Flexagons strike me as objects that would admit investigation in a first course in modern algebra. I'm surprised to be unable to find a reference discussing flexagons using modern algebra language. Could someone please provide a reference?
Perhaps the reason is that the "motions" between "states" of a flexagon might not form a common algebraic object like a group, although these motions and states admit a Cayley graph-like state diagram, an example of which can be seen here.
The first part of my question is an odd pedagogical one, in that the flexagon is what draws students in, and would be the gateway to teaching topics in introductory modern algebra:
Question A: Do flexagons admit study by algebraic structures with well-developed theories, e.g. groups, or do they simply appear as an isolated oddity?
This suggests a second part of the question, which I think is really only a rephrasing of A:
Question B: Does the "algebraic structure" found in the study of flexagons appear in any serious, or at least more mainstream, mathematics?
I should mention this post, although it also doesn't employ group theory in any satisfying way.
Flexagons by C. O. Oakley and R. J. Wisner. The American Mathematical Monthly Vol. 64, No. 3 (Mar., 1957) (pp. 143-154) looks like the first hardest look at them. Somewhere it says they are analyzed into sets of "recursively defined permutations." If you look at that, and then trace forward other things that reference it, I think you'll get somewhere.
I saw a good bit of combinatorics in these articles so far, and there is a definition of an "abstract flexagon" on page 146. The article is labeled as free access at JSTOR.
As for question B, I see 11 hits in mathscinet, lots of which are in the Monthly. They seem like decent articles, but I didn't notice any bigger journals. Two of the hits are from 2012.