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Scrolling flexagon

\put(0,0){\epsfxsize 120pt \epsffile{fig21.eps}}

The last of these maps was discussed already, being a nice example of a design which can be extended indefinitely. In other words you just wind the map around and around its center, which means that the squares in the middle map overlap and the Tukey squares have to be figured out carefully. On the other hand, once it is clear how to make the polygon list and place its signs, it is no longer necessary to draw the Tukey squares in detail.

It also has the nature that by turning the flexagon over after each flex, the segments can be built up in fanfold style which means that many more colors can be incorporated into the flexagon than otherwise possible. Of course, if the precaution of turning the flexagon over is not taken, the result may be so bulky that no more folding is possible, and the mistake will have to be undone before proceding. Turning the flexagon over reverses all arrows in the map; that way it is possible to pass through junctions which otherwise would be blocked.

The first of the four maps, the straight line, was also mentioned. It is indeed possible to work through all the colors by judicious turning over, but the flexagon it defines is not particularly elegant. Still, the sequence works.

Making a flexagon from the second map is a useful exercise, but the result doesn't show any noteworthy symmetry. On the other hand, the third map, which was not yet discussed, leads to a scrolling flexagon. Making one of several turns leads to a thorougly unmanageable wad of paper, but the principle is most definitely interesting. And it works well enough for the map shown.

The Tukey squares and their duals are shown in the following diagram:

\put(0,0){\epsfxsize 240pt \epsffile{fig22.eps}}
with polygon list

+ + + - - - + + - - +
1 3 3 9 7 7 9 1 5 5 1
2 2 4 8 8 6 10 10 6 4 2

In turn, the frieze for one of the two required segments is

\put(0,0){\epsfxsize 180pt \epsffile{fig23.eps}}

To get scrolling, not all the colors are exposed, but rather six of the ten are taken in the sequence 3 - 4 - 1 - 6 - 9 - 8. With that restriction they can just be rolled out in sequence.

It is interesting to observe the duality which persists through all the flexagons made with differing polygons - triangles, squares, pentagons, and so on - that a frieze which runs along at a natural angle folds up into a flexagon which has a complete tree, but if the map runs along at this natural angle, the flexagon meanders back and forth before it is folded up, and scrolls. But if the map runs around in a circle, the frieze is still made of natural segments, but they join at a kink. And the flexagon can be made to flex by fanfolding.

With experience, one acquires a repertoire of folds and maps, which can be used to get combinations of fanfolds, scrolls, and full behavior from some particular level. All this is particularly striking for triangle flexagons, but they are hard to type in ASCII. But the square flexagons perform just as well, and with patience so do the pentagonal flexagons. There is nothing wrong with higher polygons, but they begin to approximate circles, which makes them both bulky and harder to manipulate.

Since these special cases are so regular (of course, that's why they have been chosen) it is not overly difficult to work out simple mathematical formulas to express their properties. For example, with the spiral Tuckerman Tree we have

length plus minus
4 5 5
5 8 5
6 5 7
7 10 6,
a pattern which can be checked by looking at the Tukey squares.

The best way to get this information is to construct some examples, make several members of the family, and compare them to see how going from one member of the family to another changes both the map and the frieze.

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Pedro Hernandez 2004-01-14