Triangle Flexagon Flex Notation

This notation can be used to describe a series of flexes on a flexagon. There's a concept of a current side and a current vertex. A flex is done relative to the current side and vertex and defines what should be considered the current side and vertex once the flex is done. The notation also includes a way to change just the current side or vertex. Typically the backs of the two faces adjacent to the current vertex are folded together when starting a flex, as you would do when performing a pinch flex.

The following video demonstrates several flexes being performed on a hexaflexagon, shows how the flexes relate to the current vertex and gives examples of describing a series of flexes with flex notation:

A shape preserving flex starts and ends with the same shape. Most of the flexes listed below are shape preserving flexes, with the exception of the pocket flex and slot flex.

Changing current

   ^ Change the current side. Flip the flexagon over while keeping the same current vertex.
   < Change the current vertex. Step one vertex counterclockwise (left if vertex is at the top).
   > Change the current vertex. Step one vertex clockwise (right if vertex is at the top).

Alternately, ^ can be represented as U, > as R and < as L.

Flexes

   P Pinch flex
   P(a,b,c) Pinch flex where the numbers describe how many vertices to step along when determining which vertices to pinch. For example, the standard pinch flex as performed on a hexaflexagon could be described as P(2,2). Some of the pinch flexes that work on a dodecaflexagon include P(2,2,2,2,2), P(4,4) and P(6).
   T Tuck flex
   S Pyramid shuffle
   V V flex
   F Flip flex
   St Silver tetra flex
   Lt Slot-tuck flex
   Ltb Slot-tuck-bottom flex
   Tk Ticket flex
   K# Pocket flex, flexed # times
   L# Slot flex, sliding the strip # times

The following illustrates many of the flexes. The hexaflexagon is used as an example, though all the flexes work on a wide variety of triangle flexagons. The first column lists the full name of the flex and the corresponding notation. The second column shows the initial position of the flexagon, with the left image showing the current side and the right image showing the back side. The * marks the current vertex. The third column illustrates how to initiate the flex. The final column shows what the flexagon may look like after the flex.

flex notation

Other notation

   (...) x N Repeat a series of flexes N times. For example, (P<) x 2 means take the sequence of doing a pinch flex followed by rotating to the left and repeat it twice.
   X' Perform the inverse of flex X, i.e. the exact opposite.

It's also important to note that the underlying structure of the flexagon determines where and when flexes can be performed, and this varies as you apply flexes. This means you can't always perform a sequence of flexes. Because of this, if you find two different flex sequences that change a flexagon in the same way, you may not always be able to do one sequence when you can do the other. To capture this concept, there are two different ways you can say flex sequences are equal: they're always equal regardless of the structure of the flexagon, or they're equal as long as the flexagon structure supports the sequences.

   A = B The flex sequences A and B always transform the flexagon in the same way.
   A ≈ B A and B transform the flexagon in the same way as long as the pat structure allows it.

Identities

Note that some of these are not strict identities, since a particular flex may not be possible given the current structure of the flexagon. However, if a flex is not possible, you can modify the flexagon by adding the necessary leaves and hinges to make it possible and to make the identity work.

Some of these identities assume a particular handedness for the flexagon. If a flexagon has a different handedness, simply swap > and < and the identity should hold.

n refers to the number of triangles per side, e.g. n=6 for a hexaflexagon.

Inverses

In most cases, performing a flex in reverse (its inverse) accomplishes the same thing as flipping the flexagon over and performing the original flex. The extra flip (and sometimes vertex shift) at the end of the following sequences is included so the current side and vertex are exactly the same.

   ^' = ^
   <' = >
   >' = <
   P' = ^P^
   T' ^T^
   S' = ^>S^>
   V' = ^V^
   F' = ^F^
   St' = ^St^
   Lt' = ^Lt^

General identities

In the following, A and B refer to either a single flex or a sequence of flexes. I is the identity flex, i.e. the operation that leaves the flexagon unchanged.

   I = AA'
   I = AB => A = B'
   (AB)' = B'A' i.e. you can undo two flexes by undoing the second flex then undoing the first.
   I = ABC => A = C'B'    C = A'B'

Flex equivalences

   I ≈ SP<T>P' S ≈ P<^T<P^ T ≈ <P^SP^>
   I ≈ V>>>TP’>>> V ≈ >>>PT’>>> P ≈ >>>V>>>T
   I ≈ F^S^>T<<T> F ≈ <^T<<T^S< S ≈ F>T<<T>
   I ≈ F^SSt^ St ≈ F^S^ F ≈ St>S< S ≈ FSt'
   St ≈ >T'<<T'>

Pocket flexes

   P = K>>>K>>> on a hexaflexagon
   P = (K>>>) x (n-2)/2 in general
   PP = (K2>>>) x (n-2)/2
   PPP = (K3>>>) x (n-2)/2

Others

   PP ≈ T>>T>>T>> on a hexaflexagon
   PP ≈ (T>>) x n/2 in general (or, loosely, T=PP)
   P^>P^>P ≈ S<<S<<S<< on a hexaflexagon
   P ^>P^>P ≈ (S<<) x n/2 in general (or, loosely, S=PPP)
   PP ≈ (>St'>>) x n/4 on right triangle flexagon (or, loosely, St=PP)

Interesting Sequences

   T<T^<T>T<T changes only 1 leaf on 1 side
   F>T^>>S changes a single pat, with the same number on both sides
   I ≈ (S>T'>^T^>>) x 2  S-cycle. Whenever a pyramid shuffle is possible on a hexaflexagon, this sequence can be performed to cycle back to the starting position
   ((T>) x (n-1)) ^<T^<T>T on an even-ordered flexagon, this transforms from sides a/b to sides that alternate cd/ef
on an odd-ordered flexagon (e.g. the heptaflexagon), this transforms from sides a/b to sides c/d
   Q>Q<Q'<Q>Q on an enneaflexagon, where Q = P(3,3) P(3,3), this transforms from sides a/b to sides c/d
on a dodecaflexagon, where Q = P(3,3,3) P(3,3,3), this transforms from sides a/b to sides that alternate cd/ef

Triangle Flexagon Flexes

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