Flexagon Definitions

A single polygon
A collection of polygons where each polygon is connected along exactly two edges to mirror images of itself

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.

A stack of polygons in an f-linkage folded together such that the mirrored sides all align
valid pat:
A pat that's been created using leaf splitting, resulting in only a single way to unfold a pat into two sub-pats along any given hinge, where both sub-pats are themselves valid pats

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.

A pair of adjacent pats
A configuration of an f-linkage consisting of valid pats
An f-linkage with a given f-folding
A series of modifications to a flexagon that take it from one f-folding to another, consisting of combining, splitting and sliding pats

The most well known flex is the pinch flex, but there are many others. See flexes for more examples.

<flex> class:
All the states reachable from an initial state using a given flex or set of flexes

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.

<flex> traverse:
A sequence of flexes that touches every state in the corresponding <flex> class

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.

Main flexagon page

© Scott Sherman 2011 send comments to comments at this domain