Edvard Rudolf Neovius was a student of Hermann Amandus Schwarz. His surface, published in 1883, is by far the most complicated minimal surface discovered in the 19th century.

It has the same symmetries and straight lines as Schwarz P-surface, but here, complicated Neovius-handles (i.e. plumbing X-pieces) extend to the edges of a cube, making it a genus 9 surface. Nevertheless, the surface becomes surprisingly simple as soon as one realizes that it is tiled by 96 congruent minimal triangles. Twelve of them form eight minimal 12-gons, parametrized in polar coordinates above.

The conjugate surface is not embedded, but the conjugate 12-gons has a simple polygonal contour, following the edges of a 2x2x2 cube. The Enneper-Weierstraß representation is also quite simple: The flat structures of the holomorphic coordinate 1-forms corresponding to a 12-gon are just slit rectangles. The two branched point at the ends of the two slits correspond to the critical points of the coordinate function.

Not much is known about the deformation space of the surface. A simple 1-parameter deformation in a box lattice over a square leads to limits with a pattern of Callahan-Hoffman-Meeks surfaces on one end, and to a nodal limits on the other end.

Below is an animation of this family.

Resources

E.R. Neovius: Bestimmung zweier spezieller periodischer Minimalflächen, Akad. Abhandlungen, Helsingfors, 1883

Mathematica Notebook and CDF

PoVRay Source

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