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.

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