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Explorable Flexagons

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Flexing Flexagons

Flexagon Intro

Exploring the hidden sides of hexaflexagons

By: Scott Sherman

Applying the pinch flex to a hexaflexagon:

Some triangle flexagons:

Flexagons are interesting puzzles made from folding strips of paper. You can make them from a wide variety of polygons: various triangles, quadrilaterals, pentagons, and so on. Once assembled, you flex them, i.e. fold them in different ways, in order to find new faces and rearrange the polygons.

The most common flexagons are made up of 6 equilateral triangles arranged in a hexagon; thus they’re called hexaflexagons. And the most commonly described flex is the pinch flex, which can reveal previously hidden faces. But it’s not obvious at first where to apply the pinch flex in order to find every face.

This explorable will teach you how to make hexaflexagons and how to use the Tuckerman Traverse to find all the hidden faces. Along the way, it also introduces you to flex notation, which is used to describe a series of flexes, and pat notation, which is used to describe the internal structure of a flexagon.

But, first off, if you haven’t seen a flexagon before, watch this introduction by Vi Hart to learn their history and see what folding and flexing look like...

3-faced hexaflexagon

We’ll start with the simple 3-faced hexaflexagon (or trihexaflexagon). The hexagon below has numbers and colors on each triangle, which change as you perform a flex.P is the flex notation for a pinch flex. The clickable buttons below labeled with a P indicate the corners where you can perform a pinch flex. Since a pinch flex involves doing a mountain-valley fold alternating every other vertex, the exact same pinch flex can be achieved by picking corner 1, 3, or 5 (numbered around the hexagon).^ is the flex notation for turning the flexagon over, which is also represented with a button in the figure below.

Try clicking on the buttons and watch how the flexagon changes and how this is tracked in the diagram on the right.

This diagram tracks the faces you’ve visited as you pinch flex the flexagon. The green circle marks the original face, and the red circle marks the current face.

But, of course, everything on this page is just a simulation of an actual model. The fun part is in creating and flexing the real thing. Following each simulation, you’ll find a strip of triangles (AKA template, frieze, or net) you can print, fold, and optionally decorate to create a real flexagon. Here’s the unfolded template for the 3-faced hexaflexagon.

These instructions apply to all the triangle templates on this page.

You might want to start by right-clicking on the image and picking 'Save Image' in order to view the template separately from the rest of the 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 small number represents the number on the back. Copy the small number from each triangle onto its backside.

Fold: Find the largest pairs of adjacent numbers and fold those numbers together. Then find the next largest pairs 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 hexagon consisting of 6 equilateral triangles. Tape the edges of the first and last triangles together to complete the flexagon.

4-faced hexaflexagon

Pick to start over, or to undo the last flex.

Next is the 4-faced hexaflexagon (or tetrahexaflexagon). After your first pinch flex, you find yourself in the middle of the diagram to the right, with multiple paths you can take. What happens when you try pinch flexing at different corners? How can you visit every point in the diagram in the fewest number of flexes? Do you notice a pattern?

Pat notation: Something else to pay attention to is the notation beside each edge of the flexagon above. This describes the internal structure of each pat, or stack of triangles.- indicates that there’s only a single leaf in the pat, while [- -] indicates two leaves folded against each other in a pat. Three leaves can be arranged either as [[- -] -] or [- [- -]]. If you flex a real flexagon while you flex the simulation, you can start to get a feel for what this represents. And if you put a unique number on every leaf, you can use this notation to describe where each leaf is as you flex your flexagon.

5-faced hexaflexagon

If you figured out a simple way to visit every state for the 4-faced version, does this same pattern work when you try to visit every state in the 5-faced variant? If not, perhaps a bit of experimentation can lead you to a simple pattern that works for both.

History:

(This history description uses flex notation to describe the series of flexes you’ve performed.)

Flex notation is used to describe a series of flexes so that you can later reproduce the exact same sequence. Flexes are performed relative to the ‘current vertex’, which is indicated by a * in these flexagon simulations. Shifting the current vertex one vertex clockwise is represented by >, and shifting counterclockwise by <. Thus P>P says to do a pinch flex at one vertex, rotate so the vertex one step clockwise is the current vertex, then do another pinch flex. Recall from earlier that ^ indicates turning over the flexagon while keeping the same current vertex.

