Zero Angles

Amid all our concern for face degrees equal to and greater than $180^{\mbox{o}}$, the reader may have noticed the conspicuous absence of $0^{\mbox{o}}$-faces, corresponding to $180^{\mbox{o}}$ leaf angles, which would complete the range of possible values.

Two equivalent types of $0^{\mbox{o}}$-faces can be built both of which involve placing the hinges in line with one another. The first method superimposes the hinges, as in fig. 11.1a. The second puts them side by side (fig. 11.1b). The only advantage of one over the other is that the second method allows us to reverse the position of the $0^{\mbox{o}}$ pat (or leaf) in respect to the two leaves it joins (See fig. 11.1b. Forcing is still necessary).

Clearly, a $0^{\mbox{o}}$-face will be made up of a set of flaps, joined along a line at the center of the flexagon. Since this is like the folded-together position in non $0^{\mbox{o}}$-face flexagons, we are tempted to open it out. Nothing prevents doing this, since the flaps of the $0^{\mbox{o}}$-face cannot lock sides out of sight. Thus most 0$^{\mbox{o}}$-faces are not only $0^{\mbox{o}}$-faces, but exhibit a third side as well, which may have any face degree; if $0^{\mbox{o}}$, we can open out again, until one face degree $\neq 0^{\mbox{o}}$ is obtained. In a sense, then, $0^{\mbox{o}}$-faces can be completely ignored in working the flexagon. On the other hand, since they are incapable of locking shut, $0^{\mbox{o}}$-faces tend to lessen the flexagon's stability greatly. For this reason most of the research done heretofore seems to have avoided them. However, Dr. F. G. Mannsell and Miss Joan Crampin ^11.1, present a notable exception; the irony of the situation being that their interest was apparently due to a misunderstanding of the significance of the regular triflexagon of order 6, which was presented in an article by Miss Margaret Joseph^11.1. The order 6 flexagon they discuss will, however, serve ideally for introducing $0^{\mbox{o}}$-faced flexagons. It is built from a straight strip of 18 equilateral triangles, folded up in the constant order 1 2 3 4 5 6. Thus its map is that shown in figure 11.2a. It is, of course, a hexa-flexagon. We were told that these could not be made of triangles without hinges overlapping: this is precisely what happens in this flexagon. A simple calculation shows which sides will appear together as $0^{\mbox{o}}$-faces: those connected by dotted lines in the figure. When we have built this flexagon, we are surprised to see that, except for numbering, it is identical with the triflexagon of order 3, made with 6 units. This, too, is explained by the map. Suppose we erase the numbers off the leaves. Then sides making up $0^{\mbox{o}}$ faces will be indistinguishable, and, if they are drawn as one point, we get the map in figure 11.2b, from the map in fig. 11.2a. This second map is much more easily read than the first, but is difficult to use in making the flexagon. It is therefore recommended that each form be used for the better-suited purpose, making or operating, only.

Miss Crampin points out that any combination of the sides at one vertex of a map such as that in fig. 11.2b and the sides at an adjoining vertex can occur. Not only is this true, but the sides at one vertex can be mixed, so that a number appear at one time. This is a further reason to consider 0$^{\mbox{o}}$ face sides as identical. An interesting figure that demonstrates how the sides at each vertex can mix is shown in fig. 11.3. It should be marked according to the number sequence $\frac {1\;3\;3\;5\;5\;1}{2\;2\;4\;4\;6\;6}$. The flap on this object travels around the object's center. The 0$^{\mbox{o}}$ faced flexagon may be made to do the same thing, with its three flaps.

Aside from the points already mentioned, 0$^{\mbox{o}}$ faced flexagons have little to distinguish themselves from other flexagons. Both types (either may be used in most cases) are made using only slight alterations of the usual construction method. The map requires no changes at all. To see how any 0$^{\mbox{o}}$ faced flexagon can actually be put together we consider the flexagon in figure 11.4. The first thing that strikes one is the comparative simplicity of the second map. The first map, however, shows each possible pairing of the sides into faces separately. Since this is true, we also notice in the first map that paths leading to opposite ends of 0$^{\mbox{o}}$ face from the same side are of equal degree. This is characteristic of 0$^{\mbox{o}}$ faced flexagons, and, if no 0$^{\mbox{o}}$ faces are present in the $0$-cut cycle, so that their signs are readily seen, the presence of $+n$ and $-n$ signs in the sign sequence, for one cycle, may help indicate their presence. Another sure indication of the presence of 0$^{\mbox{o}}$ faces in a flexagon is that its class may be less than its cycle. This forces some of the hinges to overlap, so that 0$^{\mbox{o}}$ faces result. The $n$ used in the maps given indicates that any angle will be adequate in place of $n$. Let us choose 120$^{\mbox{o}}$, so that the map becomes like fig. 10.12.

