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Mapping the flexagon construction

Given a flexagon, the middle map level is the one that would probably get drawn up, because the flexagon sprawled out on the tabletop is the first thing you would see, unless you were already holding it in your hands. And even then, whatever was visible would look like a face'' and would be the thing you would want to keep track of. Coloring would aid this considerably, although setting down numbers would be more readily available and appeal to the mathematically minded: face 1, face 2, ... .

Writing the first number - say 1 - on a piece of paper would be the way to start the map. Doing whatever can be done to the flexagon will make another face, and give the chance to write another number, both on the flexagon itself and on the map. On the map it could be joined to the other number by an arrow, to remember which came first. If the flexagon can be adjusted in several ways, several numbers will result, all connected to the first by arrows, but not (yet, anyway) to each other.

The flexagon can be abused, and by beginners often is. That is to say, changes in its layout can be forced, it can even be torn or ripped; so some judgement has to be used as to what is a natural movement and what is not. For example, with pentagonal flexagons, a certain amount of coercion is required, but is is very minor in comparison to breaking the flexagon. And even if no stretching or tearing is involved, ploygons can sometimes be slipped sideways relative to one another, to get something which looks and feels reasonable, even though it is more congested. Such have been called slipagons'' and discussed in articles and patents.

Laying out the map can continue, by setting down new points with their labels and connecting them to the previous points. Two things can happen during this process; no way to continue, or arrival at an earlier place. A place where no continuation is possible is called terminal, whereas coming back to an earlier point makes a loop (or cycle). For flexagons it turns out that there are only loops, no terminals, as a consequence of their recursive construction. But for maps in general, there can be points at which there are no arrivals as well as there being no points without continuations. There can also be groups of points internally connected without connections to each other. But with flexagons, patience will lead from each face to any other.

Just drawing the map with reference moving the flexagon around, lines may cross over but that would be the result of a first attempt. Redrawing the map can eliminate crossings, but that is characteristic of flexagons; there are maps for which it is not possible. Try connecting three points with each of three others.

Once it is apparent that there is a map of mutually tangent loops, it is time to draw the higher level map where the cycles are replaced by points with lines connecting cycles which have edges in common. For historical reasons this map, or graph, is called a Tuckerman Tree. Even though using it gives a good overall picture, there are still rules for connecting the cycles together, which are a little more complicated than just drawing linking lines, or even lines with arrows on them. That is because the middle level has arrows, and they have to be respected. So a number can belong to several cycles, but the only allowed cycle switching is one which respects the arrows.

The map which comes the closest to the actual polygons is the third, or lowest, level map. That one takes into account what is visible both on top of and underneath the flexagon, and is the one which reflects both the base of a cycle representing a polygon stack and the recursive step of removing o polygon and replacing it by its counterspiral. That is the diagram made by diamonds in the examples.

Although there are maps which help with designing and understanding flexagons, it is still necessary to pay attention to them and learn the details of their use. For example, to get a flexagon for which it is possible to run through a sequence of colors without repetitions, there are four different ways to put four cycles in line:

Actually there are more than four, but the new ones that you can get by rotation or reflection don't lead to essentially different flexagons. That is, the results are also either rotated or reflected. So what is important is the angle between the links, because that decides which portion of the loop has been extended, so that it has another tangent loop. Flexing is accordingly different in maps where the angles are different.

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Pedro Hernandez 2004-01-14