It may be noted that in order to flex twice successively you must rotate the flexagon 60° about its center (see figure 7) before the second flex. This gets the flexagon into the right position for the next folding-together. If the flexagon is folded together without the intermediate rotation, all the hinges are folded the other way, the important symmetry is lost, and we can't flex.
What is there about the rotated flexagon that allows it to flex, while the unrotated one cannot? It is the slit between the two triangles in the double-thickness pats. This slit has been fancifully named the ``thumbhole,'' because, naturally enough, it is the hole where the thumb may be pushed through the front of the flexagon to emerge at the back (see figure 2 A). Each thumbhole enters at one side of the double pat and leaves at the other side. What is more, all thumbholes pass through in the same direction (clockwise or counterclockwise) in any given flexagon.
If each single pat could be given an extra triangle, it would no longer be necessary to rotate between flexes. There would be a thumbhole available to make the triangle-shift no matter where the flexing folds should be made. All the pats would be the same.
Adding in these extra triangles necessarily involves adding extra twists. When we have finished the job, our flexagon resembles figure 8. Now, at last, we can perform the long sought-after flexing operation, and we are probably no longer able to be too surprised when a wholly new side is exposed.
Suppose the side exposed when the extra thumbholes were added was labeled ``1.'' Sides may, of course, be labeled with color, a marking code, or any similar thing, but numbers are most useful. Instead of going on to the old side, ``2,'' we detoured to a new side which we will call ``4.'' Another flexing brings us back onto familiar ground with side ``3.'' This is not too startling, because the position that would flex to side ``2'' from ``1'' and the position that flexed to the new side from ``1'' are identical. Therefore, when we rotate between each flexing, as we did in the unaltered flexagon, our new cycle operates just as the old one did:
when turned over. Each time we pass between sides ``3'' and ``1,'' we are given a choice where to go next: ``2'' or ``4.'' All of this is concisely shown in the map of the new flexagon, which appears in figure 9.
When we use one cycle of this flexagon - either 1 2 3 ... or 1 4 3 ... - the twists and extra triangles which are used in the other cycle remain together, undisturbed. The only changes that occur in the use of one cycle are between the two halves of the thumbhole-pat in use and the other pat.
We should note at this point that, if the new flexagon is unwound, it is seen to be made from a crooked strip of paper (see figure 10). This in no way impairs its function as a Moebius band, for any topological figure can be bent, squeezed, stretched, or shrunk. In this case it is actually advisable to use a crooked band, so that the finished flexagon will fit together. Otherwise, it turns out looking like an octahedron with a pair of opposite faces knocked out. The fact is that most flexagons must be made from non-straight strips, or ``plans.'' The ``plan'' is the layout of the unfolded flexagon. Some plans, as we will see, take on rather amusing forms.
What can be done once can be done again. Therefore, each time a side is exposed there is a choice of which way to flex next - whether to rotate or not. And each time, if necessary, a new set of triangles can be added in to permit further flexing. Here, then, is a method of building a flexagon of any size. We can add in sides as long as we want to do so.
In order to have a thumbhole, there must be a hinge opposite the central vertex of the pat containing the thumbhole. Only in cases where there is no such hinge already in place (that is, an already existing thumbhole) can one be added. If there is no hinge here, the only hinges must be those leading to the adjoining pats, and the given pat must therefore be of a single thickness.
With this small fund of knowledge, let's return to the map. As has been seen, we have but two choices, in any case, in leaving any given side. These consist of going to one or the other of the two ``sides'' on either side of the map line which was last traversed (see figure 11). The rule of thumb is that the last side visited, the side next to be visited, and the side being visited must be the vertices of a triangle in the map (figure 11).
Then the positions at which we can add new sides are those at the edge of the map, where each map line is a member of only one triangle. Adding a side in the flexagon is represented by adding a new map triangle at the corresponding map line, which will be found at the edge of the map. All outer edges of the map are representations of positions in the flexagon at which three pats are of single thickness.
Suppose we want to expand the four-sided flexagon by adding in a new side, ``5,'' between sides ``1'' and ``4'' on the map, as in figure 12. (We can omit arrows from the map henceforth, since they only clutter it up).
The first step is to flex to side ``1.'' Then travel along the line leading to ``4.'' From here we should be able to reach the hypothetical side ``5,'' according to the rule of the map. That is to say, ``1,'' ``4,'' and ``5'' form a triangle over which we can travel. Three of the pats at this position will be found to be of a single thickness. Mark them, cut open and unfold the flexagon, and make them double. This method, shown in figure 13, is about the easiest way to add in new sides.
The new side which will turn up will be formed from the newly created faces of the doubled triangles. If the old flexagon was numbered or painted, the pattern will have to be repaired, since we did not actually produce the new thumbhole by splitting a triangle. The new addition will have disturbed the coloring scheme.
To fold the flexagon back up, first fold together the new triangles, as in figure 13 B. This will require a kind of winding motion. Then, being careful to continue winding the folds in the same direction, fold the flexagon up the same way it was dismantled. If you feel at this point that the flexagon has become too messy in the doubling process, it is relatively easy to make a new five-sided flexagon plan from the pattern of the old one.