A "flexagon" is a way of folding and rejoining paper to make a flat object which appears to have more than two sides. It can only show two of those sides at once, so you "flex" it - folding it and unfolding it in a particular way - to switch out which two sides appear at a time.
The word "flexagon" was coined by one of four people, all of whom were graduate students in physics or mathematics at Princeton in 1939. They were Arthur Stone, who apparently discovered the hexagonal format you're going to see in this demo; Bryant Tuckerman, Richard Feynman, and John Tukey. All four of them achieved some fame in their respective fields and one of them achieved a fair amount of fame, period. These four did all the initial "research" into the flexagon (read: they farted around with them a lot when they should have been studying.) You can find the whole story, and a lot more detail than this demo goes into, in Martin Gardner's The Scientific American Book of Mathematical Puzzles and Diversions (Simon and Schuster, 1959) - if you can manage to find a copy.
We're going to make two hexaflexagons (the hexa- just refers to the shape of the finished object; there are other kinds). We'll start with the simpler three-sided "trihexaflexagon," because it is easy and shows the basic principles; then we'll try for the six-sided "hexahexaflexagon."
You will need:
- a roll of adding machine paper (not thermal paper)
- a means of measuring a sixty-degree angle
- something to cut the paper with
- something to glue two surfaces of paper together with that will only glue those surfaces (e.g. won't seep through and glue anything else in a stack of layers); I used double-sided tape because 1) I had some and 2) you don't have to wait for that to dry.
Begin by cutting off the end of the tape at a sixty-degree angle. Discard the scrap end or use it as a bookmark.
Using the sixty-degree angle you cut, you can now start folding the tape into equilateral triangles in a zigzag, using each triangle as a guide for where to fold the next one. The first two are shown in the picture. Although this is not precision science, do try to keep the corners en point and the triangles as exact as possible (which will get harder as the stack gets taller). Sloppy triangles make the structure harder to flex when it's done.
Of course you could take your protractor and mark out triangles on the tape and then fold them, for extra precision, and if you prefer that, be my guest. I'm lazy and this way is much faster.
You'll want to make at least ten triangles this way.