New Angles

The theories concerned with the flexagon committee's ``flexible hexagons'' seemed at first to be a fairly complete description of the flexagon family. Each assumes a more or less clear-cut definition of the flexagon, from which it proceeds to deduce certain properties. However, it has been found that, by altering the definitions used slightly, whole new ``dimensions'' of flexagons can be discovered.

It may be best to clear up the haze surrounding the definitions of flexagons now before they are extended any further. It should be clear that the conditions of any definition will be determined by the object itself. It will be impossible to give an absolute definition of flexagons since it is a matter of taste what one wishes to include in this category. As we proceed, we will come to count more and more strange figures under the general heading ``flexagons''. Most of these are not included by definitions already considered; their only justification is the interesting results they produce. Of course, reason dictates that only figures related to flexagons or including flexagons as a special case should be considered.

The first extension to be added in this section is hinted at by the pat structure definition: A flexagon is an ordered pair of pats. This pair is repeated twice to form the actual paper flexagon. Why then can we not build a flexagon with the pair of pats in duplicate? Or in quadruplicate? It is soon found that there is no reason at all, flexagons can be made with any number of units. More than four units cause appreciable overlapping of pats and produce relatively unstable models, but are theoretically practicable. The four-unit flexagon does not lie flat, but this makes it all the more fascinating, One-unit flexagons must be operated in the imagination, or else cut $N$ different times to reveal all the faces.

The two-unit ``cup'' flexagon is built by joining the pats to form a tetrahedral angle. Since the vertex cannot be ``pushed through'', placing the inner surface on the outside, the outer side remains constant. Flexing is always accomplished by pinching together pairs of adjacent pats and occurs between the constant outer side and the variable inner side. The map, therefore, presents the appearance of spokes emanating from the central point representing the constant side, since no outer faces or paths can be used. Because arrows adjacent along the spokes point in different directions, the flexagon must be turned over after each flexing to use the next spoke. The obvious way to take advantage of this is to build a two-unit fan flexagon, which will have no unused sides.

The construction and operation of this fan flexagon suggest a further possibility. During the entire operation of the flexagon, the stacks of leaves remain on the outer side, slowly shifting between alternate pats. These pats, due to the fashion in which they have been built, can be constructed to incredible length. Until now, the physical limitation of paper thickness had made it impossible to construct flexagons of orders greater than 48 or so. However, utilizing the principle of the cup fan flexagon, we can now build 2- (or 3-, if flexing is restricted to the ``cup'' pattern) unit flexagons of unlimited size. A three-unit flexagon of order 658 has been built by the authors; all the sides can be exposed but only $N-1$ paths may be used, instead of the usual $2N-3$.

Another possibility, related to the cup flexagons, is the mixed flexagon. Mixed flexagons are based on the observation that the units used need not be identical. If they are different, each will have its own map, and the map of each will partly coincide with those of the others. If a11 three coincide at any face, this face will open out and lie flat. For faces where only two of the maps coincide, the third unit will not open out, and the flexagon will look like a cup flexagon. If only one map occupies a face, it will not open out at all. To build mixed flexagons, the three maps must be drawn with differently colored pencils, overlying in the desired way. In copying down the sequences, all proceeds as usual except that after each trip about the Tukey triangle network, one must change maps. The tukey triangle network is drawn over all the maps, but only the portion corresponding to the map being used is followed at a given time. A simple mixed flexagon is shown in fig. 5-1. Mixed flexagons may be extended considerably by the use of more than $3$ units.

2	2	3	1	5	3	4	4	6	6	2	2	4	4	5
3	1	5	6	6	1	5	3	1	5	3	1	5	3	1
+	+	-	+	+	-	+	+	+	+	+	+	+	+	-

Figure 5.1

Just as the pat structure definition does not mention the number of units in the flexagon, so also it neglects to say that the leaves must be equilateral triangles. In dispelling this requirement, we open up another new dimension.

The relative position of the vertices of the leaves within any given flexagon is constant. Therefore, varying the angles of the triangles used as leaves should make no difference in the operation of the flexagon. For instance, a 45$^{\mbox{o}}$ - 45$^{\mbox{o}}$ - 90$^{\mbox{o}}$ flexagon of order 6 may be constructed quite easily, by the doubling-over technique, from a strip such as the one shown. (fig. 5.2) We have seen, by means of ``dots'', that various angles occupy the center of the flexagon at various times. Varied angles provide an excellent form of ``dot''.

When the 45$^{\mbox{o}}$ angles of the 45$^{\mbox{o}}$ - 45$^{\mbox{o}}$ - 90$^{\mbox{o}}$ flexagon are at the center, four units must be used, in order to prevent cup-flexagon-like operation. When the 90$^{\mbox{o}}$ angles are pointed toward the center, there is, therefore, a surplus of 360$^{\mbox{o}}$. A number of things may be done about this. The most obvious is to ignore it and flex on. In this particular case, however, opposite units may be laid flat and, with a half twist, the two middle units may be opened out flat. The final product is a bicolored rectangle (fig. 5.3).

