Definitions

# Flexagon

[flek-suh-gon]
In geometry, flexagons are flat models made from folded strips of paper that can be folded, or flexed, to reveal a number of hidden faces. They are amusing toys but have also caught the interest of mathematicians.

Flexagons are usually square or rectangular (tetraflexagons) or hexagonal (hexaflexagons). A prefix can be added to the name to indicate the number of faces that the model can display, including the two faces (back and front) that are visible before flexing. For example, a hexaflexagon with a total of six faces is called a hexahexaflexagon. The trihexaflexagon is an example of a Möbius strip. A flexagon whose hexagonal faces are each divided into twelve right triangles as opposed to six equilateral triangles, and which can consequently flex into nonhexagonal shapes, has recently been christened a dodecaflexagon ().

Harold V. McIntosh also describes nonplanar flexagons folded from pentagons called pentaflexagons and heptagons called heptaflexagons

## History

The discovery of the first flexagon, a trihexaflexagon, is credited to the British student Arthur H. Stone who was studying at Princeton University in the USA in 1939, allegedly while he was playing with the strips he had cut off his A4 paper to convert it to letter size. Stone's colleagues Bryant Tuckerman, Richard P. Feynman and John W. Tukey became interested in the idea and formed the Princeton Flexagon Committee. Tuckerman worked out a topological method, called the Tuckerman traverse, for revealing all the faces of a flexagon. Tukey and Feynman developed a complete mathematical theory that has not been published. Feynman also invented a diagram for analysing flexagons.

Flexagons were introduced to the general public by the recreational mathematician Martin Gardner writing in Scientific American magazine. The columns were reprinted in several books including:

• Mathematical Puzzles and Diversions (1959; Pelican, UK ISBN 0-14-020713-9)
• More Mathematical Puzzles and Diversions (1961; Pelican, UK ISBN 0-14-020748-1).

## Tetraflexagons

### Tritetraflexagon

The tritetraflexagon is the simplest tetraflexagon (flexagon with square sides). The "tri" in the name means it has three faces, two of which are visible at any given time if the flexagon is pressed flat.

It is folded from a strip of six squares of paper like this:

To fold this shape into a tritetraflexagon, first crease each line between two squares. Then fold the mountain fold away from you and the valley fold towards you, and add a small piece of tape like this This figure has two faces visible, built of squares marked with "A"s and "B"s. The face of "C"s is hidden inside the flexagon. To reveal it, fold the flexagon flat and then unfold it, like this The construction of the tritetraflexagon is similar to the mechanism used in the traditional Jacob's Ladder children's toy, in Rubik's Magic and in the magic wallet trick or the Himber wallet.

There is also a method of creating a more complicated hexatetraflexagon. To make it, take a piece of square paper and cut a square hole in the middle. Make sure all edges are straight. Then from the left hand edge, make a valley fold towards the middle. From the top, make another valley fold towards the middle. Now make a valley fold from the right hand edge towards the middle. Finally, make a valley fold from the bottom towards the middle. You now have the hexatetraflexagon. Please note that you do not need any tape or paste to make this flexagon. The most exciting thing to do with this one, is to colour both faces, then keep on flexing and colour the faces as you find them, until you get back to your starting position.

## Hexaflexagons

Hexaflexagons come in great variety, distinguished by the number of faces that can be achieved by flexing the assembled figure. See below, "Inventory_of_hexaflexagons"

### Trihexaflexagon

A hexaflexagon with three faces.

While this is the simplest of the hexaflexagons to make and to manage, it is a very satisfying place to begin. It is made from a single strip of paper, divided into ten equilateral triangles. Patterns are available at The Flexagon Portal

It is possible to automatically section and correctly place photographs (or drawings) of your own selection onto Trihexaflexagons using the simple program Foto-TriHexaFlexagon

### Hexahexaflexagon

This hexaflexagon has six faces.

Make a mountain fold between the first 2 and the first 3. Continue folding in a spiral fashion, for a total of nine folds. You now have a straight strip with ten triangles on each side. There are two places where 3's are next to each other; fold in both these places so as to hide the 3's, forming a hexagon with a triangular tab sticking out. Lift one end of the hexagon around the other so that the 3's near the ends are touching each other. Fold the tab over to cover the blank triangle on the other side, and glue it to the blank triangle. One side of the hexagon should be all 1's, one side should be all 2's, and all the 3's should hidden.

