These translation invariant minimal surfaces are limits of triply periodic minimal surfaces discovered by Werner Fischer and Elke Koch.
They can be obtained by taking limits of Plateau solutions to closed 8-gons in space as indicated above. The two vertical segments on the left become the central axes and the surface can be extended by rotations about it and about the horizontal straight lines:
In order to obtain an embedded surface, the angle between the horizontal lines needs to be of the form π/k, and the gap between the vertical segments needs to be k times as large as the length of a vertical segment of the original contour, where k is an odd integer. For even k, the surface obtained from the Plateau solutions are not embedded, as the normals at opposite ends are equal by constructions, which is impossible in this case.
Below are the surfaces for k=5 and k=7.
- Screw motion invariant Fischer-Koch surfaces
- Screw motion invariant Fischer-Koch-Freese surfaces
- Translation invariant helicoid with handles.