Cycles and Traverses

A cycle is flex sequence that brings you back to the initial state. A traverse is a cycle that includes every state in a given flex class, where a flex class is every state accessible from an initial state using a given flex or set of flexes. This page uses flex notation to describe series of flexes.


Whenever you can perform a single pinch flex on an even ordered triangle flexagon, you can perform a series of three pinch flexes to bring you back to the original state. The flex sequence is P>P>P>.

On quadrilateral (or higher order) flexagons, being able to do a single pinch flex does not guarantee a cycle. If it's possible to perform the entire sequence, (P>) x n (where n is the number of sides on a leaf), will cycle back to the initial state.

Tuckerman Traverse

Named for Bryant Tuckerman, the Tuckerman Traverse is a technique for traversing all the states of a hexaflexagon in a given pinch class. The recipe is very simple: keep performing a pinch flex at the same corner until you can't continue, then shift to the next corner and repeat.


The S-cycle is the flex sequence (S>T'>^T^>>) x 2. Whenever you can peform a pyramid shuffle on a hexaflexagon, you can do the S-cycle to return to your initial state. This is easy to demonstrate by trying it on the simplest hexaflexagon that permits the pyramid shuffle (see minimal pyramid shuffle hexaflexagon).

The following video shows the S-cycle, starting at around 0:28. (The flexagon it's showing is this 12 leaf hexaflexagon.)

Traverses on a 12 leaf hexaflexagon

This 12 leaf hexaflexagon has two different simple traverses. One involves the slot-tuck and pinch flex: (LtP'<P'<) x 12. The second involves the slot-tuck and tuck flex: (LtT'^<T^) x 12.

Main flexagon page

© Scott Sherman 2011 send comments to comments at this domain