Schoen S’-S”

This genus 4 surface is maybe the geometrically simplest among the new surfaces that appear in Alan Schoen’s 1970 Nasa report.

In comes in a 1-parameter family. Above we are approaching a nodal limit, consisting in a translational fundamental piece of two planes joined by 1 large catenoidal neck in one slab and four small catenoidal necks in the other slab. The small necks are placed at the vertices of the square tiling, the large neck at the centers of the same squares.

The other limit consists of singly periodic Scherk surfaces with eight ends. The name S’-S” derives from the structure of the skeletal graphs of the complement of the surface: Both graphs have horizontal square grid graphs, but at different sizes.