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

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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.

Piece

 

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.