H. B. Meyer gives a second basic model type; I started with this one, but it turns out that it's very sensitive to the edges being the same length, which isn't a property that the dual to a randomly chosen triangulation can easily be arranged to have. (I could make Goldberg solids this way, I guess.)

I've marked two strips, so you can see how it works; if I draw lines along the folds, the structure is obscured.