The origin of monkey saddles in gyroidal graphenoid

For the simplest gyroidal graphenoid, the monkey saddle is created by six octagons that connected to the central phenyl group. Due to the strain among them, three octagon are pushed upward and connected to the phenyl group lying above; while the other three octagons are pushed downward and connected to the phenyl group below. This is similar to a three-layer cyclophane with three tethers connected each two of them.

For the next gyroidal graphenoid, the central phenyl is replaced by coronene, other structural features remain the same. Imagine now that we have instead of one unit cell, but an infinite number of them arranged at each lattice site. The geometric feature of this creature may easily be seen.

By the way, using this construct, the normal of each layer is along (111) direction. However, this is only one diagonal, there are three major diagonal in this system. The gyrodial graphenoid looks exactly the same along these three diagonals.I.e. threre are two other sets of monkey saddles lying on the layers whose normals are the other two diagonals.