Playing Cards

The past few sections have been devoted mainly to the generalization of the previously rather strict notion of class. In this section we find the culmination of this tendency, in the complete abandoning of class as a relevant flexagon characteristic.

We have seen that face degree is determined by class, and that is all. Thus, where the hinges between two leaves are placed in relation to the other hinges to those leaves makes no difference whatsoever in the shape of the map. Nor, for that matter, does the shape or size of the leaves involved. Any variation in this direction has been shown acceptable; any shape of plan may be assembled into a flexagon with any given kind of map, so long as the hinging - or, what is equivalent, the numbering of the leaves is done correctly. Let us, then, make the shapes and sizes of the leaves completely arbitrary, irrelevant. This is difficult to do in practice, so we make use of the fact that hinging and numbering are equivalent by using one in place of the other. Using numbering only gives a ``plan'' made up of separate leaves, to be arranged in specified order, and to be rearranged only by the allowed operations: flexing, rotating, etc. The hinging is left to the imagination of the operator. What is the class of this flexagon? As first proposed, the class makes no difference at all (within limits: if the ``class'' is ``lead plates 10 ft. square'', practical difficulties may arise in performing the various operations). For the sake of convenience, we may use a set of rectangular cards, numbered on both sides. Also we may limit ourselves to one unit per flexagon, since the flexagon no longer has angles meeting at a center. If desired, a holder (which fixes the constant cyclic order) may be made for the cards, so that they are not rearranged by accident.

Now that we have a general idea of what the ``cards'' look like, we must learn to operate with them. This will be slightly more difficult than it was in flexagons where there were hinges to serve as guides, but it will also demonstrate far more clearly the nature of each operation. First we notice that all the cards actually do is give us a rearrangeable ordered set of ordered number pairs. Thus we can easily represent all the essential details of a card flexagon by a sequence of numbers. This may bring to mind the flexagon representation of C. O. Oakley and R. J. Wisner (see section 4) and the temptation to call the number sequence the flexagon, the object itself a ``flexagon model''. As far as the present authors are concerned, the two are equivalent.

As for the numbers actually used on the cards, it must be recalled that there are several systems for labelling flexagons, all of then acceptable: the number sequence, constant order, and pat structure systems.

The first of these, the number sequence system, results in a set of cards labeled with the numbers $(1,2), (2,3), (3,4),\ldots (N-1,N), ( N,1)$ and arranged so that like numbers are together, in the order shown above, but with the cycle broken in two spots to form a pair of pats. (All the card sequences such as the one above, must be considered cyclic: the first term is understood to follow the last.) The trouble with this system is that any two flexagons of order $N$ are indistinguishable unless further information is given. If we are given the map, we can follow it in operating the flexagon, but we would prefer to be independent of the map.

The constant order system incorporates all the information given by the map into the cards. To construct a constant order card flexagon, the outer faces in the map are numbered in the order in which they are approached by the traversal of the hinge network (large numbers in fig. 14.1). This establishes a correspondence between this system of numbering and the number sequence numbering system. To change system, we can now easily substitute the numbers of the constant order into the positions of the terms of the corresponding number sequence cards. The card flexagon for fig. 14.1 would be $(1,8), (8,5), (5,4), (4,3), (3,2), (2,6), (6,7), (7,9), (9,1)$, again broken into two pats. As can be seen, all the process really amounts to is the copying down of the constant order numbers from about the edge of the map. Since in both the constant order and the number sequence systems adjacent leaves (number pairs) have like numbers facing one another, we can eliminate one number in each pair and let a leaf be represented by only one number, with the understanding that the surfaces of two adjacent leaves that face together are actually to be colored alike (they make up the same side). Then the flexagon of fig. 14.1 becomes, in constant order cards, $1,8,5,4,3,2,6,7,9$ and customary flexagons, with hinges, may be so labeled by simply numbering the leaves in the plan from 1 to $N$ in the order in which they are attached to one another.

The pat structure system, unlike the other methods, views the flexagon not from the point of view of a fixedly numbered unit of leaves, but as an ordered pair of structures of leaves, We have seen (section IV) the relationship between the pat structure system and the constant order system in triflexagons; it is analogous in the general case.

