Two Hexaflexagons

Flexagons have been around since 1939.  The most popular iterations are the Hexaflexagons, or those shaped like a hexagon.  By judicious folding of a model, one can reveal all of its faces, and in different orientations.  The effect seems almost magical, and creative coloring or drawings on the faces can make for some highly entertaining results.

There is considerable detailed historical information and examples of these devices, so I'll not redundantly repeat it here.  If you are unfamiliar with Flexagons, then this is a good place to start reading:

[https://en.wikipedia.org/wiki/Flexagon]

I have been constructing hexaflexagons for friends, most of whom never had seen such a toy, since before Martin Gardner made them famous in one of his marvelous books.  Back in high school, my brother came up with a mimeographed, type-written sheet of instructions for making many different models.

To that end, sturdy materials are required.  A flexagon made simply by folding a strip of paper will not last long, and such a unit can be impossibly difficult to fold or even construct if it has a lot of faces.

My tried-and-true design is as follows:

1. Get some sheets of poster board at your local hobby store, in colors of your choice.
2. Get a roll of ½-inch Filament Tape, which is very strong.  Regular Scotch Tape will break after just a few folds.
3. Cut the equilateral triangles for the faces, 2½-inches on a side, from the poster board.
4. Tape the triangles together in the requisite pattern, leaving a ⅛-inch gap between them; this enables them to flex without binding.
5. Affix 2-inch lengths of tape to both sides of the pieces, so that the sticky sides of the tape are covering each other.
6. After folding as per the instructions, taping the last two pieces together can be a bit tricky; but you can do it.
7. Using this method, the end-piece marked '(blank)' is unneeded; do not include it in the pattern.

Using the recommended materials, I have created models of up to twelve sides without any flexing issues.  Be aware, however, that because of the gap (hole) in the middle, it is easier than usual to jam the unit in mid-fold if not manipulated carefully.  Having done so, there is no recourse but to unravel the thing and re-fold it.  To help with such a reconstruction, I pre-mark the spot where the unit is taped last, because that is the best place to 'break' it for repair.

I have included specifications for two models — the first because it is my favorite, and the second because I believe that nobody else has posted instructions for it.  The numbers are provided for folding purposes; they can be written on the faces with a light pencil, then erased later.

SEPTAHEXAFLEXAGON — 7 FACES

• Fold each pair of 7's together.
• Fold the first pair of 4's together, then the first pair of 5's, then the first pair of 6's.
• Turn the whole thing over and fold the pair of 1's together.
• Turn it back over and fold the next pair of 4's together, then the next pair of 5's, etc.  Continue like this to the end.
• (Do not fold together any pairs of 2's or 3's).
• Glue the two blank sides together; or if not using blanks, tape at Sides 1 & 6.

If you employ a color-scheme similar to the one shown, then each of the seven faces will appear with a unique pair of alternating colors on the pieces.  I like to put red and green on Side #7, because I frequently place a Christmas-related greeting there.

Here is the Tuckerman Traverse (cheat-sheet) of the possible moves on this design:

HENDECAHEXAFLEXAGON — 11 FACES

• Starting at the left, 'roll' the strip by folding the 7's, 8's, 9's, 10's, and 11's together.
• Fold the 4's, 5's, and 6's together.
• Fold the pair of 1's together.
• Put some sort of clip on this section to prevent it from unraveling, then continue like this to the end.
• Glue the two blank sides together; or if not using blanks, tape at Sides 1 & 6.

The Tuckerman Traverse:

I also have instructions for Hexaflexagons of 4,5,6,9,10, and 12 faces; but those already have been published elsewhere.