Blame Timur for inspiring this question.
Big cubes can be generalized into 2 groups based on their geometry: the "even" group and the "odd" group. Of course, the "even" group tends to be a little more complex.
What about Pyraminxes? If you make them larger and larger, how do you generalize them?
My instinct was to divide them into three types: the "3n" group, the "3n+1" group, and the "3n+2" group. Why? Because the 3n group has no centers, the 3n+1 group has upward-pointing centers, and the 3n+2 group has downward-pointing centers. But because I do not own and have never solved any Pyraminxes except for the regular 3-layer one, I can't figure this out easily.
Has anyone explored this sort of thing?