NEOVIUS SURFACE

 Surface studied by Neovius in 1883. Edvard Rudolf Neovius (1851-1917): Finish mathematician. Websites: Wikipedia www.indiana.edu/~minimal/archive/Triply/genus9/Neovius/web/index.html www.susqu.edu/brakke/evolver/examples/periodic/periodic.html#neovius

 The Neovius surface is a triply periodic minimal surface the fundamental patch of which, reproduced opposite, has 12 openings centered on the edges of the cube, hence at the vertices of a cuboctahedron (compare to the fundamental patch  of the Schwarz P surface, which has 6 openings centered on the faces of a cube). The figure opposite was made thanks to the equation  which gives a surface, said to be "nodal", that is non minimal, close to the true Neovius surface. The fundamental patch is composed of eight skew dodecagons of the type opposite: The complete Neovius surface splits the space into to connected components, isometric to one another, like the Schwarz P surface (a translation with vector  leaves the surface globally invariant but swaps the faces).   Above, at the center, imagine an cell identical to its neighbors, embedded in the hollow.
 View of an elementary cell (intersected by a sphere instead of a cube) with its 12 openings. View of its connections of 6 of its 12 neighbors. View of the cubes containing each cell. Production: Alain Esculier.

 Another minimal surface, discovered by Alan Schoen, has a fundamental patch with 8 opening located at the vertices of the cube (abbreviation I-WP) Its approximate nodal equation is . The complete surface is composed of cubes connected by their vertices.  The interior zone is no longer isometric to the exterior zone (the translation with vector  leaves the surface globally invariant, without swapping the faces).

Also compare to the gyroid.

 Patch of the Neovius surface, by Patrice Jeener, with his kind authorization. Anaglyph 3D image of the Neovius surface, by Alain Esculier, to watch with red (on the left) and blue (on the right) glasses.