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# Strips of Paper

If one is going to paste and paint flexagons, it is worth adding an old newspaper or piece of carton or whatever to the list of supplies. That keeps from messing up the table top where one is working.

The best way to get started is to begin with something simple. The historical beginning was with a strip of paper folded into equilateral triangles. At first a simple loop made a hexagon which could be folded into a three bladed structure which would close up at one end and open out from the other. While doing so the faces of some triangles which had been hidden from view became visible, while others disappeared. Numbering the faces, or coloring them, revealed a cycle of three, which could be repeated over and over again indefinitely. Coloring with the primary colors fits in well with further experiments.

It is not too hard to make isosceles right triangles instead of equilateral triangles. The same technique of back-folding keeps good angles, and when the strip is finished, it can be twisted in one direction over and over again to make a square. It takes a little more care to join the ends than with the sixty degree triangles, but it can be done. Interestingly, this was a way to fold up a tabloid sized newspaper into a wad which could be thrown up on a porch without stopping when riding by on a bicycle. What the thickness of the wad obscured was the joining and subsequent flexing. Also, some new creases are required in order to fold the square into a hollow cylinder.

The outside of the cylinder can be painted, the figure unfolded back into a square, and then folded again on the other diagonal. This time the painted surface will be inside the cylinder, and the outside can be given another color. Coming back to a square, half will be painted and the other half not. This time fold the diagonal backward instead of forward (or the other way around, depending on the initial choice), and the cylinder will reappear with all the colored faces hidden. So a third color can be used, and then a fourth after returning to the square and changing diagonals. Finally everything has been painted and there is nothing more to do than to watch the cycle of folds.

Martin Gardner [2] described this under the heading of ``flexatube'' in one of his flexagon columns, emphasizing the way that the folding sequence could be used to turn a paper box (topless and bottomless) inside out without tearing or stretching it. But it is also part of the flexagon family.

There are other artifacts which can be gotten by braiding two or more strips of paper together (using rope, you get Celtic knots and the like). Most likely anyone who has sipped Coca Cola or a milkshake through paper straws has idly braided them back and forth to have something to do after the drink has been finished. Again, the figure is rarely ever closed into a circle, nor studied further, even then. Crossing the strips at sixty degrees gives one result, but crossing them at right angles makes another interesting structure.

It is not hard to take a very long braid and close it, but the shortest which seems to work uses a sequence of eight squares in each strand. Actually it is better to take a strip of nine, and paste the ninth onto the first to get a ring of eight. The first strip can be closed at once; but it is better, in fact essential, to intertwine the ring with the free strip, which goes well enough until the last two (plus tab) squares have to be put in place. Pressing the figure down into a plane, carefully making diagonal creases as needed in the squares (better to crease them all from the beginning, before doing any folding or joining), the end of the strip can be threaded into the folds in the ring, and the ends pasted together.

This is the structure which the flexagon book calls a bregdoid; although it is interesting enough to play with, it is not really part of the lore of flexagons. It also makes a hollow cylinder of squares, and so is related to the flexatube. More complicated arrangements result from adding multiples of four squares to the two starting braids.

Still working with paper strips, the original flexible hexagon (= flexagon) was prepared by folding the paper strip double, before joining the ends. So instead of a string of nine (plus a tenth for joining), a string of 19 (18 plus the joining tab) was first doubled up into 9, then closed into a hexagon. This is a process which is hard to resist, it is just a question of how the string of triangles is twisted. However, this is the point where one has the makings of a puzzle, because it is not so easy to locate and paint all the faces. In fact, Martin Gardner used it as a magic trick by pasting a photo on one of the faces and cutting it up according to the folds. It was then shown to someone with the photograph hidden, with the challenge to find it (``cherchez la femme'') and bring it into view.

Once the strip is folded double, there is no reason not to double it again (36+1) and again (72+1) and again. But at that point or even sooner, the folds of paper have become so thick that the flexagon is unmanageable.

So there are quite a few things which can be done just starting from a strip of paper. One of them is, having built a flexagon, to make a map of the faces, just as one would do when exploring Mammoth Cave; or in real life, finding your way home from the woods. It is also worth opening up a flexagon after it has been colored, to see just how the colors are arranged along the strip, and on which sides. So don't paste it too tightly at first, or use a paper clip instead of glue. Knowing that, you could color the strip beforehand, and avoid a certain amount of messiness. Or have the strip printed up, ready to cut out and fold.

Next: Stacks of Polygons Up: A Quick Flexagon Survey Previous: Materials and Motivations   Contents
Pedro Hernandez 2004-01-14