By the method described in the last section (sect X, part D), we find, for the sign sequence, $++ 0^{\mbox{o}} +-+$, where $+=120^{\mbox{o}}$. The number sequence is of course $\frac {1\;3\;3\;5\;5\;1}{2\;2\;4\;4\;6\;6}$, with three units. The only irregularity comes in the handling of the 0$^{\mbox{o}}$ leaf. The leaf following this leaf can be placed either above it or below it; unless we either use the double hinge position type of zero flex or superimpose the new hinge in the correct place, the flexagon will not fit together. To determine the correct position; count either up or down away from the number on top of the 0$^{\mbox{o}}$ leaf, just so long as the number on the bottom of the leaf is not encountered. When a number occurring in either the last leaf before the 0$^{\mbox{o}}$ leaf or the next leaf is encountered, stop. If the number encountered was on the leaf before the 0$^{\mbox{o}}$ leaf, the new leaf is placed underneath. If the number was on the new leaf, put the new leaf above. The proof of this little theorem is not difficult, but it is awkward, so that it is omitted here.

When it has been placed correctly, the new leaf must be folded over (or under) the leaf preceding the 0$^{\mbox{o}}$ leaf, before other leaves are added on. This puts the right surface on the top side of the leaf and orients it correctly for use with the sign sequence. The remainder of the assembly is routine. When we are finished, the face (1-5, 3-4-6) should look like fig. 11.5.

The main consequence of the inclusion of 0$^{\mbox{o}}$ faced flexagons in our discussion is the enormous increase in the number of flexagons we can build. For example, there are now 13 possible 4-flexagons or order 4, made from squares. All of these have been made; few prove interesting to any great extent, since most of the type II maps (See fig. 11.4b) are trivial. When we get around to building 0$^{\mbox{o}}$ faced flexagons of more than one cycle, though, a new difficulty arises: flexagons in which any two cycles have a 0$^{\mbox{o}}$ face in common require two hinge positions at the 0$^{\mbox{o}}$ face. Even when these are given, as when we use the 2-hinge position type of 0$^{\mbox{o}}$ face, the change of cycle cannot occur. To see why all of this is so, and to find a remedy for the situation, we must consider the past structure of an intercycle face. There are two pats, one for each cycle. Each of these is made up of a pile of subpats (and/or single leaves) arranged in the pat structure $A B C D \ldots N$, where each letter represents a subpat, and $A$ is connected to $B$; $B$ to $C$, etc. When these are joined, they form the constant order $A B C D \ldots N N'M'L' \ldots B' A'$, where $A$ is connected to $N'$ and $N$ to $A'$. However, if we try to use only one inter-pat hinging position during the 0$^{\mbox{o}}$ face, the hinge $A-N'$ blocks off $N$ from $A'$, and the two hinges must intersect. (See fig. 11.6, upper half). The figure shows, even if the hinges are allowed to intersect, by using a double hinge position type of 0$^{\mbox{o}}$ face, one pat ($ABC \ldots N$) will be locked shut. This is what prevents further flexing in one of the two cycles. Yet we can see that if the intersection could be moved, so that it would occur in front of the subpat $A'B'C' \ldots N'$ as shown in fig. 11.6, rather than in front of $ABC \ldots N$ then flexing could occur. The only way to move the intersection is to rotate both pats about themselves through a half twist, in opposite directions, so that the crossover is unwound off of the second pat onto the first. Doing this would add no twists to the strip, but it does require flexible leaves. Specifically, it requires that the leaves be creased along the perpendicular drawn to the intersecting hinges at their midpoints. The ends of the hinges toward the top of the flexagon during the 0$^{\mbox{o}}$ face may then be folded down next to the bottom ends, after which the bottom ends are to be folded up to replace the top ones. This operation produces the desired result. Fig. 11.7 gives the picture for the 4-flexagons; other cases are analogous.

Even this does not clear up all of the difficulty. In the flexagon shown in fig. 11.8, for example, each of the three hinges $a$, $b$, and $c$ will intersect the others. Suppose that we cut away the bottom $2/3$ of hinge a to make room for hinges $b$ and $c$. Then we must cut away the top $(1/3)$ of the hinges $b$ and $c$. However, to make room for hinge $b$, we must the cut away the bottom $1/3$ of hinge $c$, which completely severs it. Hence it is impossible to have a $2n$ cycle of 0$^{\mbox{o}}$ faces, each of which joins to another cycle. This difficulty will disrupt the counting of the number of possible 0$^{\mbox{o}}$ faced flexagons quite a bit.

Since each pat of a 0$^{\mbox{o}}$ face has a sign sequence sum of zero, the flexagon could be slit all the way along each 0$^{\mbox{o}}$ face hinge and put back together into two smaller flexagons. The reverse is also possible. Thus the two flexagons in fig. 11.9a, when each is cut while folded together at side 1, can combine to form the 0$^{\mbox{o}}$ faced flexagon in fig. 11.9b.

