Hexaflexagon Maker

A hexaflexagon is a flat paper hexagon made of six triangles. Pinch it, push the middle, and it opens like a bud into a face that was not there a moment ago. Do it again and a third face arrives.

It looks like sleight of hand. It is closer to bookkeeping: there is more paper in the hexagon than the hexagon needs, the surplus is folded out of sight, and a twist in the band decides which part is hidden at any moment. Once that is clear, the right place to design a flexagon is not the hexagon at all. It is the strip.

A strip of spare paper

The story of the discovery has been told the same way since 1956, when Martin Gardner wrote it up for Scientific American. Arthur Stone, an English graduate student at Princeton in 1939, found that American notebook paper was too large for his English binder. He cut strips off the sheets and, for amusement, folded the offcuts. One strip, folded at every third crease with its ends joined, made a hexagon that opened to a new face when pinched.

Stone then worked out a longer strip that gave the same hexagon six faces instead of three. He showed both to friends, and a Flexagon Committee formed: Stone, Bryant Tuckerman, Richard Feynman and John Tukey. Conrad and Hartline, writing in 1962, say the four had laid the groundwork of a theory by the early 1940s and that it was still unpublished. Then the group disbanded, and the flexagon was nearly forgotten for ten years.

The name began as the group’s flexible hexagon. Hexa is for the six triangles of a face, flexagon for the ability to flex, and a prefix counts the faces: the first model is a trihexaflexagon, the second a hexahexaflexagon.

A few very brief articles appeared in mathematics magazines in the early 1950s, and two Pennsylvania inventors applied in 1955 for a patent on it as an amusement and advertising device, granted in 1959 and now listed as expired. The toy they sold did not catch on. What made the object famous was Gardner’s article of December 1956. It was received so well that the publisher asked him for a regular column, which began in January 1957 and ran for twenty-five years.

Why a twisted band has a face to spare

Count the paper. A trihexaflexagon is folded from a strip of ten equilateral triangles, of which one is a tab glued onto another. That leaves nine leaves, each with two sides: eighteen triangle sides. A hexagon shows six on top and six underneath. The remaining six are folded against each other inside, and they are exactly one more face.

The fold that manages this is simple to state. The strip is folded at every third hinge, always turning the same way, and the ends are joined. Conrad and Hartline, whose 1962 report is still the clearest technical account, insist on the consistency: fold one hinge the other way and the result loses its symmetry and will not hold together. Pressed flat, the band has six piles of leaves round the center, alternately one leaf thick and two.

What you have made is a relative of the Möbius strip, the one-sided surface that Möbius and Listing found independently in 1858. An ordinary Möbius strip has one half twist. The trihexaflexagon has three, so it too has one side and one edge. The six-faced model is different. Its strip keeps two distinct sides, with faces 1, 2 and 3 on one and faces 4, 5 and 6 on the other.

Flexing moves the twist along the band. Pinching folds the hexagon into a three-bladed shape in which each blade is a small stack of leaves. That stack can be opened flat again in more than one place, like a book opened at a different page. Open it where it was closed and you are back where you started. Open it at the other place and the leaves that were face to face inside are now the top.

Six faces from nineteen triangles

The hexahexaflexagon doubles everything. Its strip has 19 triangles, again with one tab. First it is wound on itself: the triangles are folded in pairs, back to back, all the way along, which coils the paper into a spiral that looks just like the ten-triangle strip of the simpler model. Then that double strip is folded into a hexagon exactly as before. The piles are now two and four leaves thick, there are eighteen leaves and thirty-six sides, and thirty-six is six faces.

Most templates share one numbering. One side of the strip is marked 1, 2, 3, over and over. The other side is marked in pairs, 4 and 4, 5 and 5, 6 and 6, so that the first winding hides every pair. If the folding is right, one side of the finished hexagon is all 1 and the other all 2, and the first flex opens to 3 or to 5.

Here is a worked example. With triangles 40 mm on a side, the ten-triangle strip is five and a half edges long, 220 mm, and fits along a sheet of A4. The nineteen-triangle strip is ten edges long: 400 mm, far more than the 297 mm of the sheet. The hexagon it folds into is only 80 mm across. A strip that long has to be printed in two pieces and joined, or printed smaller; with triangles of about 25 mm it fits on one sheet.

Straight strips are only the beginning. Longer straight strips give face counts in multiples of three, and strips that are not straight fill in the numbers between, such as four or five. Collectors still publish templates for these, and for seven-faced versions that fold two different ways from one pattern.

The map, and why three faces keep returning

Random flexing soon shows that the six faces are not equals. Faces 1, 2 and 3 turn up constantly, while 4, 5 and 6 seem to hide. A casual flexer can go round the same three for a long time.

Tuckerman’s answer was a procedure. Flex at the same corner until the flexagon refuses to open, then move to the next corner and carry on. This Tuckerman traverse visits all six faces in 12 flexes, and in those twelve the faces 1, 2 and 3 each appear three times while 4, 5 and 6 appear once each. Written out, one published cycle runs 1, 3, 6, 1, 3, 2, 4, 3, 2, 1, 5, 2.

A drawing explains the imbalance better than a list. Mark a point for each face and join two points when those faces can be front and back at the same time. The trihexaflexagon gives a single triangle. The hexahexaflexagon gives a triangle 1, 2, 3 with a smaller triangle built on each of its sides, and 4, 5 and 6 are the outer tips. Each line is one state of the paper, nine in all. The three inner faces sit on four lines each, the outer ones on two. Where a state lies on two triangles, the flexagon opens at either set of corners, and that choice is the only way out to a tip.

What happens to a picture

Put a single color on each face and none of this shows. Put a picture on a face and something new appears: the picture does not always come back the way it left.

Each triangle of a face has three corners, and different flexes bring different corners to the center. Conrad and Hartline suggested marking the corners with dots to watch it happen, and mentioned pictures used the same way. On the trihexaflexagon every face has two arrangements. On the hexahexaflexagon there are 18 conceivable combinations of face and central corner, and the straight-strip model reaches 15 of 18: three for each of the frequent faces, two for each of the rare ones. In the other arrangements the six wedges of a picture are all present, but each has swung round so that the old rim is at the middle.

So a design has to choose. A picture can be whole in one arrangement, which should be the one seen most easily. A pattern made of rings or of six equal parts turns into a different but still orderly pattern. Text is unforgiving. This is the reason to plan on the strip: each face is cut into six triangles, and each triangle has one exact place on the front or the back of the paper, at one exact rotation. Get one wrong by a sixth of a turn and the face never assembles.

Clean folds, and the limits of a diagram

The paper matters more than the pattern. Crease every line between triangles in both directions before any folding, because every hinge will bend both ways in use. Ordinary office paper works; stiff card resists and cracks. Use little glue, only on the two blank triangles, and let it dry before the first flex. A new flexagon is tight. It loosens after a dozen flexes.

Two-sided printing is the usual weak point. Front and back must line up closely, and a printer can shift the second side enough to show, so print one test sheet and hold it to the light before cutting. If the printer cannot manage it, print front and back side by side on one face of the sheet and fold the strip over on itself along its long edge.

A diagram of states also leaves things out. It treats paper as having no thickness, and it follows only the pinch flex. Real flexagons can be forced through other moves by bending the leaves, and those lead to arrangements, and sometimes to muddles, that the neat map of nine states does not contain. The map tells you where the pinch flex can take you. Your fingers will sometimes find somewhere else.