One of the text book examples of conformal maps is given by the complex function f(z) = z+1/z which maps the upper half disk bijectively to the lower half plane. This is the main ingredient for the following problem:
We would like to have a conformal map from a rectangle to a half disk (say of radius r) so that the left and right sides of the rectangle are mapped to half circular arcs around two points a and b.
The inverse of this map can be obtained by composing the text book map above with a Möbius transformation that sends the images of a and b to 0 and infinity, and then use polar coordinates on the upper half plane. Details are in this notebook.
What is this good for? Examples are 4-ended surfaces like the ones above that have sufficient symmetry that one can place the ends as four punctures on the real line, symmetric with respect to a circle.
This can also be useful more generally when existing symmetries cut the surface into pieces with four ends along the boundary of the pieces, as above.
One of many ways to construct triply periodic minimal surfaces is via conjugate surfaces. You start with a polygon in space, solve the Plateau problem, construct the conjugate surface. Instead of straight edges, this conjugate surface piece will allow extension by reflecting across its boundary edges, and with a fair amount of luck, you obtain a nice triply periodic surface. There are two disadvantages of the Plateau method: You are numerically limited to minimizing Plateau solutions, and the whole approach gives little theoretical insight. Here is a variation of this approach:
We start with a minimal polygon inside a box with all edges perpendicular to the faces of the box. Reflections at the faces will produce 8 copies, which constitute a translational fundamental piece of a triply periodic surface. If we look at the boundary of the polygon in the vertical faces, we note that at the corners the Gauss map will be vertical. We encode this in a sequence of + and – signs. For the left boundary component in the example, we have two points with normal pointing (say) up, encoded by +. In the second component we first point down at the upper point, and then down at the lower point, encoded by +-. Both sequences give the symbol (++|+-).
The same information is also contained in the shaded rectangle above, with the red dots labeled a and b corresponding to the corners in the left boundary edge, and c and corresponding to those in the right edge. The entire rectangle then represents the torus quotient of the surface under the 180º rotation about the vertical axis. The vertices are the zeroes and poles of the Gauss map.
Together with the additional reflectional symmetries at the horizontal box faces (the vertical green lines in the rectangle), this information determines the Gauss map. The height differential on the quotient torus is just dz, so we have the entire Weierstrass representation of the surface, except that we do not know the values of the parameters a,b,c,d and τ.
A linear combination of the parameters a,b,c,d determines how the Gauss map rotates in horizontal symmetry planes. For 8-gons as above, one usually is then left with a 2-dimensional period problem, resulting in a 2-dimensional family of examples. This approach is useful for three reasons: One can use the Enneper-Weierstrass representation for theoretical and numerical purposes, investigate limits easily, and extend the method by forsaking the horizontal symmetries, as we will see at a later point.
This page shows examples for these seven types, you can find more under the individual surface pages, listed under the genus 5 box types section in the triply periodic minimal surfaces page.
One of the most active areas of minimal surface research concerns embedded triply periodic minimal surfaces. They are periodic with respect to a 3-dimensional lattice of translations, and roughly classified by the genus of the quotient surface. The simplest case is that of genus 3, and even that is far from being understood. The earliest set of examples is due to Hermann Amandus Schwarz, and he already understood well that they come in parameter families. Below you can see the P surface, the D-surface, and members of the H and CLP-families.
So, how many are there? A casual parameter count tells that if such a surface is reasonably generic, it should come in a smooth 5-parameter family, not counting rotations, translations, and scalings. Five is a fairly large number. The space of possible lattices is also 5-dimensional, and in the best of all possible worlds, there would be precisely one surface in every lattice. Unfortunately, it is not even true that small surface deformations are in 1:1 correspondence to small lattice deformations, and things are dramatically more complicated.
Alan Schoen added to the Schwarz examples the Gyroid (left) a member of the associate family of the P surface, and Sven Lidin the Lidinoid (right), an associate member of the associate family of a particular H surface.
Key in their construction was an understanding of the Gauss map: This is a map of degree 2 to the sphere, branched over 8 points. Schwarz had already realized their significance; above is a reproduction of Plate VI from his 1871 monograph. Figures 43 and 44 show the branched values of the oP and tP deformations of the P surface, figure 45 of the CLP surfaces, figure 46 represents a special case of the rPD family, and figure 47 the H family.
Bill Meeks realized in 1970 that for any choice of 8 branched values of the sphere that is antipodally symmetric, there are two (conjugate) embedded triply periodic minimal surfaces with these branched values. This Meeks-family covers all then known examples, except for the H-surfaces, the Gyroid, and the Lidinoid.
Since then, progress has been made in small steps. Above are two new Non-Meeks surfaces (called oΔ and oH) due to Hao Chen and myself that can be deformed into different Meeks surfaces. It is not clear at this point whether they are related. Hao Chen has also recently proven that the Lidinoid and Gyroid belong to a 1-parameter family. I am sure this year will see more progress.
Here are two immediate questions that I would love to have an answer for:
For what triply periodic minimal surfaces are there embedded surfaces in the associate family that are not conjugates? For genus 3, the only examples so far are the Lidinoid and the Gyroid.
We know that the H-surfaces belong to a 5-dimensional family (work of Martin Traizet) that intersects the Meeks family (work of Hao Chen and Matthias Weber). Are the Gyroid/Lidinoid part of this family, or do they belong to a third, separate family?
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.
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):
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 . 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:
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.
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).
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.