Prismatic Triply Periodic Minimal Surfaces

The pages here showcase some 40 triply periodic minimal surfaces from a special viewpoint: They all have right prisms as a fundamental piece under reflectional symmetries, and admit a rather simple Weierstrass representation. All surfaces in each family can be generated by a single Mathematica notebook, which you can download. This is still work in progress (jointly with Shoichi Fujimori), and more families will come.

Recent conversations with Alan Schoen allowed me to identify some of these surfaces with ones he had discovered in the 70s.

Right prisms over triangles

Type (1|) classical surfaces
Type (1|1) classical surfaces with a horizontal handle
Type (1|-1)
Type (1|-1,-1) (includes Neovius)
Type (1|1,-1)

Technical comments

hese triply period minimal surfaces have a simply connected fundamental piece in a right prism, with all faces lying in symmetry planes. There are two boundary arcs in the vertical planes, and all points with vertical normal are assumed to occur along these arcs. Then we say a surface is of type say (1|1,-1) if in one boundary component, there is one point with normal pointing up, while in the other boundary component, there is one point with normal pointing up and one with normal pointing down.

Note that in the overview pages for each series, the piece shown is twice the size of the fundamental piece, using two copies in the vertical direction so that the piece admits a vertical translation.