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This content is taken from the Davidson Institute of Science Education at the Weizmann Institute of Science's online course, Flexagons and the Math Behind Twisted Paper. Join the course to learn more.
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Davidson Institute of Science Education at the Weizmann Institute of Science

Skip to 0 minutes and 0 seconds The tetra-octa-flexagon, has eight triangles on its face, hence, octa. It is shaped like a square, and has four faces. This is a straight strip flexagon, and perhaps we should say is more like the tri-hexa-flexagon family. First, download and use the four-colored template below. Print out and glue together the back and the front, so that the paper is printed on both sides. Make 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.

Skip to 0 minutes and 39 seconds Folding comes in two stages, just like the hexa-hexa-flexagon. We first fold the yellow triangles together face to face, and in the new strip, fold the green triangles face to face, and tape the flaps together. to get this square, with one side blue, and the other side red. Notice that the inner side colours are yellow and green, and the outer side colours are red and blue. You’ll notice that flexing this flexagon is really puzzling. If you pinch flex red face up, blue face down, you go through this intermediate stage, which can be opened up into a red and green rectangle, to get blue face up and green face down.

Skip to 1 minute and 20 seconds Pinch flexing this, goes straight back to red up blue down, without the intermediate stage, and this cycles. But wait a minute! Where is the yellow face! Only a special reverse-pass-through flex, can reveal that face. Make a pinch flex from a face that goes through the intermediate rectangle stage, like this, but just before you open up, switch all mountain folds to valleys and vice versa, like this. and there is is - the yellow face!

The tetra-octa-flexagon

Watch the video and follow the instructions to build the tetra-octa-flexagon:

• Download the file below which has the templates for the back and front of the paper strip that will be folded into a tetra-octa-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 squares on one side from left to right should alternate blue, red-green, four times; On the other side, when you turn over the strip from top to bottom you should have, from left to right: one green triangle, then alternating squares: yellow, red-green three times and then one yellow square and one red triangle.
• Crease the paper along the lines, back and forth, so that the paper is flexible and easy to work.
• Fold the paper strip into a square. To assemble, fold each yellow triangle on top of its adjacent twin. Then fold the pairs of green triangles in the same fashion. A square will form with red on one side and blue on the other. Position the white triangles face to face and fix with adhesive - and that’s it!

Perform the pinch-flex. You will only be able to expose three of the four faces: red, blue and green. These are the dominant faces. The yellow face is the hidden face and it can only be exposed if you can manage to do the reverse pass through flex.

Discussion

Try and draw the Tuckerman diagram for this flexagon. Post your diagram on your social media with the hashtag #FLflexagon, and then post a link to it in the comments below.