In principle, we need to make a program so you can put in the machine settings and it spits out the triangle (with the physical roots). As you know, the point is that these roots have physical meaning and in fact, these are what they determine when using an NV as a sensor. The NV guys told me that they like this triangle view point a lot. It's just a cubic, but connecting the geometric picture to physics is pretty fun!
When I'm further along into the article, I'll as you James to please take a look at the band structure arguments. Clearly I'm sort of out of my depth with most of this stuff. This is actually a good thing in some ways, since it means I have to really think about how to explain it, even to myself :D