But when you stop rolling on the floor holding your stomach, you might ask yourself 'where did the 3rd side come from'? and, well, I might have an answer.
A hexaflexagon is a 2 dimensional object in some sense. As you fold it up into a triangle, preparing to turn, it becomes 3 dimensional, and when it does, you can open it up. What you find is that the triangular pockets formed by the folds held the 3rd side. This 3rd side is inaccessible until you fold it, but when you open it, you are opening those pockets, revealing the hidden triangles. The former front becomes a symmetrically inverted back, and the former back side moves to the inside of the newly-formed pockets.
Another side-effect of the pockets is that if you keep folding and unfolding, effectively turning it inside out over and over again, it will rotate in the plane, without you turning it.
If you're trying to fold one, here's one tip:
You can estimate a 60° angle by carefully lining up the top corner with the bottom edge of the paper strip, as in the pink circle above. At other angles, the corner is either onto the paper or hanging off the edge, but at 60°, it will line up exactly, so long as the sides are straight.
Chirality is important. Make sure you've got three diamonds visible. If you don't, there is probably an up where there should be a down or vice versa.
I haven't quite got the hexa-hexaflexagon down yet, but we'll get there. The description at Hexaflexagon portal is very helpful, particularly the variation A hexa-hexa-flexagon, available as a PDF.
Oh, yeah, and while you're contemplating your notebook paper:
You didn't think that 9 1/2 x 11 inches was an international standard, did you? Guess what. Most of the rest of the world uses a different 'system', um, like an actual systematic system. The equivalent 'letter' size is A4, but we also have A1 (poster sized), A7 (index card), and other variously-sized characters in between. These sizes have the nicely chosen aspect ratio so that, $L/W = \sqrt{2} = 0.707 $... but that would be irrational, so they have to round off a bit.
This doesn't look very useful until you take the ratio of $\frac{\sqrt{2}}{2}$. Remember how to divide fractions involving roots? The bit you need to recall is that $2$ is just $\sqrt{2}\times\sqrt{2}$. This means that $$ \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{\sqrt{2}\sqrt{2}} = \frac1{\sqrt{2}} $$ when you simplify by canceling like terms on top and bottom. Then, taking the ratio of the long: short sides gives $$ 1: \frac1{\sqrt{2}} $$ Multiplying both sides by $\sqrt{2}$ gives a simpler form, which happens to be the same ratio as the original, large piece of paper: $$ \sqrt{2}:1$$ And no matter what the paper size, the math still works. Now that's a system. Folding an A4 and rotating 90° gives an A5, etc. As always, wikipedia is your friend.
A4 is 21.0 x 29.7 cm, so it's narrower than US Letter paper by 1.23 inches. Which is a perfectly sized strip for hexaflexagon folding.
I haven't decided on whether or not to hold a hexaflexagon party. It might have to wait until next October.
