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This content is taken from 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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## Weizmann Institute of Science

Skip to 0 minutes and 0 secondsNow, we’re going to make a hexa-tetra-flexagon. As usual, download the template. 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. This time we’re going to cut out the blank square in the middle. By cutting through this bold line here, between the star and the smiley. Throw it away - we don’t need it. Now, fold the paper along the lines, back and forth,

Skip to 0 minutes and 33 secondsFold the flexagon like this: globes to globes, flowers to flowers, moons to moons, and doves to doves. And that’s it! You should have one side showing stars, and the other side showing smiley faces. Tape the top right wad of smiley faces together. You have made a non-cyclic hexa-tetra-flexagon with six faces. It flexes from the center just like the tetra-tetra-flexagon. but this time, it flexes in both directions! Up and down, right and left. Going horizontally, we get smileys, stars, moons, turn over and see the flowers. Going back to the smiley faces and now folding vertically, we get stars, doves and on the other side, globes!

# The non-cyclic hexa-tetra-flexagon

Watch the video and follow the instructions to build the non-cyclic hexa-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 hexa-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.
• Cut along the bold line into the blank centre square, and then cut out the blank centre square.
• 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 2x2 square making sure that you always fold identical symbols back-to-back.

You should end up with four smiley faces on one side and stars on the other. Tape together the top right horizontal wad of paper. You’ll be able to see two faces or sides of the flexagon, once it’s assembled. You can use the ‘book’ flex, both vertically and horizontally to expose the other 4 faces. Did you find them all?

### Discussion

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