Construction of a $G$-Flexagon of General Order $N$.

If, after flexing a $G$-flexagon of the kind described above once, the same flexing axis is maintained, the flexagon will not flex since there is but a single leaf in the left pat. This pat is joined to the upper leaf of the right pat by a hinge in position $(G-1)$, where $G$ is the cycle of the flexagon. This single leaf in the left pat may, however, be slit in the manner used with the tri-flexagons. That is, another leaf may be hinged by a 1-hinge underneath the single leaf ^6.4.

This allows a flex which will expose a new side. Note that the new hinge must be in the ``one'' position, otherwise, the flexagon still will not flex. In general, if the new side is added at side $a$, having come from side $(a+1)$, the new side will become $(a+1)_2$ and the sides $(a+1)_1, (a+2)_1, (a+3)_1 \ldots G$ (where $G$ is the old order) must be renumbered (the subscript refers to which of two cycles the side belongs). If we now rotate the flexagon and attempt to flex, the flexagon instead opens up using a set of hinges other than the 1-hinges and becomes a closed strip of four polygons in which the hinges do not meet in the center. This phenomenon is called tubulation. When an order five tetra-flexagon tubulates, it resembles a cube with two opposite faces missing (see figure 6.7a). A flexagon which is tubulating has the side which was tubulated from, $(a+1)$, on the outside and another side, $(a+2)$, on the inside. This relationship is shown on the map with a dotted line drawn between the two sides as in figure 6.7b and c.

If the flexagon is turned over and flexed back to $(a+2)$ it may then either flex to $(a+3)$ or tubulate to $(a+1)$, depending on whether the flexagon was rotated or not before flexing again. The operation of tubulating is very much like that of flexing, for it is seen that tubulating removes leaves from the left pat and deposits them on the right pat. Indeed if the effective hinge, that is, the hinge (not the zero-hinge) of each unit which is being used for the operation, were in a one-position, the operation would be a flex. If we were to cut the effective hinge used in this tubulation, which in this case is in a $(G-2)$ position, turn the flexagon inside out and tape the hinge back together, we would have side $(a+2)$ on the outside and side $(a+1)$ on the inside and could then close the tubulation, open to side $a$, and flex back to side $(a+1)$. This particular sequence requires only three operations (flexings and tubulations) to return to a given side without retracing a path, and thus may be looked upon as an attempt by the flexagon to become a tri-flexagon. However, we do not want to have to cut the hinge every time we run into a tubulation.

We want to have a complete cycle of ordinary flexes. We may do this in the same manner as before, by slitting one of the leaves which is present at the face $(a+1)$, $a$. But in this case we find, oddly enough, that there are three possible ways of doing so. First of all, having come to side $(a+1)$ from side $a$, but after rotating and before the flexagon is tubulated, there is a single leaf, which can be slit, in the right hand pat. Also, tubulation removes all but a single leaf from the left hand pat. The third possibility is the middle subpat. This may be slit, in much the same manner as the single leaf. Slitting the right hand or middle leaves produces a cycle with a mixture of right and left flexes and will be considered in chapter 9. Any flexagon made so that it always flexes left (or right if it is wound the other way) will be called a proper flexagon. Any flexagon which does not flex consistently left or right and whose subpats are not hinged consecutively $1 \;2 \; 3 \;\ldots \;G$ is an improper flexagon. Since all our previous flexings have been from the left hand pat, at the present we shall consider the slitting of only the single left hand leaf arrived at after tubulating. If, after slitting this leaf and numbering the new side we rotate and try to flex again, we find that again the flexagon tubulates, this time using a $(G-3)$ ^6.5 hinge. Now we again have a number of choices for slitting the leaves, it is possible not only to slit either the left or the right leaves, but to slit any one of the subpats in between them, hinging the new side in a number 1 position. The subpats are for all intents and purposes single leaves in this case. However, in order to be consistent, keeping the flexagon proper and flexing left, we will choose to slit the left hand leaf. Each successive time we slit, flex, and rotate, the hinge position of the tubulation's effective hinge decreases by one. After $(G-2)$ slittings, the effective hinge will be a 1-hinge, and the flexagon will flex normally through the new cycle. The new sides may be numbered in succession counter-clockwise about the map $(a+1), (a+2) \ldots (a+G-2)$ where $G$ is the cycle. The other sides through side $G$ may be renumbered, starting with $(a+G-1)$. By the method just explained, a cycle $G$-flexagon of any order may be constructed. The map will be made up of polygons with $G$ sides, which will be joined to one another by single edges (see figure 6.8).

The ability to tubulate is extremely important and is deserving of extra study. If we add a new side by slitting the single left hand leaf in a tubulating flexagon and hinging the new leaf in a number one position, we find that we have not destroyed the tubulation but rather we have hidden it by making it easier for the flexagon to flex using a 1-hinge. This can be seen by clipping closed the side which would normally turn up next in any given flexing operation. If this if done the tubulation again appears. Tubulations which have been concealed by the addition of new sides are called ``hidden'' tubulations, while those which are a normal part of a series of flexes are called ``exposed'' tubulations. In any flexagon, there are hidden tubulations from any given side to every other non adjacent side of a given cycle. These hidden tubulations may be shown in the map as in figure 6.9a but since they clutter up the drawing, they are often omitted. Each hidden tubulation from a given side to each non adjacent side uses a different hinge. In all, there are $(G-3)$ possible tubulations using $(G-3)$ hinges originating at a given side. The hinges in positions 1 and $(G-1)$, are used by forward and backward flexes respectively (see figure 6.10a). Furthermore these hidden tubulations may be considered as short cuts for if the tubulation were cut, turned inside out, and normal flexing resumed, one or more sides would be omitted from the cycle (see figure 6.9b).

It is convenient to call all tubulations flexes and to give them a number which corresponds to the position of their effective hinges. Thus a normal flex which uses a 1-hinge will be called a ``1-flex'' while a tubulation which uses a 2-hinge will be called a ``2-flex'' and so on. The tubulation which we first encountered in slitting leaves to add a new side (figure 6.7c) was a $(G-2)$-flex. The backward flex, which use a $(G-1)$-hinge, is a ``$(G-1)$-flex'' (see figure 6.10b). The proper octaflexagon is a striking proof that a tubulation should be considered a flex. The angle between the input and output hinges in a 2-flex is 90$^{\mbox{o}}$ and in a two unit flexagon the sum of the angles about the center is 360$^{\mbox{o}}$, so the tubulating flexagon will lie flat. Furthermore, since the tubulating flexagon does lie flat, it is not necessary to force the tubulation. Two separate yet complete cycles of 2-flexes can be made, the operation resembling very closely that of 1-flexes in a tetraflexagon (see figure 6.11).