This is typical of a second type of distortion of the flexagon. A flexagon with more than 360$^{\mbox{o}}$ about its center may always be forced to lie flat by the following procedure: Simply fold a unit together, then fold together the next two pats on either side of the first two, etc., just as was done to distort the structure of the 3-equilateral flexagon,.

With care a flexagon of any number of sides, any number of units, and any shape of triangle may be built. However, beyond order 3 in flexagons of many units the structure is generally so easily distorted that the flexagons are not worth the trouble of construction.

The irregular strip used in making a scalene or isosceles flexagon of order 3 corresponds to the straight strip in equilateral flexagons. A process has been developed to simplify the making of what will be called by definition a ``straight'' strip; i.e., the strip of leaves corresponding to a straight chain of equilateral triangles. Draw any triangles and label the vertices $A$, $B$, and $C$. Then construct

$$\begin{array}{lcl} \triangle A' B C & \cong & \triangle A B... ... \triangle A'' B' C' & \cong & \triangle A' B' C' \end{array}$$

etc. as in figure 5.4. This is continued until enough leaves have been produced. To find by construction the number of leaves required for all faces to have at least 360$^{\mbox{o}}$ about their centers, encircle the vertex of the smallest angle and measure off around the circle to find the minimum even number of arcs, equal to the arc of the smallest angle, which will cover the circumference. $3/2$ of this number is the required number of leaves. One of the most interesting of these flexagons, because of the distortions possible with it, is the 30$^{\mbox{o}}$ - 60$^{\mbox{o}}$ - 90$^{\mbox{o}}$ flexagon of order three. The five unit star-shaped flexagons are also amusing.

Notice that as any one of the angles in the leaves approaches zero, the number of units required to make all faces have at least 360$^{\mbox{o}}$ about their centers becomes larger and larger. It soon becomes most practical to abandon some of the faces by using some arbitrary small number of units, but at least 2. Suppose, then, that we let the arms of one angle swing all the way around to a zero degree angle. This should not disturb us, with, say, 2 units, any more than a 1$^{\mbox{o}}$ angle would. The problem now is how to operate with leaves having parallel edges infinite in length. This is easily solved by chopping off the 0$^{\mbox{o}}$ vertex at infinity, just as we chopped off vertices of ordinary flexagons to prevent binding during flexing. The infinite vertex may as well be chopped off so as to leave each leaf square, as in figure 5.5. In building flexagons like this out of squares, one must remember that one edge of each square leaf is theoretically a cutting off, and cannot be used for a hinge.

When an infinite vertex tries to come to the center of the flexagon, its parallel hinges will prevent it. It will form instead an infinitely long cylinder, or, in the case of our 2-unit flexagon made of squares, an open-ended cube (see figure 5.6). Needless to say, such faces will not ``push through'', or reverse inward and outward sides. Other faces of such a flexagon will appear as in figure 5.7.

There is no reason to stop at zero. Leaves with negative angles (duly chopped off) are completely acceptable, although the flexagons they make may at first seem confusing. These flexagons will actually be made from trapezoidal leaves. When the negative angle in not toward the center the flexagon will look like figure 5.8, and when the negative angle in ``toward the center'', it will actually be pointing away. In fact, since it points away, the flexagon may be made to lie flat, upside down, during such faces, as in figure 5.9. Thus these faces can be ``pushed through''.

A little experimentation will show how this can be the case, improbable as it sounds. An interesting example is the regular flexagon of order 6, built from the leaves in the unit plan shown in figure 5.10. It must be remembered, in flexing such a flexagon, that, when the negative angle is ``toward the center'', the flexagon is upside down: the ``upper'' side shows underneath, the ``lower'' one on the top of the flexagon. Still, in flexing, the ``lower'' side must be folded together, so that the ``upper'' side will remain during the next face; i.e., we must flex backwards.

It will be noted in some faces of this flexagon, as in all flexagons with more than 360$^{\mbox{o}}$ about the center, that these faces are rotated only with difficulty, and could not be rotated if made of rigid leaves. Thus, to rotate the hinges $AB$ of the face shown in figure 5.8 to the high position of hinges $BC$, the leaves must be bent considerably.

It is not hard to see how leaves with negative angles could be chopped up into pentagons, hexagons, and so forth (see figures 5.11 and 5.12). We have already used hexagonal leaves, made by cutting off corners of triangular leaves (see fig. 5-13). Thus we have already begun to relax the restriction of leaf shapes to triangles. However, so far neither the pat theory definitions, the map, the tree, nor the triangle network has needed alteration. Only the original descriptive definition, ``flexible hexagon'', needs changing. Now that we have seen how polygons of all kinds can be used to build flexagons, we will break the last remaining restriction in this direction, by allowing the polygons to have hinges on more than 3 sides. Doing this will give us another larger field of flexagons in which, at last, not only will leaves not necessarily be triangular, but map polygons will not necessarily be triangular, and neither will hinge network polygons. Moreover, flexagons will not necessarily be pairs of pats, as heretofore defined.