Our next problem is to interpret the flexagon operations in terms of operations upon the sequences of numbers that we have obtained. The simplest of these is rotation, which reverses the order of the two pats, without altering them structurally. Turning over the flexagon inverts the structure of each pat, without reversing their order. These operations can be clearly interpreted in all the systems. All remaining operations involve alteration of the pat structure. Of these other operations, which include distorting, flexing, and any other operations that one should choose to allow, only flexing will be considered here.

In describing a flex, the first thing to notice is that it acts upon three sets of leaves, each of which may or may not change pats, may be inverted, etc., but is not broken up. One of these sets is an entire pat; the other two make up the second pat, and are separated by a thumbhole. To recognize a thumbhole, we notice that it must be any spot at which the leaves above, connected to another pat along one edge of the given pat, are connected to the leaves below, which connect to another pat along a different edge of the given pat, by a single hinge. For this to be the case, the leaves in the pat must be divided by the thumbhole into two sets of leaves, each of which is made up of leaves lying together in the plan. Here we find our relationship to the constant order system, for this means that one must be able to arrange all the numbers between the thumbholes in each pat consecutively. There will be just $G-2$ thumbholes per cycle, not including the two thumbholes separating the two pats in each unit. The fact that pats must be separated by thumbholes lets us know which groupings of leaves are possible as faces.

Now, flexing is the operation which folds together two pats, thus forming a $(G-1)$st thumbhole, and then removes a different thumbhole, so that the face is changed. The critical position is that at which there are $G-1$ thumbholes, in the folded-together unit. We must first know which of the sides is to be folded together; where the extra thumbhole is to be formed. The formation of this thumbhole will eliminate any thumbholes that might be present from another cycle. Then we pick any one of the $G-1$ thumbholes remaining of the thumbholes previously present, in the folded together unit, and using this, lay the flexagon flat again. The reason why the thumbholes must now be thought of in terms of the folded-together unit, or in terms of a single cycle, is that in flexagons of $G>3$ there will be a mixture of left -and right- flexes, so that thumbholes of other cycles cannot be separated by assigning them all to, say, the left-hand pat, as we did in triflexagons. However, there is one difficulty in finding which of the thumbholes remain in the folded-together unit. Since the flexagon plan is cyclic in structure, we have no way of knowing the order of the two pats, unless we invent some way of distinguishing the two different hinges joining the two pats. If we cannot distinguish these hinges, we will be unable to tell which of the spaces between two pats has become a thumbhole when the flexagon has been folded together, and, if the ambiguity persists, the folded together structure will be divisible into thumbholes at any point, due to the flexagons cyclic construction. To prevent this we say that the hinge between the highest-numbered leaf in the left-hand pat, as seen written out in numbers, and the lowest-numbered leaf in the right-hand pat is the hinge that will be folded together to make the extra thumbhole. That is, the constant order numbers of the right-hand leaves follow those of the left-hand leaves. In the number sequence system there is no possible way to tell the thumbholes without a map, anyway. In the constant order system, the remaining break in the constant order will sort out the desired thumbholes. To keep in mind that it is a break in the cyclic constant order, the two leaves connected by the unfolded-together hinge can be encircled: the lowest term of the left-hand pat, and the highest term of the right-hand pat. In the pat structure system, we need merely keep in mind that, in folding together, the right-hand pat must receive higher numbers than the left-hand pat in the pat structure of the folded-together unit. Then the pat structure need not be considered cyclic. Although the method used in the pat structure system may seem simpler, it does require renumbering of leaves, and therefore, while well-suited to work with sequences of numbers, is not well suited to card flexagons.

Supposing that we are able to eliminate the thumbholes of any other cycle, we can set up the actual mechanism of flexing in all three systems. Suppose that a comma represents a thumbhole, a semicolon the space between two pats. To represent some specific permutation $P_i$ of the set of $y$ consecutive integers $x, x+1, x+2, \ldots, x+y$, we will use the notation $P_i\{y_x\}\cdot P_i^*\{y_x\}$ will indicate the permutation $P_i\{y_x\}$ with the terms taken in reverse order. Then, for the three systems, we have the following, where $k+m+n=N$, the cycle is arbitrary, and all addition is mod $N$.