G4G10 - Flexagons
kite sample flex

This page contains supporting material for Scott Sherman's talk at Gathering for Gardner 10 in March 2012.

Flex Theory

The Flex Theory paper was my gift for the gathering. It includes a description of Flex Notation, useful for describing sequences of flexes, and Pat Notation, useful for describing the internal structure of a flexagon as it changes during flexes.

10 flex, 12 leaf kite hexaflexagon

The following flexagon was designed to show off 10 different flexes (in honor of the 10th Gathering) with only 12 leaves (because the year is 2012).

To make this 5-sided kite hexaflexagon:

  1. Click on the following two thumbnails to get the full sized versions. Note that the two separate shapes below are mirror images of each other.
  2. Print each one separately, cut them out and paste them back to back.
  3. Cut along all the solid black lines and pre-crease along the dotted gray lines. Note that there's a solid line in the block of 4 kites on the left.
  4. Start by folding adjacent 5's together.
  5. Next fold together the adjacent 4's, then the adjacent 3's.
  6. Repeat this process until only the 1's and 2's are visible, tucking the final 3's back-to-back.
  7. Finish by taping together the adjacent edges of the first and last kites in the template.
template for 12-leaf kite hexaflexagon, side a template for 12-leaf kite hexaflexagon, side b

Alternately, you could use the following black and white template. Rather than pasting anything to the back, simply copy the small numbers to the back and follow the remaining directions above.

template for 12-leaf kite hexaflexagon

Whether or not you can perform a flex depends on the internal structure of the flexagon. Thus, in order to perform each of the 10 flexes, you need to do them in a location that supports the flex. You can try out each of the flexes by following this recipe:

With all 1's on the front and all 2's on the back, you can do the pyramid shuffle if you start next to the small 3 in the middle of the flexagon.

pyramid shuffle

Next, turn the flexagon over and you can do the flip flex.

flip flex

Turn it over again and you can do the inverse tuck flex.

inverse tuck

When you're finished with the inverse tuck, turn it over and you can do the tuck flex. Note that I'm counting the tuck and inverse tuck as two different flexes because they change the internal of the structure in different ways, unlike most of the rest of the flexes listed here.


You should now be back to the starting position, with all 1's on one side and 2's on the other. You can now do the silver tetra flex if you put the edge with the tape on the left hand side.

silver tetra

Next you can v-flex. Note that the flexagon won't lie flat after this flex.


After turning it over, you can do the pinch flex twice to bring you back to the starting position once more.

pinch flex

You can do the slot tuck flex if you start near the small 4 in the middle of the flexagon.

slot tuck

Turn the flexagon over and you can do the slot tuck bottom flex. This is just like the slot tuck except that you open it up from the bottom rather than the top. The flexagon will not lie flat at this point.

slot tuck bottom

Finally, a pair of pocket flexes brings you full circle. Note that the small 2 in the middle of the flexagon is on the left-most leaf at the beginning of this flex sequence.

pocket flex

Main flexagon page

© Scott Sherman 2012 send comments to comments at this domain