Trihexaflexagons and Hexahexaflexagons

After over a year of inactivity on Instructables, I am happy to say I will be making more in the future. Today's topic will be regarding a very intriguing paper novelty, the hexaflexagon.

A standard flexagon is a folded piece of paper in which the shape can be contorted by folding and flexing the shape such that it can reveal a side that was not originally visible. This side is hidden within the folds of the other two sides; the front and back. Flexagons can reach different orders of complexities with the most basic being three sides and the more complex ones reaching twelve sides and higher. By adding a prefix to the root 'flexagon' we can denote the type of flexagon such as a tetraflexagon or a hexaflexagon. This Instructable will deal with hexaflexagons primarily although other types of flexagons can be achieved.

Arthur Stone is credited with the discovery of the first flexagon: the trihexaflexagon. As denoted by the prefix, it has three sides. He had bought his binders in the United States and his paper in the United States. However, due to incompatibilities of the different paper sizes (the British used A4 paper while the United States used Letter). He began to fold them into various shapes and at one point, he formed a trihexaflexagon. Arthur Stone showed his new-found discovery to Bryant Tuckerman, Richard Feynman and John Tukey. Tuckerman worked out a method of revealing all the faces of a flexagon; known as the Tuckerman Traverse. Although the discovery was made in 1939, it would take another 17 years before it would be introduced to the general public by Martin Gardner in his Mathematical Games column for Scientific American.

You will need:

1. Scissors
2. Glue Stick
3. Hexaflexagon Template (The file is attached below and was designed by Scientific American)

You will probably want to print two as this Instructable covers both trihexaflexagons and hexahexaflexagons.

Cut the template out along the outside outline and then make a mountain fold (line side out) down the middle line. Fold on each line back and forth so that you make assembly easier. Try to be as exact as possible so that the trihexaflexagon looks neat. Once all the folds are made, then apply glue to the inside of the triangles and stick it together.

Now for the tricky part: the hexaflexagon assembly. If done incorrectly then it won't make a hexaflexagon. It is stressful when it doesn't work the first time but don't be discouraged. Follow along with the pictures and it should help clear up the assembly.

How the 10 triangles are oriented is important: hold the strip of paper with the left hand triangle's top corner away from yourself. It should look like the first picture.

Then, near the right side, there should be a dotted fold line 4 triangles in on the right side, fold this forward or backward depending on what you want; personally, I folded it backwards. On the left side, fold the 3 triangles on the dotted line the opposite of what you did on the right side. I folded it forward.

Finally, tuck the left side underneath the right side (or vice versa) and there should be a triangle sticking out on top. Apply glue to the top two triangles and then fold the triangle down (see pictures for reference).

This is also very tricky the first time you do it. First, pick up the hexaflexagon and number the top side 1. Make a mountain fold on one of its edges. That will create a sort of bulge on the other side. With your other hand, push the bulge in and make a valley fold. This should give you a piece that looks something like the third picture. Then, put your finger between the folds of the paper and it should open up nicely to a black side. Number this side 2. Repeat the same process and number the last side 3. Now have some heaflexigating fun!

The next step is the the beginning of the hexahexagon which has six sides instead of three. You may want to stop here to continue enjoying the trihxaflexagon or continue on to make an even more intricate flexagon.

This is the more advanced version of the flexagons that this guide will cover. This is even harder to make and is more complicated to flexigate but it is a very intriguing novelty. It has six sides instead of the standard three but, all six sides are not as easy to get to as they were with the three sided one.

The template is still the same as the other one so you don't have to get a new template. The assembly is very different, though. With the trihexaflexagon, the paper is folded in half and doubled up to make it thicker and stronger. With the hexaflexagon, the paper becomes hard to flexigate when doubled up so it is only a single layer.

It is strongly recommended that you make a trihexaflexagon first to understand how to make it and how to flexigate it considering the fact that the hexahexaflexagon is several orders more complex.

Start by cutting the template out similar to the trihexaflexagon but this time, cut down the mid-line to split the paper in half. This will create two strips with 10 triangles each. Take the two strips and apply glue to the back of one of the triangles (side with nothing printed on it). The triangle should be facing with its top corner facing away from you. See the third picture for reference. This will result in 19 triangles in a long row. Then with the left side's triangle pointing away from you, start to number the triangles 1,2,3,1,2,3,1,2,3....

At the end, there will be one extra triangle at the end; put the letter 'G' on that triangle. Flip the triangle over on its long edge and then start with 4,4,5,5,6,6,4,4,5,5,6,6...from right to left (the first 4 should be on the back of the G) all the way to the end where your write 'G' again. (See the pictures for reference).

I made a separate step because this step is very complicated and takes some time. Start by taking the full strip and folding it back and forth, similar to the way you did with the trihexaflexagons. This step was somewhat necessary with the trihexaflexagons but definitely necessary here. The folds facilitate the making of the hexhexaflexagons as well as flexigation later on. When you reach the glued triangle in the middle, it will be a little hard to fold but try to fold it on the line.

Once that is completed, make a fold between the first 2 and 3 on the left hand side of the strip (Picture 2). Repeat this fold between the next 1 and 2. Essentially, you are folding every second triangle. This should go quickly if you folded the triangles back and forth first.The structure will try to unravel but try to keep it down.

At the end, fold the glue triangle tab as well.

You will notice that once this is done you will have a strip very much similar to that of the trihexaflexagon.

This part is very,very tricky. If done incorrectly then it will make what I like to call a pseudo-hexahexaflexagon; it is a shape that looks and feels like a hexahexaflexagon but it does not form a hexahexaflexagon and does not hexaflexigate properly.

On the right hand side, it should say 1,1,2,2,3 and then it should fold over. Similar to before, fold the right hand side between the 1 and the 2 with a mountain fold so that it is folded backwards.

On the left hand side it should say 1,2,2,3,3. Make a valley fold (fold it forward) between the 2 and the 3 and the tuck the strip under the other one. The glue tab triangles will line up. Apply glue and paste down. You have completed the hexahexaflexagon.

To flexigate this hexaflexagon, you repeat the same thing you did with the trihexaflexagon except that this time, there are six sides to find. You can use the Tuckerman traverse if you want to find the fastest route to all 6 sides. You will begin to notice that you seem to be able to find 3 numbers at a time in triples: try to find all the triples. If you want to read my findings you can read on but:

Challenge Yourself:

Try to find all 6 sides of the hexahexaflexagon and try to figure out exactly what triples show up.

Since you made the hexahexaflexagon you can vouch that there are 6 numbers on the strip of paper. They should all be accessible if you made the hexahexaflexagon properly. To prove that they are all accessible, I have attached pictures of all 6 sides.

I like to call these triples; similar to the way we have Pythagorean triples, I think of these as hexahexaflexagon numbers that always show up together:

1,2,3

1,3,4

1,2,5

2,3,6

The way I wrote the triples is that if you can find 1, you can access 3 and therefore, 4. You can also access 2 and 5 from 1.

I hope you enjoyed reading about the complex subject of hexaflexagons. It is not an easy topic to explore and can be very frustrating. You can try to draw on your new trihexaflexagon or your hexahexaflexagon and see what happens. Enjoy.

Here is one lass thought for you to think about:

If you trihexaflexigate too many trihexaflexagons you might get trihexaflexaperplexia or trihexaflexadyslexia or even develop trihexaflexadementia but if you hexahexaflexigate too many hexahexaflexagons then you might get hexahexaflexaperplexia or hexahexaflexadyslexia or even develop hexahexaflexadementia.