A chapter from
Explorable Flexagons

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Generating Sequences

Flexing Flexagons

A short tour of different ways to flex a flexagon

By: Scott Sherman

The previous chapter showed how to use the pinch flex to explore a flexagon. But there are lots of other ways to flex a flexagon, all of which rearrange the leaves in different ways. On this page, we’ll cover flexes like the tuck flex, pyramid shuffle, and v-flex.

The best way to learn a new flex is on the minimal flexagon that supports the flex, which is the flexagon that has exactly the minimum structure necessary to perform a given flex. On this page you’ll find a variety of flexes, and the option of trying out the minimal flexagon with a variety of flexagons.

For flexagon names, prefixes are used to reflect the number of polygons per face, e.g. hexaflexagon indicates there are 6 triangles per face. Additionally, these names are often proceeded by a word that describes the shape of the triangle, e.g. right for a right triangle.

The early naming for flexagons reflected this approach, e.g. hexaflexagon for 6 triangles per face and tetraflexagon for 4 squares per face. But you’ll also occasionally see the prefix used to describe the polygon used in the flexagon, e.g. pentaflexagon to indicate that the leaves are pentagons. My preference is to use pentaflexagon to mean 5 leaves per face and pentagon flexagon to mean pentagonal leaves.
  • 4: tetraflexagon
  • 5: pentaflexagon
  • 6: hexaflexagon
  • 7: heptaflexagon
  • 8: octaflexagon
  • 9: enneaflexagon
  • 10: decaflexagon
  • 12: dodecaflexagon
  • equilateral: all sides equal
  • silver: 45-45-90 triangle
  • bronze: 30-60-90 triangle
  • right: one angle is 90 degrees
  • isosceles: two sides are equal (the default)



You may also see an additional prefix used to indicate the number of faces in the flexagon. For example, a pentahexaflexagon is a 5-faced hexaflexagon. But, as you’ll see below, the concept of a “face” of a flexagon only makes sense when using the pinch flex, not when using any of the myriad other possible flexes.

Pyramid shuffle

Step through the images to see the pyramid shuffle on an equilateral hexaflexagon. The green circle on the first and last slides indicates where the current vertex starts and ends for the flex.

The flex notation for the pyramid shuffle is S. Performing the flex in reverse is S'. You can also undo the pyramid shuffle by turning it over, doing the pyramid shuffle one vertex to the right (clockwise), then turning it back over again, summarized as S' = ^>Sh<^.

The pyramid shuffle requires at least 5 leaves per face (i.e. a pentaflexagon or greater). It requires mild bending of the leaves on the pentaflexagon, but no bending for most other flexagons.

Here are some flexagons you can use to try the pyramid shuffle:,,,,,,,,,,,, or.

Every leaf (triangle) is labled with a unique id. The backside of a leaf has the same id as the front, only negative (e.g. 1 and -1). The flexagon to the left shows you where the labels should be after folding, both on the front (the large flexagon) and the back (the small flexagon). If you click the S button, you can see what the labels will look like after a successful pyramid shuffle.

These instructions apply to all the templates on this page.

Cut: Start by printing out the template and cutting along the outside edges.

Prefold: After you’ve cut out the shape, fold and unfold along every dashed line to prepare it. Note that the first and last triangle in the template have dashed lines to indicate where to tape the edges after everything is folded.

Number: The large number on each triangle represents the number on the front of the triangle, while the smaller number next to it represents the number on the back. Copy the smaller number in the center of each triangle onto its backside. Note that the small numbers in the corners represent the leaf’s unique id, which corresponds to what’s showing in the flexagon simulator above.

Fold: Find the largest pair of adjacent numbers and fold those numbers together. Then find the next largest pair of adjacent numbers and fold them together. Continue in this way until the only numbers still visible are the 1′s and the 2′s.

Tape: You should now have a polygon consisting of the appropriate number of triangles, e.g. 6 for a hexaflexagon, 7 for a heptaflexagon, etc. Tape the edges of the first and last triangles together to complete the flexagon.

Flex: The Sh’s on the first and last leaves correspond to the * in the flexagon simulator above, which represents the current vertex, This is where the flex should be performed from.

Flip flex

Step through the images to see a flip flex on a silver octaflexagon:

The flex notation for the flip flex is F. As with the pyramid shuffle, F' = ^F^.

The flip flex requires at least 6 leaves per face (i.e. a hexaflexagon). It works well on right triangle flexagons. Isosceles triangle flexagons require at least 7 faces, however.

Here are some flexagons you can use to try the flip flex:,,,,,,, or.

Tuck flex

Step through the images to see a tuck flex on an equilateral hexaflexagon:

The flex notation for the tuck flex is T. Or, to specify which additional flap needs to open, you can use T1, T2, etc.

The tuck flex requires at least 6 leaves per face (i.e. a hexaflexagon). It requires mild bending of the leaves on an equilateral hexaflexagon, but no bending for most other flexagons.

The tuck flex is different than the other flexes on this page in that it requires an additional opening in the flexagon in order to allow enough freedom to perform the flex. On the hexaflexagon, this can only happen in one place. On the heptaflexagon and above, however, there are multiple ways you can add this additional opening. That’s why there are multiple variants for some of them, e.g. the two heptaflexagons.

This is also why, unlike many other flexes, T' isn’t always the same as ^T^. This is summarized as T' ~= ^T^, where ~= means the sequences are the same only if the structure of the flexagon allows it.

There are two bronze hexaflexagon variants because there are two distinct places you could potentially perform a tuck flex: at a point or at the vertex in the middle of an edge.

Here are some flexagons you can use to try the tuck flex:,,,,,,,,,,,, or.

V-Flex

Step through the images to see the v-flex on an equilateral hexaflexagon:

The flex notation for the v-flex is V. V' = ^V^

The v-flex requires at least 6 leaves per face (i.e. a hexaflexagon).

Silver tetra

Step through the images to see a silver tetra flex on a right decaflexagon:

The flex notation for the silver tetra flex is St. St' = ^St^.

The silver tetra flex requires at least 4 leaves per face (i.e. a tetraflexagon). It works best on right flexagons, though it works on others with enough triangles per face.

It gets its name from the fact that the simplest flexagon it works on is the silver tetraflexagon.

Here are some flexagons you can use to try the silver tetra flex:,,,,,,,, or.

What next?

That was a quick tour of some of the many flexes that are possible on a variety of different flexagons. A lot of them work on standard hexaflexagons, rearranging the leaves in far more ways that just the pinch flex alone, e.g. on the 5-faced hexaflexagon. Or these flexes can be used to make an interesting puzzle or maze.

View the next chapter, Generating Sequences, or look at the Table of Contents.

Copyright © 2018-2020 Scott Sherman