Occasionally some fascinatingly mysterious device manages to creep out of the obscurity of mathematics to claim a place for itself in the everyday world. Such an object is the flexagon. This odd contraption has emerged from one of the most intriguing fields of today's mathematics - topology1. Topology is that study whose play-things are one-sided bottles, one-edged paper bands, bridges, maps, etc. Now, to this list of things topology has given us, we may add the flexagon.
The flexagon has a great advantage over most scientific toys in that it requires a ridiculously meager material investment to reap immense results. All that is needed is paper, scissors, pencil, and tape. The embellishments - paint, special paper, carrying envelopes, etc. - can, of course, be allowed to raise the investment considerably. No such extravagance, though, is necessary, so dig out some scissors and paper, and let's get started.
I say that flexagons (and there are an infinite number) belong to topology. This is because they are, basically, modified Moebius bands. We will let ourselves be introduced to the flexagon by means of the Moebius band. The Moebius band is the strip of paper mentioned previously, which has but one edge and one side (see figure 1)2. If the strip is given two half-twists, it will have two sides and two edges. That is, it is no longer a true Moebius band. However, the flexagon does not depend upon the number of sides or edges. In its case, it is the number of twists that is critical.
``And just what is a flexagon, anyway?'' you ask. To this I could give you any of various answers - ``A flexagon is a Moebius band of 4n - 6 half-twists'' - or - ``A flexagon is a flexible hexagon'' - or - ``A flexagon is an ordered pair of parts'' - each of which leaves us just as much in the dark as we were before. Therefore, let's begin by making a flexagon or two and then you will be able to see for yourself.
The simplest flexagon is a squashed Moebius band of three half-twists. Figure 2 A shows how the metamorphosis is enacted. The resulting hexagonal object, which is the flexagon, has to be folded along all the dotted lines. Rather than go through the squashing process, we will use a shortcut.
Rule off a straight strip of paper, about six times as long as it is wide, into nine equilateral triangles. Cut off the excess, as shown in figure 2 B. Crease the paper between each pair of triangles (see figure 2 B), then lay the paper flat again. Following the figure, fold together the top sides of triangles 2 and 3 and of triangles 8 and 9. Fold together the lower sides of triangles 5 and 6. Triangles 1 and 2 should now be in such a position that they can be taped together. The hexagonal object resulting should look like figure 2 A.
By the way, if you doubt that this band has only one side, try painting one side red and the other green.
The strip out of which the original Moebius band was built was, of course, uniformly straight. Hence it would seem that the twists could be placed anywhere along its length. Not only is this true, but the twists can be moved along the band. It is the remarkable ability of the squashed-flat band - the flexagon - to move its twists along its length that makes it so interesting. The reason for folding off equilateral triangles will now become apparent, for these folds are employed in the twist-moving, which in this case is called ``flexing.'' The process of flexing is accomplished without bending or folding the triangles, by the method that will be shown presently.
First note that the hexagonal shape is divided into six piles of triangles, each of which is known as a ``pat'' (see figure 2 A). Those pats which enclose twists are made up of two triangles. The twist-moving turns out to be identical with the moving of one triangle in each double pat to the next pat, which would make the next pat double.
In order to flex, start by folding adjacent pats together. Triangles which were adjacent in the strip from which the flexagon was made should be folded so that they are next to one another (the whole process may be followed in figure 3). The flexagon is now in a three-bladed position, and is symmetrical in this respect: It looks exactly the same viewed from either end of the axis along which the three blades meet.
The preceding process could, of course, be performed in reverse, laying the flexagon flat. But, due to the flexagon's symmetry, this can be done from either end of the axis. If we open out the flexagon from the other end (see figure 3), we find that the desired triangle shift has been made. Since the flexagon is now structurally the same as it was before, it may be flexed again without going back. Moreover, we can flex repeatedly, going on infinitely, without once flexing backwards.
The overall effect is one of indefinitely turning inside out. This phraseology is soon seen to be even more apt than might at first be suspected, for if we hold the flexagon flat at any given time and mark or paint the top surface, then flex, the painted part turns up on the bottom. Another flex hides it completely, and a third restores it to its original position. This suggests coloring or numbering each surface - or as it is properly called, ``side'' - of the flexagon when it appears. It will be found that there are three sides. These follow one another in one of two ways:
depending on whether the flexagon is upside-down or not. When a flexagon is flexed, the side that last showed is then found on the underside.
To represent this cyclic change of sides, we employ what is called the ``map'' of the flexagon (see figure 4). If we turn the flexagon over, the directions of all the arrows are reversed.
Let's go back for another look at the flexagon itself. Its ability to open out at either end after having each pair of pats folded together is due to double hinging, as is found in doors that swing both ways. But the hinging edges are not parallel, as they are in the door, and therefore use of the flexagon hinge should shift the triangles 120° about their midpoints, as in figure 5. To observe this effect, fold the flexagon together into one large pile of triangles and dip each corner into colored ink. One color is used for each corner. When the flexagon is opened out, the central angles will all be the same color (see figure 6). When the flexagon is flexed, the central angle changes color. The third color appears after the second flex. After the third flex (and a rotation of 3 × 120°=360°), the first color is again shown.