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Stacks of Polygons

Having folded some paper strips into flexagons, painted their faces, and made a map of their interconnections, one might wonder about the range of possibilities. One of them is to try out different polygons, although small regular polygons such as equilateral triangles or squares are the easiest to work with and a good place to begin. Just folding a long strip invites using equilateral triangles, or with more care and patience, right isosceles triangles.

Since the individual polygons are folded over, preferably as a fanfold, to get the flexagon, it is not hard to see that successive polygons are reflections of one another. One way to build a more general, and possibly irregular, strip is to start with a polygon, preferably made by drawing one on a piece of carton and cutting it out. Then, it can be traced on a sheet of paper to get more copies of exactly the same form.

It is a good idea to number the edges in order, in the same way on both sides of the carton cutout, although turning it over will reverse the clockwise or counterclockwise sequence. Because the eventual string of polygons will be folded along the edges, the polygon ought to be convex. For three or four vertices they can't be made any other way, but from pentagons onward they could have concavities. Actually it is possible to make concave quadrilaterals, so we should say there are unusable polygons from three vertices onwards.

The reason for numbering the edges is to foresee that the polygons will be folded in sequence, so that running along a string there will be a last (that is, previous) one and a next one adjacent to whichever one is held at the moment. They don't have to be joined along consecutive edges, so the numbering helps to describe the jumps if there are any.

One of the best ways to begin is to use regular pentagons about two inches in diameter. The size should be something conveniently held in the hand, and for which several can be drawn on a sheet of normal letter size paper, which is the most likely material to have at hand. Of course, sheets of brown wrapping paper or anything else at hand can be used. Newspapers aren't printed on very strong paper, so they wouldn't work as well. Pentagon flexagons won't lie flat when folded, but almost so and strike a balance between simplicity and complication which makes their working much easier to understand.

To build the flexagon, trace out the first polygon near one corner of the paper using the template. Copy the numbers onto the tracing, preferably putting ``1'' at the bottom left, and tilting so that ``2'' will make the strip run off at a diagonal.

Now, putting the template back with the same side up, turn it over along edge ``2'' and trace out a new copy. The numbers which are visible should be transcribed onto the copy, being sure to use the ones which are now visible but which used to be on the bottom of the template. Put the template back, turn it over along edge ``3'' to get the top side back again. Trace it from the new position, and once again copy the numbers. Because of pen or pencil thickness, the template has to be positioned so that the connecting edge is traced in the same position both times. A little sloppiness won't matter, but reasonable care should be taken so that everything will come together smoothly in the end.

Some practice may be needed to get the figure to fit neatly onto the sheet of paper where it will be drawn. The result should run off in pretty much a straight line, even though it will be a little crinkly. Just how many pentagons should be drawn? To begin with, five or six; six will permit closing a ring of five by pasting the sixth over the first. It will turn out that closing the ring is something you won't want to do, and using a paper clip for the closing instead of pasting will show you why. As the construction goes on, you will want two rings, so it is better to make them both from the beginning from strips of six pentagons, and to join them later.

Once the strip, or frieze, has been drawn, it should be cut out along the edges and the extra paper discarded. The connecting edges should then be folded in both directions, so as to make the paper more flexible, and to locate the places where the frieze is going to be folded. Once that is done, the frieze can be fanfolded to get a paper spiral. Notice that when five pentagons have been so folded, if the first edge was marked ``1'' then the last edge is also marked ``1'' and is parallel to that first edge. The sixth pentagon can then be folded back up on top and glued (or clipped).

What has been constructed is a spiral of pentagons, making one turn after each five pentagons because the angle between folds was 360/5 or 72 degrees. That is the exterior angle; the interior angle of the pentagon is 180-72, or 108 degrees. If the frieze had gone on and on, the spiral would have kept on turning. The important thing to notice is that edges become parallel again after five folds, so that different full turns could be separated by folding the place where they are joined back into a plane. Attempting to fold before that won't work because the axes are skewed. That becomes obvious when one turn of the spiral can't be opened out because the axes block each other, whereas two or more turns can be opened out.

Upon opening, there are two spirals lying side by side. If the eleventh pentagon was pasted over the first, a turn consisting of five pentagons spiralling in one direction lies next to another turn of five spiralling back up in the opposite direction. The eleventh can now be pasted to the first without any reservations. At this point, we see that the joint to make the pair is operative, and that some movement is possible since the bottom of the first spiral now connects somewhere else, no longer blocking the other hinges.

Here is where the third dimension comes in handy, because the top pentagon from one stack can now be transferred to the bottom of the other, and the other way around. That is the same as if the stack had been started with the second pentagon rather than the first. Repeating the operation, each of the pentagons will be exposed in turn. Therefore if they had been given different colors, it is possible to fold the stack and run through the succession of colors.

On the other hand, if they hadn't been colored, now is the time to do so, either of which will make the changes which occur while flexing more apparent.

If three or more turns had been used, the result is still a spiral stack for which the turns can be grouped together and flexed. But two turns is enough. Had triangles been used, three turns would be required, which is why the explanation of this construction wasn't based on triangles. Interestingly enough, the first flexagon to be discovered historically is more complicated than the ones which came after, although it is much easier to construct and fold up.

Once the idea behind this construction has been grasped, it is easy to work with any (convex) polygon, so it is worth trying it out with squares, regular hexagons, and so on. For the ``so on'' it is better to begin with regular polygons, bringing up the question of how to make a regular heptagon, for instance. Also, beyond hexagons, the reflected polygons overlap which means slicing the overlap, or cutting them out in pairs individually, then pasting. But that all comes later on; one should make an effort to understand the pentagons first.

next up previous contents
Next: Folding and Flexing Up: A Quick Flexagon Survey Previous: Strips of Paper   Contents
Pedro Hernandez 2004-01-14