Neovius surface

 ( Differential Geometry ) 

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).E. R. Neovius, "Bestimmung zweier spezieller periodischer Minimalflächen", Akad. Abhandlungen, Helsingfors, 1883. http://resolver.sub.uni-goettingen.de/purl?PPN591417707Eric A. Lord, and Alan L. Mackay, Periodic minimal surfaces of cubic symmetry, Current science, vol. 85, no. 3, 10 August 2003

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,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 and crystallography of soft materials.AL Mackay, Flexicrystallography: curved surfaces in chemical structures, Current Science, 69:2 25 July 1995

It can be approximated with the level set surfaceMeinhard Wohlgemuth, Nataliya Yufa, James Hoffman, and Edwin L. Thomas. Triply Periodic Bicontinuous Cubic Microdomain Morphologies by Symmetries. Macromolecules, 2001, 34 (17), pp 6083–6089

$3(\backslash cos\; x\; +\; \backslash cos\; y\; +\; \backslash cos\; z)\; +\; 4\; \backslash cos\; x\; \backslash cos\; y\; \backslash cos\; z\; =\; 0$

In Alan Schoen 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)Alan H. Schoen, Triply Periodic Minimal Surfaces (TPMS), http://schoengeometry.com/e-tpms.htmlKen Brakke, C-P Family of Triply Periodic Minimal Surfaces, http://www.susqu.edu/brakke/evolver/examples/periodic/cpfamily.html