Pinch Flex Variations

Relatives of the pinch flex on different flexagons

As shown in the Flexagon Intro, when you do the standard pinch flex, you alternate mountain and valley folds along every corner.

What’s interesting is that there’s a whole class of flexes that are all similar to the pinch flex, except that you vary the pattern of where you do mountain and valley folds. So instead of having a mountain fold every other corner, you might do a mountain fold every three or four corners.

The biggest difference is that after a single such flex, the flexagon generally doesn’t lie flat and the triangles won’t all meet in the center. But if you continue doing a second flex in the same place, you’ll get back to a flat flexagon where all the triangles meet in the center. But that’s just a bunch of words. Let’s see what this actually looks like.

P444

This flex is represented as P444 in flex notation. Each 4 indicates that from one mountain fold to the next, you skip to the fourth vertex from the previous.

The first example is on an isosceles dodecaflexagon, doing a mountain fold once every four corners.

Here’s a template you can use to create a working dodecaflexagon that supports the P444:

P334

This flex is represented as P334 in flex notation. The 3′s indicate that from one mountain fold to the next, you skip to the third vertex from the previous. The 4 indicates that one of those steps is to the fourth vertex.

The next example is on an isosceles decaflexagon. Here you do a mountain fold, skip to the third vertex, do another mountain fold, then skip to the fourth vertex for the final mountain fold.

Here’s a template you can use to create a working decaflexagon that supports the P334:

Minimal flexagons

octaflexagon:
enneaflexagon:
decaflexagon:
dodecaflexagon:

Here are the minimal flexagons for many of the pinch flex variations:

Template:

More complex flexagons

An interesting generating sequence pattern you can use with these pinch flex variations is X* > X* < X'* < X* > X*. If you start with one pair of faces visible, then apply that sequence, it will reveal two new faces. But all the intermediate states will be a mixing of the four faces.

Pick flexagon and base flex:

Generating sequence: P333* > P333* < P333'* < P333* > P333*

Cut out each of the shapes, cutting along the solid lines. Tape ‘b’ to ‘b’ and ‘c’ to ‘c’ (and ‘d’ to ‘d’ if present). Then you can follow the same folding rules as for the other templates, finishing by taping ‘a’ to ‘a’.

What next?

What if you tried other generating sequences using these pinch flex variations?

The pinch flex variations described here all do a flex followed by a second flex at the same vertex because the intermediate position doesn’t lie flat and the triangles don’t all meet in the center. But what if you stopped in that intermediate state and tried flexing from there? What other interesting flexes can you find?

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