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Primary Flexagons

One of the best ways to work with square flexagons is to make up a good collection of dominoes - that is, 2 x 1 rectangles folded in the middle. The idea works with any polygon. Even though it requires pasting every polygon on top of another one, folding the structure is manageable enough with the advantage that the frieze can wander around every which way and even overlap itself. Note that if the polygon is not regular, different dominoes may have to be prepared, according to the edge that will be folded or have been reflected while constructing the frieze. Multiple dominoes can be made simultaneously by tracing outlines on a sheet of paper and then cutting through several sheets placed on top of one another. Up to four seems to still cut nicely.

By now, fairly short strips made with several different polygons should have been made and tested, with emphasis on strips just long enough to make two turns for a polygon stack, although triangles may require three turns to get good results. More turns are possible, leading to rosettes whose blades will have a single turn, and they are all interesting. In the beginning, strips should still be kept short. It should also have been noticed that when an extra polygon has been included in the stack, it can be pasted onto the top of the stack rather than being left on the bottom. This is supposing that we look at the stack from top down, which is convenient.

If the stack is not too high, the bottom can be pasted to the top while keeping a single stack, but the stack can be opened in the middle leaving two half-stacks lying side by side and the extra polygon in a favorable position to mate it with the first and glue them. Being able to do this is the reason that two turns of the spiral were called for rather than just one. There is no difficulty in seeing how to connect multiple turns if a longer frieze was initially prepared.

It should be emphasized that the whole stack should have been prepared by fanfolding. That is hardly the only way the stack could have been folded, but it is the one that we want to start with. Fanfolding the minimal number of turns gives a primary (or first level) flexagon, of which several kiinds should have been constructed and experimented with.

The experimentation consists in noticing the behavior of the pair of stacks in three dimensions. For some flexagons, additional hinges will have become unblocked and four stacks can be laid out side by side in a plane, surrounding a central point. This is the point where the rule about angles summing to 360 degrees comes into play. If they do not, the sum may still be approximate and one can work with a puckered flexagon. Such is the case when using regular pentagons, for example.

Another possibility is to begin with more turns in the original polygon stack; a necessity when using regular triangles, for example.

Without reference to whether the figure is planar or not, there is always enough flexibility to take the top polygon off one side and make it the top polygon on the other side, which has been turned over. Except for the turning over, it would have been the bottom polygon. The overall result of this operation is that the place where the single spiral has opened out to become double has advanced. Repeating the maneuver will advance any polygon that one wants to the top of the stack and make it visible. If all the polygons had been painted with different colors, the effect would be one of running through all the colors in sequence.

There is a nice diagram which summarizes this result, as well as taking the color into account. It is easier to number the faces of the polygons than to remember colors. Consider using regular pentagons with five sides. The diagram reads

1 . 3 . 5 . x
. 2 . 4 . 1 .

The reason that it has this strange form (two rows separated by a horizontal line) is that before folding the faces of the polygons in the frieze would be numbered 1, 2, 3, 4, 5, x on the top side of the frieze. The x is for connecting the first turn to the second turn. But after fanfolding, every second number goes to the bottom (as seen by the person doing the folding) of the visible polygon rather than the top. So, the reason for two rows is to show both what would be on the top and what would be on the bottom of each polygon, and that is why the numbers alternate from top to bottom.

What numbers should be written on the opposite sides of each polygon? That depends on the fact that two turns are needed to get a stack that can be unfolded, and after the two turns have been laid out side by side and there ends joined, we want to see the same color on the top of each spiral. Or on the bottom, if we look at it from below. But these pairs are consecutive polygons in the starting frieze and so should have consecutive numbers. Thus x should be what follows 1, or 2. In similar manner, 1 should be added to each number to get the number written on the other side of the dividing line. In other words,

1 3 3 5 5 7
2 2 4 4 6 6

This means that x would be 7, but since one turn of a spiral has five polygons, numbering should start over again with the sixth to give the sequence 1, 2, 3, 4, 5, 1, 2, ... . That means x = 2, not 7. With this correction, the table reads

1 3 3 5 5 2 $\cdots$
2 2 4 4 1 1 $\cdots$

The table could be continued for as many turns as needed, but it is customary to show it for a single turn with that vertical line marking where the cycle ends; plus the next polygon's label for convenience.

All of this probably seems terribly simple and obvious which, for first level flexagons, it is. Almost. Using two rows is important, and adding 1 is important. Even though the flexagon is rudimentary, these details should be checked out and verified until one feels comfortable with them.

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