Photos 1-6 below show the construction of a hexaflexagon made out of cardboard triangles on a backing made from a strip of cloth. It has been decorated in six colors; orange, blue, and red in figure 1 correspond to 1, 2, and 3 in the diagram above. The opposite side, figure 2, is decorated with purple, gray, and yellow. Note the different patterns used for the colors on the two sides. Figure 3 shows the first fold, and figure 4 the result of the first nine folds, which form a spiral. Figures 5-6 show the final folding of the spiral to make a hexagon; in 5, two red faces have been hidden by a valley fold, and in 6, two red faces on the bottom side have been hidden by a mountain fold. After figure 6, the final loose triangle is folded over and attached to the other end of the original strip so that one side is all blue, and the other all orange.

Photos 7 and 8 show the process of everting the hexaflexagon to show the formerly hidden red triangles. By further manipulations, all six colors can be exposed. Faces 1, 2, and 3 are easier to find while faces 4, 5, and 6 are more difficult to find. An easy way to expose all six faces is using the Tuckerman traverse. It's named after Bryant Tuckerman, one of the first to investigate the properties of hexaflexagons. The Tuckerman traverse involves the repeated flexing by pinching one corner and flex from that exact same corner every time. If the corner refuses to open, move to an adjacent corner and keep flexing. This procedure brings you to a 12-face cycle. During this procedure, however, 1, 2, and 3 show up three times as frequently as 4, 5, and 6. The cycle proceeds as follows:

1-3-6-1-3-2-4-3-2-1-5-2

And then back to 1 again.

Each color/face can also be exposed in more than one way. In figure 6, for example, each blue triangle has at the center its corner decorated with a wedge, but it is also possible, for example, to make the ones decorated with Y's come to the center. There are 18 such possible configurations for triangles with different colors, and they can be seen by flexing the hexahexaflexagon in all possible ways in theory, but only 15 is flexed by the ordinary hexahexaflexagon. The 3 extra configurations are impossible due to the arrangement of the 4, 5, and 6 tiles at the back flap. (The 60-degree angles in the rhombi formed by the adjacent 4, 5, or 6 tiles will only appear on the sides and never will appear at the center because it would require one to cut the strip, which is topologically forbidden.)

The one shown is not the only hexahexaflexagon. Others can be constructed from different shaped nets of eighteen equilateral triangles. One hexahexaflexagon, constructed from an irregular paper strip, is almost identical to the one shown above, except that all 18 configurations can be flexed on this version.

#### The elusive 3 extra combinations in a hexahexaflexagon

The argument that a hexa-hexa-flexagon, like the one illustrated above, has only 15 different available combinations is a mis-conception. It seems to stem from the use of very simply labelled flexagons in the analysis which hides the three elusive combinations and perhaps an unqualified reliance on the topology diagram which indicates 15 nodes, but does not fully describe the appearance of each face described by those nodes or on the basis that certain faces can only be shown in two configurations and others in only three configurations.

Before the proof, a closer look at a regular hexa-hexa-flexagon: Once you've built a flexagon from a strip of paper as illustrated above and have coloured the faces or even labelled the centre corners and have gone through it completely using the Tuckerman traverse technique you'll discover that there are distinctly two different types of faces that appear: those made of six triangles and those made of three parallelograms—each parallelogram formed by two joined triangles. You'll notice that the faces formed by the three parallelograms can appear in only two configurations and that the flexagon is always configured so that there are five layers of paper making half the parallelogram and one layer of paper for the other half. The faces formed by the six triangles seem to appear in three configurations if you only look at the symbols used to identify the corners of the triangles in the centre of the flexagon as shown below and as is commonly represented in the literature:

Such simple labelling will hide the fourth combination that each six-triangle face has because it shares a similar centre but a different ordering of the triangles.

