the only ones.

The situation is not yet resolved for the screw motion invariant surfaces. Conjecturally, these surfaces should be Hermann Karcher’s twisted saddle towers.

The case of doubly periodic surfaces of genus 0 has been settled by Bill Meeks and Hippolyte Lazard-Holly in 2001. These surfaces have non-parallel top and bottom ends.

Doubly periodic surfaces with parallel top and bottom ends can only occur in genus one and higher. Again, the genus one case has been classified: Joaquín Pérez, Magdalena Rodríguez and Martin Traizet have shown in 2005 that these are the KMR surfaces.

The main open question is that of a classification of triply periodic minimal surfaces of genus 3. To describe the state of the art will deserve several dedicated blog posts.

Likewise, I will outline in future posts the state of the art in the next difficult (open) cases.

Finally, I should mention that there are other, equally valid viewpoints for classification, using different assumptions about the topology.


F.J. López and A. Ros, On embedded complete minimal surfaces of genus zero, Journal of Differential Geometry 33 (199), 293–300

J. Pérez, M. Traizet: The Classification of Singly Periodic Minimal Surfaces with Genus Zero and Scherk-Type Ends, Transactions of the American Mathematical Society
359 (2007), 965-990.

W. H. Meeks III, J. Pérez, A. Ros: Uniqueness of the Riemann minimal examples, Invent. Math. 133 (1998),107–132

H. Lazard-Holly and W. Meeks: Classification of doubly-periodic minimal surfaces of genus zero, Invent. math. 143 (2001), 1–27.

Joaquín Pérez, M. Magdalena Rodríguez, and Martin Traizet, The classification of doubly periodic minimal tori with parallel ends, J. Differential Geom. 69 (2005), 523–577

Derived from Scherk’s Examples

During my last semester as an undergraduate student at the Technical University in Berlin in 1984, Dirk Ferus mentioned in his Algebraic Topology class that there would be a geometry conference over the weekend, which he recommended attending. Stupid me, I didn’t go. I could have met my future advisor (Hermann Karcher), and I could have seen a future collaborator (David Hoffman) present the first images of the Costa surface.

This conference is also mentioned in the introduction of another paper from my list of highly influential papers with new examples of minimal surfaces, namely Hermann Karcher’s 1988 Embedded Minimal Surfaces Derived from Scherk’s Examples.

During the academic year 1984/85, I had attended two semesters of Karcher’s Differential Geometry. At the end of the second term he announced that while the third semester would only be for those specializing in geometry, we all should come for the first two weeks, because he intended to spend them with explaining the basics about minimal surfaces, which he had completely neglected. I was a little disappointed, because I was eager to learn about the darker arts – symmetric spaces, Einstein manifolds, Finiteness Theorems…

Karcher didn’t just spend the first two weeks on minimal surfaces, but about half of the semester, developing and presenting what would become the paper mentioned above.

The images here represent only a selection of the surfaces described in that paper: There are the saddle towers, the toroidal half plane layers, and the helicoidal saddle towers. Besides all these example Karcher develops a method to derive the complex analytic Enneper-Weierstraß data from geometric features of the surface, which, ultimately, has led to the enormous zoo of examples we are dealing with today.

Not Just a Special Surface

If I had to sum up the content of Hermann Amandus Schwarz’ price winning monograph Bestimmung einer speciellen Minimalfläche from 1867, I would do so using figures from plate VI from the Nachtrag, conveniently compiled in his Collected Works in a single figure:

What is shown here are polyhedra whose vertices are the branched values of the Gauß map of five families of triply periodic minimal surfaces that Schwarz is investigating. 

Schwarz spends most of the over 100 pages discussing a single surface, now called the Diamond or D-surface. It solves the Plateau problem for four consecutive edges of a regular tetrahedron. The details Schwarz provides are overwhelming, and it is easy to overlook that the methods Schwarz develops reach far beyond this special surface, and that he was fully aware of it.

What was keeping mathematicians busy these days? Bernhard Riemann had died in 1866 and left a legacy of new concepts and open problems. Complex analysts and geometers were working towards proofs of the Riemann mapping theorem, the uniformization theorem, and the Plateau problem. Schwarz had its own approach: Solve simple cases first, understand them as well as possible, and then apply the developed methods to solve the general case. Both for the Riemann mapping theorem and the Plateau problem, Schwarz looks at polygonal boundaries. He develops the Schwarz-Christoffel formula, and tries something similar for minimal surfaces.

Schwarz uses cutting edge technology: The Weierstraß representation for minimal surfaces, the language of Riemann surfaces, and elliptic integrals. He realizes that he can do more than just solve Plateau problems: In addition to straight lines, he can also prescribe symmetry planes. This leads to a differential equation which he can solve if the branched values of the Gauß map are sufficiently symmetric.

Competition was fierce, in particular between Göttingen (Riemann and Enneper) and Berlin (Weierstraß and Schwarz). Riemann had left a few pages of notes that hint at what Schwarz discovers. Schwarz must have been shocked when he saw the posthumous paper, with details added by Hattendorf. He also learns that Enneper had used a version of the Weierstraß representation in 1864, maybe without quite grasping its scope, two years before Weierstraß’ note from 1866. It appears that Riemann knew about this, too, as usual. How much did Enneper and Riemann talk in Göttingen? 

With the exception of Schwarz’ figure 47, representing the H-surface, all vertices are antipodally symmetric. I suspect that Schwarz would have instantly nodded if somebody had told him that his differential equation can be solved just under this symmetry assumption, an observation made by Bill Meeks in his 1975 thesis. How the differently symmetric H-surface fits into the picture, together with other, more recently found surfaces like Alan Schoen’s Gyroid, is one of the big open problems of the area.