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.

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