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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 So, we’re going to make our non-cyclic flexagon, a tetra-tetra-flexagon. First, download the 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. Now, we fold the paper along all the lines, back and forth, so that the paper is flexible and easy to work. So we’re creasing back and forth, like this, just so that later along it’s easier to fold and flex. Now, we make a window in the sheet of paper, by cutting along the three bold lines here.

Skip to 0 minutes and 39 seconds I find it easiest to fold the sheet in half like this, and then cut two horizontal slits, open up the sheet, and make the third cut. Make sure that you cut along three lines only, so that you don’t cut out the whole rectangle, and make sure you cut along the bold lines, otherwise you won’t be able to make the flexagon with the same symbol on each face. Now, to fold the flexagon. Take the flap and wrap around like this, making sure all the time that squares with the same shape on them are folded together face to face. So we fold globe to globe, and wrap around dove to dove. Now take the other side and fold it up like this.

Skip to 1 minute and 15 seconds Globe to globe and then dove to dove. …and that’s it! You should have one side showing stars, and the other showing smiley faces. Place a bit of cellotape between the two center squares to connect the two smiley faces. The tetra-tetra-flexagon is a rectangular flexagon with four faces. It’s very easy to flex the flexagon. Just fold it like this, and then open it up from the center.

Skip to 1 minute and 38 seconds And we get the four faces: smileys, stars, doves and globes. This flexagon is not cyclic. It has a beginning - the smiley faces, and an end - the globes. This flexagon is frequently used by magicians as a magic wallet, because you can hide a dollar bill or whatever in between the creases, Here is a magicians, seven-faced non-cyclic flexagon. See! There’s nothing in the wallet. Say the magic words, abra-cadabra, Sprinkle some magic dust, and… Tara! [paper petals appear] and this can go on, again, and again, and again and again and again… [paper petals keep appearing]

The non-cyclic tetra-tetra-flexagon

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

• Download the file below which has the templates for the back and front of a rectangular sheet of paper that will be folded into the tetra-tetra-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.
• Crease the paper along the vertical lines, back and forth, so that the paper is flexible and easy to work. There is no need to crease the horizontal lines.
• Cut along the three bold lines to create a ‘window’.
• Fold the paper strip into a 3x2 rectangle by wrapping around the window and folding, making sure that you always fold identical symbols back-to-back.
• You should end up with smiley faces on one side and stars on the other.
• Put a bit of sticky tape between the two centre smiley faces to secure the flexagon. Make sure you only tape across the vertical lines of the two centre squares, otherwise your flexagon will ‘stick’ and won’t flex.

You’ll be able to see two faces or sides of the flexagon, once it’s assembled. The smiley faces and the stars. To expose the third face, tuck your thumbs slightly under either sides of the vertical crease in the middle of the flexagon and pull outwards. The flexagon should open up like a book revealing the third face, doves. Do this once again and you’ll get the fourth and final face, globes. This flexagon is non-cyclic. You won’t be able to carry on the ‘book’ flex from the fourth face, so to return to the first face you have to flex the other way back through the third and second faces. Once you get to the smiley faces, the flexing once again stops.

Discussion

What are the notable differences between the tri-hexa-flexagon and the tetra-tetra-flexagon?