### How Flexible Is Paper?

In my last origami post I showed you the weight bearing strength of paper. In this post I want to show you a fun fold I learned way back in my college days. This amazing fold creates a 3-D form that rotates on its axis. It is called a Hexaflexagon.

The first time I made one of these it took hours to complete because the directions  required you measure out the form, then line it for all the folds. After that, you had to score all the lines with a sharp knife. Only then could you fold it, squash it, and finally glue it together. After making several flexagrams the hard way, it dawned on me that it was simply a repeating pattern. Why not just fold it Origami style, since all the lines were straight? It’s just a series of diagonal and horizontal folds. I tried it and can now make one very quickly, with any sized paper, as long as it is just over three times its width, allowing for the glue tabs.

This is a rather complicated project the first time you do it. It usually takes two whole periods to teach junior high and high school students how to fold them. It’s much easier when someone is there showing you how in person. The following just tells you the basic steps of the process.

You  begin by folding the paper diagonally to create the three original squares which form at the ends of the X folds. After that you divide each square into four horizontal folds. Next,  you must fold the four corners of each square to the center of each square. At this time you can see the pattern all made up of triangles. You just need to find where the pattern is missing in your project  and add the missing folds. Once this pattern is completed you will need to crease all of the horizontals into valley folds. Next, turn the paper over and fold all the diagonals with valley folds. The paper will start to curl up when you do this. Grab one end of the fold and squeeze the outside of the form, where you see little triangles, to the center with your fingers while compressing the form. Continue all the way to the end of the form until you have a tiny square, springlike form. Open it back up and cut the extra band of paper into the glue tabs. Add glue to the top of the glue tabs and insert them into the inside of the form one at a time. Squeeze it for about a minute to allow the glue to set. Rotate the form so that you can insert the second glue tab. Again hold it for a minute or so.

You have now made a hexaflexagram. To make it rotate you must pull on the outer edges while pressing the center in. It will rotate one facet at a time. Be sure to always keep the center tightly together as you rotate it or the glue tabs may fail. After you have tried rotating it a couple times it will be easier.

Some people think this form is a good model of a black hole. One that takes you through its center to another dimension. Others liken it to a flying saucer, complete with landing gear. The saucer closes up to travel through space yet can rotate to open its landing gear.

If you make multiple units they stack and interlock like gears. Stack four on top of each other and bind together with rubber bands and you have a great indoor football to throw. I wouldn’t try kicking it though. It would probably break depending on how hard you kicked it.

Another fun way to use them is to create  3D greeting cards by adding art to the surfaces. You can even make a progressive message that appears each time you rotate  a new surface.  I once won a contest with my Christmas card design created on the hexaflexagram base.

Interestingly, there are several other types of flexagrams. If you are interested in these, check out the M. C. Escher kaleidocycles. They flexagrams come in book form with cutout models. They are printed with Escher tessellations on the outside which make wonderful conversation and art pieces.

As you have seen in these pictures, the Hexaflexagram is amazingly flexible. It is also strong enough to support weight, much like the accordion pleat in the last post. I hope you have enjoyed learning about flexagrams.