The Hexa-Hexaflexagon

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

• Download the file below (template number 4 in the flexagon booklet) which has the templates for the back and front of the paper strip that will be folded into a hexa-hexa-flexagon.

  Note: There are 3 identical strips. You can use any one of them. The others are spare….
• Print the two templates using a colour printer, preferably one that can do 2-sided printing. If this is not possible – print each side on a separate sheet and 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. Take a look at the Tuckerman map and see if you can reach all the different positions. Can you see the connection between the map and the flexagon that you just built?

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.