The Associate family of the Schwarz P surface contains besides the D surface another embedded minimal surface, Alan Schoen’s Gyroid from 1970. The image below shows that eight hexagons that tile the P-surface as part of the Gyroid, and next to it four copies that show that the pieces fit together to a closed surface, both indicating the order 4 rotational symmetry.

The angle of association is given by elliptic integrals: $\phi =\arctan \frac {\int_1^3 \frac{dz}{\sqrt{-z^4+10 z^2-9}}}{\int_{-1}^1 \frac{dz}{\sqrt{z^4-10 z^2+9}}} \approx 38.0148^\circ$

This angle can be explained by looking at the images of a fundamental (regular) hexagon under the integrals of the holomorphic coordinate differentials. These are Euclidean rectangles conformally equivalent to the regular hexagon, with edges parallel to the coordinate axis for P and D. The Gyroid angle rotates this rectangle so that one diagonal becomes vertical.

A main reason the Gyroid is so hard to visualize is the lack of straight lines or reflectional symmetry planes. Slightly easier to comprehend are pieces of the Gyroid that lie in the associate family of the rP/rD surface. These show the surface with an order 3 rotational axis. Below is an animation of the entire associate family of rP/rD/rG.

Alan Schoen did not at first discover the Gyroid by experimenting with associate families. Instead, he was looking for graphs as candidates for skeletal graphs in the complement of minimal surfaces. He came across the Laves graph, named after Fritz Henning Emil Paul Berndt Laves:

This graph has girth 10 and follows the diagonals of a cubical lattice. With patience and perseverence, Alan Schoen first found polyhedral and then smooth approximations of a surfaces separating two such Laves graphs in a balanced way. To establish this surface as a minimal surface, Schoen then found that it has the same Gauss map as the P surface, a strong hint that it lies in its associate family.

Embeddedness of the surface was established in 1996 by Karsten Große-Brauckmann and Meinhard Wohlgemuth.

##### Resources

Mathematics Notebooks for rG, tG, and rPD

PoVRay Sources for rG and tG

Coxeter, H.S.M.: On Laves’ graph of girth ten, Canadian Journal of Mathematics 7, (1955), 18–23

Große-Brauckmann, K. & Meinhard, W.: The gyroid is embedded and has constant mean curvature companions, Calc. Var 4 (1996), 499-523