A concise node program to create the 12 patch gyroid boundary patch is complete. Use of some Macros makes the wiring look much better.
The Inversion center permits half of the 12 patches to be simply inversions of the other 6 patches. A simple inversion macro is included in one of the macros, which can be added to the symmetry operations repertoire.
The initial patch P0, created in Python is attached. Either the netP0, or the surfaceP0 may be used, or both, as input to the node program. The surface P0 could be extruded a bit, if desired.
When a rotation axis is pointed at the observer, for a Clockwise rotation of 90 degrees, use +90 for the Euler angle in the rotate node. This is contrary to the right hand rule and contrary to the convention that CCW rotation is positive.
The symmetry operations of Table 1 are complete. It took a long time to figure out how the symmetry operations work. The symmetry operations apply to the MoI World axes. To create say P3 from P1, notice that xprime equation of P1 needs to be negated with an initial mirror with yz plane. Then rotate about MoI x axis clockwise, by +90 degrees. This shifts (rotates) zPrime equations of P1 to yPrime equations of P3. Then rotate about MoI y axis by +90 degrees clockwise. This shifts (rotates) the old yPrime equations of P1 to the xPrime equations of P3, and the old xPrime equations of P1 to the zPrime equations of P3.
The last step is a simple move (translation) of n*eta = n*(1/4). In summary, it seems to be a lot easier to "rotate" the equations of table 1 from P1 to P3 and P11, and invert to P5, P7, and P9, and similarly from P2 to P4 and P12, and invert to P6, P8, and P10, rather than try to decompose matrices.
The new SetName node belongs in the nodeeditor extensions folder.

- Brian