In high school I usually loathed art class. But once we started an architecture project that got me excited: We were tasked to design our own house, with all bells and whistles. I decided on the rooms being regular hexagons, arranged in an annulus of six, with a center hexagon without windows.
This is the motivation behind today’s surface. Six hexagons, arranged as above, with the additional stipulation that when you exit a hexagon you re-enter another hexagon as indicated by the arrows (and extended by rotational symmetry). You can easily check that when you travers the rooms by always leaving through opposite walls, you will pass alternatingly through two rooms and return. Moreover, there are a total of six rooms around each corner, which suggest that Euclidean geometry is not suitable for this architecture.
In the hyperbolic plane, one can arrange six 60 degree hexagons around a vertex as above. The geodesics indicate which edges are to be identified, again implying silently that everything is rotationally symmetric.
Now in the above Euclidean model the identifications are done by Euclidean translations, defining what is called a translation structure. One can accomplish the same with other Euclidean polygons (that are still conformal images of a regular hexagon) like so:
or so:
In the above image we have a very short inner edge connecting the 240 degree vertices. There are a few more one can use, but they are not quite as pretty. In any case, they provide us with plenty of holomorphic 1-forms on the surface of genus 4 given by the algebraic equation w⁶=z⁶-1: This is, after all, a sixfold cover over the sphere, branched over the sixth roots of unity. The first model realized this geometrically by replacing the sphere by the double of a Euclidean hexagon.
The Inner Frame 