Neovius surface

Neovius' minimal surface in a unit cell.

In differential geometry, the Neovius surface is a triply periodic minimal surface originally discovered by Finnish mathematician Edvard Rudolf Neovius (the uncle of Rolf Nevanlinna).[1][2]

The surface has genus 9, dividing space into two infinite non-equivalent labyrinths. Like many other triply periodic minimal surfaces it has been studied in relation to the microstructure of block copolymers, surfactant-water mixtures,[3] and crystallography of soft materials.[4]

It can be approximated with the level set surface[5]

${\displaystyle 3(\cos x+\cos y+\cos z)+4\cos x\cos y\cos z=0}$

In Schoen's categorisation it is called the C(P) surface, since it is the "complement" of the Schwarz P surface. It can be extended with further handles, converging towards the expanded regular octahedron (in Schoen's categorisation)[6][7]

References

1. E. R. Neovius, "Bestimmung zweier spezieller periodischer Minimalflächen", Akad. Abhandlungen, Helsingfors, 1883. http://resolver.sub.uni-goettingen.de/purl?PPN591417707
2. Eric A. Lord, and Alan L. Mackay, Periodic minimal surfaces of cubic symmetry, Current science, vol. 85, no. 3, 10 August 2003
3. S. T. Hyde, Interfacial architecture in surfactant-water mixtures: Beyond spheres, cylinders and planes. Pure and Applied Chemistry, vol. 64, no. 11, pp. 1617–1622, 1992
4. AL Mackay, Flexicrystallography: curved surfaces in chemical structures, Current Science, 69:2 25 July 1995
5. Meinhard Wohlgemuth, Nataliya Yufa, James Hoffman, and Edwin L. Thomas. Triply Periodic Bicontinuous Cubic Microdomain Morphologies by Symmetries. Macromolecules, 2001, 34 (17), pp 6083–6089
6. Alan H. Schoen, Triply Periodic Minimal Surfaces (TPMS), http://schoengeometry.com/e-tpms.html
7. Ken Brakke, C-P Family of Triply Periodic Minimal Surfaces, http://www.susqu.edu/brakke/evolver/examples/periodic/cpfamily.html