Skip to 0 minutes and 0 secondsThe hexa-hexa-flexagon is really a very interesting flexagon. It’s based on the tri-hexa-flexagon. In fact if you take the tri-hexa-flexagon paper strip and make an exact copy of it and then attach the two copies end to end, you’ll get the strip needed to make the hexa-hexa-flexagon. However, you’ll need to recolour some of the triangles, in order to distinguish between all the six faces, so you can download and use the full six-colored template below. Print out and glue together the back and the front, so that the paper is printed on both sides.

Skip to 0 minutes and 35 secondsMake sure you do this the right way, so that one side looks like this… and when you turn it over from top to bottom … the other side looks like this. The first thing we want to do is to fold the paper along the diagonal lines, back and forth, so that the paper is flexible and easy to work. Folding comes in two stages. We first fold the purple, the orange and the green triangles, one after the other, just rolling up the strip. Then we fold with the yellow triangles face to face, make the hexagon, and tape the flaps together. And that’s it! There’s your hexa-hexa-flexagon. You’ll notice that flexing this flexagon is really puzzling.

Skip to 1 minute and 15 secondsIf you pinch flex from the same position, you’ll probably only see three faces. Play around with your flexagon and see if you can see or find all six colours. Go on, pause the video - I’ll wait! There is a more structured way to find all the faces. It’s called the Tuckerman traverse. What you need to do, is pinch flex from the same corner until you reach a position which won’t flex and then move one pat over. Like this. This way, you’ll be able to reveal all 6 faces. It’s exciting and fun to recreate the Tuckerman diagram for this flexagon, and I’ll leave this for you.

Skip to 1 minute and 51 secondsWhat’s interesting is that the inner strip colors, those that we folded first in the paper strip, the orange, green and purple, occur much less frequently than the dominant colors, red, blue and yellow. Now, I want to show you one last thing. How to mix faces to get different colored faces. For this we’re going to need to learn a new flex, the V-flex. Put your thumb along half the center line of the hexagon, and your third finger along the other half of the center line, and then bring your finger and thumb together, creating these two connected triangular pyramids. Open up to get this ‘well’, and then just pull these two triangles over.

Skip to 2 minutes and 32 secondsAnd we get one face with two triangles a different color! If you do the V-flex over and over again, You can even totally invert the flexagon, so that the green, purple and orange now become the dominant colors. With a bit of practice, you can do this as well!

The hexa-hexa-flexagon

Watch the video and follow the instructions to build the hexa-hexa-flexagon:

  • Download the file below which has the templates for the back and front of the paper strip that will be folded into a hexa-hexa-flexagon.
  • Print the two templates using a colour printer.
  • Glue together the back and the front, so that the paper strip is printed on both sides. Make sure you do this the right way, and that your back and front are correctly positioned. The order of the coloured triangles on one side from left to right should alternate red, blue, yellow, six times; On the other side, when you turn over the strip from top to bottom you should have, from left to right: one orange triangle, then 2 purple triangles, 2 green triangles, 2 orange triangles, and again 2 purple triangles, 2 green triangles, 2 orange triangles and then 2 purple triangles, 2 green triangles and 1 orange triangle.
  • Crease the paper along the diagonal lines, back and forth, so that the paper is flexible and easy to work.
  • Fold the paper strip into a hexagon. Start by folding each purple, green, and orange triangle onto its adjacent twin. You should get a strip half the length of the original. Then do the same with the now adjacent pairs of yellow triangles. Position the remaining orange triangles on top of one another. The now adjacent white triangles can be folded face to face and fixed with adhesive, and that’s it!

Perform the pinch-flex and see if you can expose all six faces. If you get stuck, try rotating the flexagon in your hand to a different triangle and try to flex again. With patience you should be able to find all six faces. Notice that the purple, green and orange faces appear much less frequently than the blue, red and yellow faces. This is because these triangles (that all appear on one side of the strip) were folded ‘inwards’ first when creating the flexagon. The reason we folded the paper this way is to reduce the strip size in half so that the right number of triangles (7) can be folded into the final hexagonal shape. The blue, red and yellow faces are called the dominant faces. The purple, green and orange faces are called the hidden faces.

You can systematically find all the faces by doing the ‘Tuckerman traverse’. Pinch flex the flexagons from the same position, until the flexagon gets stuck. Then move the position you are flexing from one place over and continue this way.

If you didn’t flex to all the faces, or you didn’t quite understand the difference between hidden and dominant faces, don’t fret! In the next step we’ll take a look at the Tuckerman diagram of the flexagon and this should be clearer.

Finally, try and do the V-flex and mix some of the faces. It takes time, but with practice you should be able to do this!

Discussion

Once you get the hang of it, can you find the ratio between the regular and hidden faces? Hint: This can also be deduced from observing the initial strip and the way we folded it… Write your answers in the comments section below.

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This video is from the free online course:

Flexagons and the Math Behind Twisted Paper

Weizmann Institute of Science