Here's how to reveal that elusive fourth combination: Take a blank flexagon with no design or labelling on it. Assemble it and put one of the six-triangle sides face-up on your desk making sure that the underside does not reveal one of the three-parallelogram faces. Label the corners of each of the six triangles in the centre of the flexagon with A's and the edges with numbers as shown:

Using the Tuckerman traverse reveal another version of that face where the "A" corners are as shown making sure the underside does not reveal one of the three-parallelogram faces and label the centre corners with B's and the edges with numbers as before:

Manipulate the flexagon to reveal another version of that face where the "A" and "B" corners are as shown. Make sure the underside does reveal one of the three-parallelogram faces. Label the blank centre with C's and the edges with numbers as shown:

Further manipulation of the flexagon will eventually reveal the fourth version of this face:

Notice that the "C" centre face has two versions: one where the edge numbers match and another where the edge numbers are completely scrambled. Also notice that the "A" or "B" centre edge numbers on the outside edge of the flexagon match or mismatch depending on which version of the "C" centre you're viewing. Also note that the "A" and "B" centred faces have exactly one configuration each.

Since a three-parallelogram face has two configurations and it has been made clear that the six-triangle face has four unique configurations and since there are three of each type, then there are truly 18 configurations that can be revealed from a regular hexa-hexa-flexagon.

### Inventory of hexaflexagons

• The unahexa. A strip of three triangles can be folded flat and a Möbius strip can be formed by joining the two edges. Some call it the unahexaflexagon, although it doesn't have six sides and obviously doesn't flex.
• The duahexa. Simply a hexagon cut from a sheet of paper. Still considered a flexagon because it's made of two hexagonal faces made of 6 equilateral triangles. It doesn't flex.
• The trihexa. It's made by marking a strip of paper that can be marked off as 10 equilateral triangles. Fold this by four sections of two and two isolated triangles one at each end, it forms a hexagon that flexes into three different faces.
• The tetrahexa. This can be folded from a crooked strip made of three connected ⅔ hexagons (Hexagons with two adjacent equilateral triangles, the ones that make up a hexagon when six is joined, cut out from it) and one spare triangle as a pasting tab. The orientation of the three ⅔ hexagons are the same, and opposite sides are joined, so that all three openings face down.
• The pentahexa. Folded from a crooked strip in the form of a backward "N", or "/". Each slash is connected at an equilateral triangle with two sides connected to each of the three parts. The circumference of the polygon formed is 15 (The third slash has one extra triangle, which is used as a pasting tab).
• The hexahexa. See the section above.
• The heptahexa. Can be folded from three strips of varying shape. One of the three forms of the heptahexa has a special property: when the Tuckerman traverse is used on this form, it will bring up the seven faces in serial order.
• The octahexa. Has 12 varieties.
• The enneahexa. Has 27 varieties.
• The decahexa. Has 82 varieties.

Also, by folding a strip of equilateral triangles twice as long as needed to make a hexahexaflexagon in the manner described above so it's "spiral" and the same length as a strip needed to make a hexahexaflexagon, we can make a dodecahexaflexagon by simply fold that twisted strip as if you are making a hexaflexagon. Tuckermann made a workable model with 48 faces in this manner, the repeated twist-twist-twist...fold procedure.

## Octaflexagon and dodecaflexagon

In these more recently discovered flexagons, each square (resp. equilateral triangular) face of a tetraflexagon (resp. hexaflexagon) is further divided into two right triangles, permitting additional flexing modes ().

## Bibliography

• Pook, Les, Flexagons Inside Out, Cambridge University Press (2006), ISBN 0-521-81970-9
• Martin Gardner has written an excellent introduction to hexaflexagons in one of his Mathematical Games column in Scientific American. It also appears in one of his early collections of this column.

Flexagons:

• My Flexagon Experiences by Harold V. McIntosh — contains valuable historical information and theory; the author's site has several flexagon related papers listed in and even boasts some flexagon videos in
• Flexagons — David King's site is one of the more extensive resources on the subject; it includes a Java 3D Simulation of a hexahexaflexagon.
• The Flexagon Portal — Robin Moseley's site has patterns for a large variety of flexagons.
• Flexagons is a good introduction, including a large number of links.
• Flexagons — Scott Sherman's site, with a bewildering array of flexagons of different shapes.

Tetraflexagons:

Hexaflexagons:

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