0$^{\mbox{o}}$ faced flexagons go through other interesting contortions. As was mentioned, they are quite unstable, so that a number of types of distortions are possible. The most interesting of these occurs in any flexagon in which a cutoff 0$^{\mbox{o}}$ face and a 90$^{\mbox{o}}$ face have a side in common. Call the 0$^{\mbox{o}}$ face $(a,b)$ and the 90$^{\mbox{o}}$ face $(a,c)$. Then consider another cutoff face, $(d,c)$, which crosses $(a,b)$ (See fig. 11.10). During this face, the leaves or subpats corresponding to $(a,c)$ and $(d,a)$ will make up one pat, those corresponding to $(b,c)$ and $(d,b)$ forming the other. The subpats $(b,c)$ and $(a,c)$ will both be either on top or on bottom. The three hinges on these two subpats will make two right angles, so that they will be able to open out away from the other two subpats. The simplest example is the map in fig. 11.10, made from single leaves rather than subpats and with $n$ a multiple of 90$^{\mbox{o}}$. The position in which the two leaves $(b,c)$ and $(a,c)$ open out is shown in fig. 11.11 $(n = 180^{\mbox{o}} )$. Since, as was mentioned before, flexagons with extra units may be considered identical with high-cycle flexagons of the same class with fewer units, this same distortion can be produced in some unsuspected places. The proper tetraflexagon of order 4, for example, can be made with 3 units. It may then be considered the same as the mixed flexagon, half of whose map is that shown in fig. 11.12.

This map satisfies the conditions to the distortion. The reader may recognize that this distortion is the same as flexing half of the order 4 flexagon frontwards, the other half backwards. By adding in still more units, the flexagon may be given more flexes backwards and frontwards at the same time.

0$^{\mbox{o}}$ faces may open out at other times, also in a most unexpected manner. The real reason for this unprecedented phenomenon is to be found in the flexagon pictured in fig. 11.11. When it is opened out, it is merely a sheet of paper with a hole in it; it is not twisted at all. The fewer the twists in a plan, the better the chance of its falling apart, With an almost arbitrary sequence of face degrees for each cycle, there is a good chance of getting very few twists. Here is a clue, at least, to instability.

A final ambiguity brought about by 0$^{\mbox{o}}$ faces may be seen in the flexagon shown in fig. 11.13. When we draw the second type of map for this flexagon, as in fig. 11.14, we find that the cycles added on to that containing the 0$^{\mbox{o}}$ faces become indistinguishable. In fact, if the flexagon is treated as an ordinary regular triflexagon with too many units, the problem sides do, indeed become a 0$^{\mbox{o}}$ face. The production of this 0$^{\mbox{o}}$ face is parallel to the flexing both backwards and frontwards at the same time in the 3 unit proper tetraflexagon or order 4.

The flexagon with 0$^{\mbox{o}}$ faces of the double-hinge-position type may also be considered as flexagons made of altered $2n$-gons, with alternate $2n$-gon vertices pushed in to form an $n$-gonal shape. Using this observation may make the planning and manufacturing of this type of flexagon slightly simpler, since they can then be treated exactly like other flexagons made of altered leaves. Certain of the faces (to be determined in the usual way) will, of course, be 0$^{\mbox{o}}$ faces. In the same way as $n$ vertices of these leaves were forced flush with the others, so any kind of leaf can be altered to give an assortment of 0$^{\mbox{o}}$ face positions; by pushing in one or more vertices.

As we have seen, the 0$^{\mbox{o}}$ face fits neatly in with the rest of our flexagon theory. Just as 0$^{\mbox{o}}$ faces can be used in conjunction with the alteration of leaf shapes, so they can be applied to the other flexagon ``dimensions''. We will not, therefore, specifically mention 0$^{\mbox{o}}$ faces, to be distinguished from other faces, in the remainder of our discussion. It is to be understood that they may be used in any possible situation as desired.

By pushing leaf vertices still further in toward the center of the leaf, we can, for the first time, produce leaves with tangible negative angles, which appear as concavities (See fig. 11.15). Previously, negative angles have been produced only by continuously altering the leaf angles through 0$^{\mbox{o}}$. Now they are produced by passing through 0$^{\mbox{o}}$ faces, which have leaf angles of 180$^{\mbox{o}}$. The only notable feature of these flexagons is that they are exceedingly difficult to handle. Since the hinges, when extended, cut the leaf, the leaf must be creased along the extension for the flexagon to open out at all the faces. Also, due to the wide variation in angles necessary to obtain negative angles, there tend to be a large number of units, always a sign of instability.

In the sign sequence we give the angles between pairs of consecutive leaf hinges. It can now be pointed out, however, that this information does not give the order of the hinge positions about the edge of the leaf. This order was determined so long as the leaves were assumed to be convex polygons. Now, though, a considerable variety of leaves may be used, all having the same sign sequences and maps in completely different-looking flexagons. For example, the leaves shown in fig. 11.16 all have same hinge positions, but in different order. The job of assigning an order to the hinge positions must be given to that old standby, the class, since the order of the hinges helps to determine the shapes of the leaves.