The 4-7-8 Measured HexaHexaFlexagon, and its State-Machine Graph

A basic hexaflexagon is a paper hexagon, made of six triangles -- but it obviously has layers, and there are odd fold-lines and paper edges showing in between the triangles. Here you can see two of them folded out of graph paper, with side 1 showing for the one on the left and side 2 for the one on the right. Each is in what I'm calling state "12", meaning that those are the two sides directly accessible.

 If you pinch one of those edges and poke the opposite edge inwards then you can fold the outside down and make it into a single thick triangle, with the center becoming the top vertex. I take the one on the right, showing the "2" side. Gently tug that vertex and you can pull it apart, revealing a third side to the flexagon; in this case it's actually numbered "3". Here:

Finish pulling it out, flatten it, and you'll be in state "23"; side 1 has disappeared, i.e. it has been folded to the inside, and sides 2 and 3 are now accessible:

 That's at least a "trihexaflexagon", and it's a "hexahexaflexagon" if there's also a fourth, fifth, and sixth side concealed in the middle. In this case, they're all right in there. For a hexahexa you can look up and use a "Tuckerman traverse" (described at the previous link) to visit all six sides, but I like thinking in terms of the graph I've put in the illustration.

   Central Cycle: Starting from side 2 and thus choosing to fold side 1 inwards, I moved to side 23 as shown, and I could then start from side 3 to fold side 2 inwards, reaching state 13...and then continue back to state 12, and round and round it goes. If you decorate the sides, you'll see peculiar things happen as the inside becomes outside, especially if you punch holes through some state (e.g., in the center, as a face's mouth) which disappear in other states. This is true even for the trihexaflexagon. This one, however, is a hexahexaflexagon, so I also have

   Peripheral Cycles: I could have chosen a different edge to pinch, and I'd have exposed side 5, reaching state 25. (Of the six edges, three will get me to side 3, three to side 5.) Similarly if I'd started with side 1 on top, I could have chosen to reach state 13 or 15. Those four states (13, 15, 23, 25) are the states reachable from state 12, as you see on the graph, and (12,15,25) makes a peripheral cycle....an epicycle. Each central-cycle state has its own epicycle: if I start in state 23 I can reach an epicycle using side 4, and state 13 gets me to one involving side 6.

This graph doesn't describe the full behavior of the flexagon; as I said, things change when the center changes places with the outside. Still it's a useful summary, and I invented it (I have no idea how many other people have done so, but I never saw it before I came up with it), and I like it, so I wanted to put it in a post.

Folding a strip of paper into one of these isn't hard, and you'll find lots of YouTube tutorials showing how to do it. Mostly they either say to measure a 60-degree angle, or simply to get a feel for folding overlapping equilateral triangles. I've done that, but I like measurement. I like numbers, and I like the fact that if you have an equilateral triangle whose sides are of length 8 (in whatever units you like) and it's pointing straight upwards, then one vertical line splits it into two equal right triangles whose sides are almost exactly 8, 4, and 7. Seven? Why seven, you ask? Well, Pythagoras says sqrt(8^2-4^2) = 6.92820... and seven is well within the margin of error you get by folding, anyway. 

Side-note: It's also nice that at least for me, it evokes my use of those triangles whenever I do Dr. Weil's 4-7-8 breathing exercise -- which definitely does not put me to sleep in 60 seconds, but like other patterns of conscious breathing gives me a way to defocus, refocus, and generally talk to my autonomic nervous system....if you have the habit of doing a specific breath pattern as part of relaxing/getting ready to sleep, then that association can help. Any ritual helps, but a breathing ritual can help more. Hooray for pranayama! So part of my own relaxation routine includes drawing those two triangles, one with each hand: inhale throughout the 4 side, turn at right angles and hold throughout the 7 side, then turn towards the beginning and exhale throughout the hypotenuse. 4-7-8.

 So does it actually help in making flexagons, at least as you're getting a feel for how to do it without any lines at all? You be the judge. Here's a piece of graph paper ruled in quarter-inch squares:

The length is 11 inches, and for a simple trihexaflexagon I want 5 1/2 base-lengths (you'll see why, as we go) so the baselength is exactly 2 inches, 8 squares. So I count 7 squares down, draw an 11" midline, 7 more squares, and slice myself a rectangle that will be (7/4)*2= 3.5"x11". Mark it at the 2", 4", 6", 8", and 10" points and add the lines you see: the ruler is there to show you that the diagonal triangle sides we're creating really are 2", i.e. the triangles really are equilateral. Number the sides, fold along the midline and glue the halves together, then slice off the incomplete triangle from each end:

As you can see, the back side of the glued strip shows the triangles that were on top.  (At this point, it would be best to fold back and forth along each of the edges to crease them, but I'm not doing that because I want to control the highlights as I take pictures.)

 The front says 1,2,2,3,3,1,1,2,2,3, and we want to hide side 3 so we fold the first pair of 3s to face each other, like so:

and then the next pair, like so:

The remaining 3 is on the other side of that final 1, and we want it to face the 3 we can see, so we slide the visible 3 under the 1:

And now we turn it over:

and fold it in, and glue the two blank sides to get:

Now I crease each edge back and forth both ways so that the edges fold readily, and even force them into a stack of equilateral triangles:

And now they can fold...so I fold back and forth for a few minutes and then decorate. As you can see, I've drawn a happy face on side 1 but there are three corners lopped off:

This time I fold through to side 2, and put a sad face....

and on to side 3...

Flexagons are fun. Finicky and frustrating, if you're as clumsy as I am, but gently folding back and forth over and over again to loosen the folds does eventually help. Just keep trying. I have made flexagons out of cereal-box cardboard...hard to fold. And out of toilet paper...even harder.

 And the six-sided version is just the same except (surprise!) for having twice as many sides...so you can either start with larger paper, or glue strips together to make a longer strip, or make smaller triangles as shown here, where I use only 10"x1.75" for 19 triangles, front and back:

After folding and gluing the halves shown, you fold the 4s together, the 5s together, and the 6s together -- you'll then have a single strip which looks much like the starting point shown above, and if you fold the 3s together it will come into a hexagon with the blank triangles ready to glue as we did here. Decorate, punch holes with a manual hole punch and see where they go, mark the insides of each side and watch them go out. Look for patterns. Always. The world needs more patterns. And the world needs more silliness. And that's all I'm gonna say....

or then again, maybe not. (But it is for now.)