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Composite pinch flexes

In this article, Yossi Elran tells us about some interesting composite pinch flexes.
Template for P333 flexagon
© Davidson Institute of Science Education, Weizmann Institute of Science, Images credit: Scott Sherman for research and educational use only

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Here is a cute little enneaflexagon:
A minimal P333 flex enneflexagon
This is the minimal flexagon needed to do a composite pinch flex known as the P333 flex. You can see the two sides of the main state of the flexagon, one side blue, the other green, and its pat structure. The flexagon has nine leaves showing in its main state and a total of 15 leaves altogether. When folded, there are only 3 nested pats with structure [[- -] -] – meaning that there are 3 leaves in the pat, the bottom leaf hinged to a pat comprised of two hinged leaves. There are two single leaf pats between every two 3-leaf pats in the main state of this enneaflexagon. Three corners are marked P333. It is from these corners that the composite P333 flex can be performed. Here’s how its done:
Left to right: P333 flexing an enneaflexagon
  • Download the p333.pdf template linked in the ‘related files’ section of this step (template number 15 in the flexagon booklet). Cut out the two templates for the back and the front and glue them back to front.
  • Fold all leaves numbered 3 face to face and then fold all leaves numbered 4 face to face until you get an enneagon and tape the two leaves numbered 2 together.
Now for flexing:
  • Start with the blue side facing up. Pinch every third corner, marked by P333 on the diagram, holding the hinge between the 1-leaf and 3-leaf pats. The meaning of P333 is to mountain fold every third hinge. There are three such hinges on the enneagram, so that’s the meaning of the three threes in P333.
  • Open up from the center. You will see a structure of six yellow triangles on the circumference, three green triangles on the inside, and an empty triangular hole in the middle. The structure is an intermediate, unstable state, that doesn’t fold flat.
  • Continue flexing from the same corners in the same direction once more to get the new state of the flexagon, in which six of the leaves have changed color from blue to red. Notice that some of the states are not flat – so “continue pinch flexing” also involves inverting the convex and concave shapes that you get after the flex.

Composite pinch flex notation

Surprisingly, although the movement made with the P333 flex is seamless and quite simple, it is a complex flex and there is another way you can do it:
P333 = T1 (>)3 T1 (>)3 T3 (>)3.
This means, do the following three times: make a tuck flex and rotate the flexagon 3 pats clockwise. The number that comes after the tuck flex symbol T is just the relative position of the vertex which opens in the first move of the flex (the extra flap), clockwise from the vertex that you hold.

Other composite pinch flexes

The P333 is really just a variation of the pinch flex. The main difference being that the definition of the pinch flex is that every other corner is pinched, all around the flexagon. For flexagons with more than 6 pats in the main state, pinching can be done at other locations. For the P333 flex, every third corner is pinched, and for the P444 flex, every fourth corner. Obviously, the minimum number of pats on the main step of a flexagon in order to do these flexes is 9 and 12 respectively. Similarly, the minimum requirement on a flexagon for a P44 flex to be done is 8 pats, for a P55 flex is 10 pats, and for a P334 is ten pats. In the latter case the pinches are made on the third, sixth and tenth corners.
You can find more pinch variations on Scott Sherman’s excellent site in the related links part of this step.

Discussion

Any questions? If so, please ask below and either myself or one of the other participants will answer.
© Davidson Institute of Science Education, Weizmann Institute of Science, Images credit: Scott Sherman for research and educational use only
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