6-faced hexaflexagons

There are three different ways to make a 6-faced hexaflexagon (or hexahexaflexagon), each made from different templates with different traversals. Try applying the same pattern you used for exploring every face of the 4 and 5-faced hexaflexagons to see if it works for the 6-faced variants. Click on,, and to switch between variations.

History:

The Tuckerman Traverse is a general pattern that allows you to easily visit every face that’s accessible using the pinch flex. Once you’ve tried finding a pattern, click on to see if you’ve found the same pattern.

Starting with any corner, repeatedly do a pinch flex until the structure of the flexagon no longer permits it. Switch to an adjacent corner and repeat the same process, doing a pinch flex until you can’t any longer. This will eventually visit every part of the diagram associated with the flexagon, visiting every configuration in the process. Furthermore, it’s the shortest traversal of every state (though it isn’t unique).

7 and 8-faced hexaflexagons

It’s also interesting to watch how the pat notation (e.g. [[- -] -]) for the pats changes as you flex, and compare that to the actual flexagon.

Now that you know the Tuckerman Traverse, you can try it on the four 7-faced hexaflexagons and twelve 8-faced hexaflexagons. Try out the 7-faced variation,,, or, and 8-faced variation,,,,,,,,,,, or.

History:

The unfolded template:

The following template takes a little bit of extra work to assemble. Cut out each of the shapes, cutting along the solid lines. Tape ‘b’ to ‘b’ and ‘c’ to ‘c’. Then you can follow the same folding rules as for the other templates, finishing by taping ‘a’ to ‘a’.

The following template takes a little bit of extra work to assemble. Cut out each of the shapes, cutting along the solid lines and the lines between ‘a’ and ‘b’. Tape one edge marked with ‘b’ to the other ‘b’ edge. Then you can follow the same folding rules as for the other templates, finishing by taping ‘a’ to ‘a’.

The following template takes a little bit of extra work to assemble. Cut out each of the shapes, cutting along the solid lines. Tape ‘b’ to ‘b’, ‘c’ to ‘c’, and ‘d’ to ‘d’. Then you can follow the same folding rules as for the other templates, finishing by taping ‘a’ to ‘a’.

The following template takes a little bit of extra work to assemble. Cut out each of the shapes, cutting along the solid lines. Tape one edge marked with ‘b’ to the other ‘b’ edge. Then you can follow the same folding rules as for the other templates, finishing by taping ‘a’ to ‘a’.

Other triangles

So far, all the flexagons we’ve looked at have 6 equilateral triangles per face, but there’s no need to have exactly 6 per face, nor do we need to stick to equilateral triangles. We can carry out the Tuckerman Traverse on a wide variety of triangle flexagons. However, this may mean that the flexagon doesn’t always lie flat because the central angles may not add up to 360 degrees. The following video demonstrates what the pinch flex looks like on a silver octaflexagon.

Try experimenting with using different types of triangles and different numbers of faces.

Switch the type of triangles to an,,,,, or.

Change the number of faces to,,,,, or.

History:

Note that some of these templates may have overlapping triangles. Compare the number of leaves (triangles) in the template to the number listed above to see if you might need to print multiple copies.

What next?

This intro showed you how to explore hexaflexagons using the pinch flex and the Tuckerman Traverse. It introduced flex notation for describing a series of flexes and pat notation for describing the internal structure of a flexagon. It then gave some examples of using different types of triangles to create octaflexagons, decaflexagons, and dodecaflexagons.

But that’s just a quick taste of the large variety of flexagons and flexes. You can make flexagons out of any polygon, including squares , trapezoids, rhombuses, and pentagons. And there are many other ways to flex them, with names like the v-flex , pyramid shuffle, and flip flex. It can be fun to decorate them, put mazes on them, or create illusions with them.

View the next chapter, Flexing Flexagons, or look at the Table of Contents.

Copyright © 2018-2020 Scott Sherman