In his posthumous paper (published 1867), Bernhard Riemann dedicates a few pages to the derivation of the explicit Weierstraß data of a minimal surface that has as boundary four consecutive edges of a regular tetrahedron.single

Independently, Hermann Amandus Schwarz publishes in 1865 a brief note discussing the same surface, and subsequently two long papers analyzing this and related surfaces in great detail. Alan Schoen named this surface the diamond surface, because its symmetries are related to the diamond crystal.

Both Riemann and Schwarz note that the surface can be extended by rotation about the edges to a periodic surface. Different extensions have dramatically different appearances.

The period lattice is the face-centered cubic lattice, and its quotient is a compact Riemann surface of genus 3, the hyperelliptic cover over the vertices of the cube, as both Riemann and Schwarz remark



B. Riemann: Über die Fläche vom kleinsten Inhalt bei gegebener Begrenzung, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13 (1867), 3-52

H.A. Schwarz: Ueber die Minimalfläche, deren Begrenzung als ein von vier Kanten eines regulären Tetraeders gebildetes räumliches Vierseit gegeben ist, Monatsberichte der Königlichen Akademie der Wissenschaften zu Berlin, 1865, 149-153.

Mathematica notebooks: tetrahedral, cubical

PoVRay sources: tetrahedral, cubical

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