Today, we are pushing things a little. We begin with a minimal heptagon with two horizontal straight edges, and the other five edges lying in symmetry planes parallel to the vertical coordinate planes. We also want this heptagon arranged so that extension by reflections/rotations creates an embedded triply periodic surface.
Above is a first example, showing a translational fundamental piece of a triply periodic surface of genus 6 that is tiled by eight such heptagons. We see that the horizontal straight segments are replicated into edges of squares, so that all surfaces we will look at today will sit one way or another between two squares.
A way to encode the possibilities for such heptagons is through the picture above. It represents the divisor of the square of the Gauss map on the quotient torus under the order 2 rotation about a vertical axis. Less technically, the grayed rectangle represents the heptagon, with the two vertical blue edges corresponding to the horizontal straight segments. The red dots are the seven heptagon vertices, and the symbols 0 and ∞ tell whether the Gauss map points up or down.
There are 12 possibilities that lead to different surface candidates. For each of these, a 3-dimensional period problem needs to be solved, leading in the case of success to 1-parameter families. In 8 of these cases, I was able to solve this period problem numerically. No existence or non-existence proof is known in any of these cases at this point.
There are several motivations behind this exploration. For one, I would like to know what limits can occur, and how the limit surfaces are combined. This suggests possibilities for general gluing constructions that would establish the existence of these and many other surfaces, Secondly, some of the limits are surfaces that are either only very difficult to obtain (like the Callahan-Hoffman-Meeks surface of higher genus that emerges above), or have also only been established numerically (like the exotic doubly periodic Karcher-Scherk surface of genus 3 below).
Finally, there is always the chance that an exhaustive enumeration leads to examples with unexpected properties, like the surface below, that in a diagonal view reveals an appealing tunnel system that indicates that we are probably having a special pair of skeletal graphs in front of us.
Resources
Main Page for all 8 surfaces
Minimal Surfaces 