Previously, I suggested that there are essentially two kinds of rules in weaving a beaded model: local and global. Local rules consist of the description of tiles such as 5-, 6-, 7-, and 8-gons and the how to connect these polygons locally. For gyroidal graphenoid, the local rules are quite clear, but the global rules are harder. I have thought about what I meant by the global rules these two days. It seems to me that the global rule is something like the weaving codes I developed for small fullerenes. Somehow this is similar to the Hamiltonian chain that specify how the weaving procedure goes. Of course, it is easy for simple object like the simply-connected fullerenes. However, it becomes not so clear for gyroidal graphenoid. What is the simplest weaving path (the Hamiltonian chain) for this system. I don't think Hamiltonian chains exist in this system. (Maybe I am wrong) But in order to weave this system effectively, maybe we should have a better global rule.
Another way to handle this is to follow Chuang's recipe and weave many unit cells and patch them together. It may also be the best way to build gyroidal graphenoid since I have played with my local and global rules for several hours without success. Then we can simply forget what I say here.