I've read the threads here and here but I am having a problem. I am trying to duplicate the cut pattern on the surface of the Dioctipoid. If I use planar cuts, what is the base shape for the position of the planes? I've tried using the FTO, but this does not seem to work.

I do not understand exactly what your problem is but I will try an answer anyway:
The planes have to be oriented like a faceturning octahedron. This still leaves the question for the cutting depth open. This illustrates that plane cuts do work:
http://www.jaapsch.net/puzzles/sphere.htm?red=100&sym=3

Thanks Andreas, I have prepared these for a secret project