Inside or Outside?

The last minimal surface that made it into Alan Schoen’s NASA report is the F-RD surface. It has genus 6 and looks fairly simple.

Tetra

A fundamental decision one has to make these days is to choose the side one wants to live on. If, for instance, we decide on the orange side, we will have the impression to live in a network of tetrahedrally or cubically shaped rooms with connecting tunnels at the vertices of each. Not too bad, but, as things stand, we will never know what life on the other side looks like.

Cubical

Luckily, our imagination is still free, and we can think about the other, green side. What we can hopefully see from the pictures above and below is that the rooms of the green world are all cubical, with tunnels towards the edges of each cube. Alternatively, we can also think of the rooms as rhombic dodecahedra, with tunnels towards the faces. That’s where F-RD got its name from: Faces – Rhombic Dodecahedron.

Double

Incidentally, the conjugate of the F-RD surface is again one of those discussed by Berthold Steßmann, with the polygonal contours having been classified by Arthur Moritz Schoenfließ

A simple deformation of F-RD maintains the reflectional symmetries of a box over a square, but allows to change the height of the box. It turns out that there are two ways to squeeze the box together.

Limit1

In both cases we get horizontal planes joined by catenoidal necks, but differently placed in each case.

Limit2

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