An 11-leaf hexaflexagon

To make this flexagon, start with the following strip. Click on the thumbnail to get a larger version, then print it, cut it out and copy the numbers to the back.

strip for 11-leaf hexaflexagon

Pre-crease all the edges. Fold 3 on 2, 6 on 5 and 9 on 8. Place 10 under 1 and fold 11 under 10. Finish by taping the outside edges of sides 1 and 11 together.

Though very simple, you can perform the pinch flex, v-flex, tuck flex and pyramid shuffle on this hexaflexagon to travel between 10 different states. The following diagrams show these states and describe how you can flex between them. The diagram on the left shows which numbers are visible for each state as well as listing how to flex between them. The pair of numbers in parentheses gives a hint as to how to perform the associated flex. You typically find the pair of numbers on the flexagon and start by folding together the backs of the associated pats, leaving the numbers visible. For the v-flex, this indicates one of the pair of vertices you fold together in back to start the flex. The diagram on the right is a simplified version with the states represented with letters.

state diagram for 11-leaf hexaflexagon

There are two 3-state cycles you can use the pinch flex to traverse, ABC and DEF. You can use a v-flex, tuck flex or pyramid shuffle to go from one cycle to the other.

On this flexagon, the tuck flex is always a one-way flex, represented by the arrows. If you pay close attention to how you did the tuck flex, you can reverse the process to go the other direction, though this is a tricky move. When you first tuck in the leaves, you may be able to open the flexagon back up either from the top or bottom. Sometimes you can open it up from either end, for example when travelling from state A to either E or F. Sometimes you can only open it up from one end, as when travelling from state G to H.

The following diagrams show which states are accessible using only a subset of the flexes. With just the pinch flex and v-flex, only 8 states are accessible. The middle diagram shows that using the tuck flex and pyramid shuffle (and the reverse tuck flex), you can get to all 10 states. The final diagram untangles the network to show a cleaner view of the middle diagram.

state diagram for 11-leaf hexaflexagon

You can see that there are two separate collections of states accessible just using the tuck flex, BCDGH and AEFIJ. You can travel between these "tuck classes" using the pinch flex, v-flex or pyramid shuffle.

© Scott Sherman 2009 send comments to comments at this domain