Hexaflexagons are paper hexagons folded from strips of paper which reveal different faces as they are flexed. The Hexaflexagon described here has six different faces, thus the name Hexahexaflexagon. An interesting chapter that describes Hexaflexagons can be found in Martin Gardner's book Mathematical Puzzles and Diversions, first published in 1959 by Simon and Schuster, and reprinted in both hard-cover and paperback by several publishers including Penguin Books (U.S., U.K. and Australia) and the Chicago Press. The book consists mainly of Gardner's articles published in the Recreational Mathematics column of Scientific American. Most public libraries in the United States either have this book, or can borrow a copy for you from a neighboring library. It is probably just as readily available in Europe and Asia. The copy I have was printed in the UK, and purchased in the Philippines.
According to Martin Gardner's account, Hexaflexagons began in the fall of 1939 when Princeton University graduate student Arthur H. Stone from England trimmed an inch from his American notebook sheets to fit his English binder. After folding the trimmed-off strips for amusement, he came upon the first Hexaflexagon, one that had three faces (now called a Trihexaflexagon). A flexagon committee was organized including Bryant Tuckerman, Richard P. Feynman and John W. Tukey. Larger structures were developed including the one variety of hexahexaflexagon described here.
The following diagram illustrates how to fold one variety of Hexahexaflexagon from a straight strip of paper.
Cut a strip of paper and divide it into 19 equilateral triangles as shown in (A) above. Number the triangles 1, 2, 3 on one side and 4, 5, 6 on the other. You may also use six colors or other geometrical shapes or symbols to identify the triangles. On the front side of the strip, the numbering pattern is 1, 2, 3, 1, 2, 3 etc., with the 19th triangle left blank. The blank triangle will be used to glue the contraption shut at the end. On the back side of the strip, the numbering pattern is 4, 4, 5, 5, 6, 6, etc. with the first triangle left blank. The blank triangles are on opposite ends of the strip, on opposite sides. If this is your first model, you might want to fold all of the edges both ways a couple of times to make folding and flexing easier later.
Next, fold the strip so that the numbers on the back side face like numbers, that is, 4 on 4, 5 on 5, 6 on 6. The result is shown in (B) above. In effect, the strip has been folded into a flattened spiral.
Fold the strip back at line A-B, and then back at line C-D in Figure (C). On the final fold at C-D, bring the tail end of the strip up in front of the head so that the triangle numbered 3 faces the other triangle numbered 3. The result is shown in (D). Finally, fold the last triangle flap over and glue the two blank faces together. If you did it right, one face of the hexagon will have all triangles numbered 1 and the other face all numbered 2.
The width of the original strip depends on how big you want the resulting Hexahexaflexagon to be, and how thick the paper is to start with. Heavier paper will require you to build bigger models for easier folding. For standard weight copier or printer paper, a strip of about 2.5 to 5 cm. (1 to 2 inches) wide should be okay. I use adding machine tape that is about 4 cm. wide. If you plan on making more than a few models, I recommend purchasing a pack of adding machine or cash register tape, and making a thin strip of metal or thin stiff plastic the exact width of the tape. If you wind the tape around the strip, you can quickly create the flattened spiral shown in Figure (B) above.
Start by pinching two triangles together along one of the edges. Push the outer corner of the opposite two triangles inward. Then, open up the opposite two triangles at the center to reveal an inner face of the Hexaflexagon. If they refuse to open, try pinching an adjacent pair of triangles. In the Hexahexaflexagon described here, if the model cannot open on one edge, it will surely open on an adjacent edge. If the model was badly constructed, it may be difficult to flex.
If you continue to flex the Hexaflexagon, it will eventually reveal all faces. For this model, faces 1, 2 and 3 will appear more often than 4, 5 and 6.
Besides being a very fascinating plaything for children of all ages, Hexaflexagons have been used for greeting cards and promotional materials. You can play tricks on some of your co-flexagators by gluing some faces together so they can't get to them. Some faces of the Hexaflexagon appear in two ways, so that a photograph cut and pasted on a face may sometimes appear correct, and sometimes appear incorrect as the corners that once appeared at the center of the hexagon now appear at the outer edge. If you are interested in other Hexaflexagon models, Martin Gardner's book is the place to start.
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Alejandro Kapauan
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Updated 1/14/96