If you generalize this to consider polygons mirrored across points instead of edges, you can create point flexagons. It can also be generalized to include mirroring across edges or points within the same flexagon to give you skeletal flexagons. Or you can generalize this definition to include n-dimensional polytopes mirrored across n-1, n-2, etc. dimensional edges. These are called flexatopes.
Note that there are a couple additional things to note about the generalization to point flexagons. One is that folding a point flexagon into a single pat (where the same number is on the top and bottom) may result in a pat that's not valid. All that's required to make it valid, however, is to unfold it along one hinge so you have two adjacent pats, with a pair of equal numbers on each side. Additionally, you may end up with nested hinges. In the equivalent edge flexagon, only one of these hinges will open.
The most well known flex is the pinch flex, but there are many others. See flexes for more examples.
For example, the pinch class for a flexagon is all the states reachable from an initial state using just the pinch flex. Most flexagon theory is only concerned with pinch classes. But you could also explore other possibilities, such as the tuck-pyramid-shuffle class that uses both the tuck flex and pyramid shuffle.
The Tuckerman traverse is a recipe for always finding a pinch traverse for an even ordered triangle flexagon. You simply keep performing a pinch flex at the same vertex until you can't anymore. You then step to the next vertex and repeat. |
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