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.

Arthur Moritz Schoenflies had classified these quadrilaterals, there are precisely six types, 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 in detail. Its contour 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.

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.

Alan Schoen observed that this surface is the conjugate of his I-WP surface, predating it by 40 years.

##### Resources

B. Steßmann: Periodische Minimalflächen, Mathematische Zeitschrift 38 (1934), 417-442

Mathematica Notebook

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