I’m not a math guy, so I don’t know if the topology of this thing is something special or if it just has an average amount of strangeness. All I do know is, even making something as simple and yet as deeply weird as a Möbius strip makes me worry that I ought not be playing around in a cavalier way with time and space, on account of who knows what portals might get cracked open, which as you know always leads to trouble.
Anyhow, as best I can figure it, a tetra-tetra-flexagon (which is apparently just one of many kinds of flexagons), works basically like this: make 3 mountain folds along the short or “vertical” dimension of the sheet, then make two parallel horizontal cuts as shown, then make one last vertical cut to form a tab. All of the folds, by the way, will flip from mountains to valleys and back again as you operate the flexagon; that’s normal behavior for this structure, but part of why it’s not super-durable. Still, it’s fun to experiment with the way it forms whole images from fragments. Plus, when you make one, note how it hides and reveals content as you unfold it.
Fold the tab back then bring it back up against the sheet to the right of the hole. Fold the square that’s sticking out back down onto the front of the sheet. You can already see at this point how the numbers will start to line up as the flexagon comes together.
You can now fold down the flap on the left side of the sheet. Bring it in under the sheet, then fold it back up so that a square of paper appears in the hole. Then fold the left side back one more time. There should now be 3 layers of paper on the left-hand side.
Flatten down the square tab of paper on the right and secure it to the matching square now visible in the hole. Use a piece of clear tape running in the direction shown along the join. Be careful not to tape anything together other than these two squares (the edge of a non-matching square is hidden just under this join).
To operate the tetra-tetra-flexagon, simply fold the two sides back along the central vertical fold. If you need to encourage the book to open the first time, you can do so by gently plucking apart the gap between the middle squares. Then repeat the procedure to access the content on “side 3.”
The fourth side is really just the back of the book when side 1 is visible. Once you reach side 3, you’re at the end of the hidden content. To get back to 1, simply flip the book over and reverse the folds twice.
I think this kind of project works best with plain old copier paper, but feel free to experiment with different weights of stock, because you never know. What’s really cool is to set up a design file with bits of imagery and text on two pages, and then print it out front and back as a mystery puzzle book. Not that it’s a huge mystery or a really hard puzzle, but this book is nonetheless always a little bit baffling even if you’ve made a ton of them. Also, of course, you can just collage a couple of flats and run them through a xeroxing machine. The PDFs below are for practice, although this is actually a really simple thing to make.