Folding and Flexing

Flexagons are widely known because of Martin Gardner's article in Scientific American from around 1960 [1], and although few people or even libraries have copies of the magazine from so long ago, the articles have been reprinted in several of his collections and can still be bought. I think I have seen a recent announcement a new collection, titled something like ``the best of the best.'' On the other hand, what appeared in those articles was merely an introduction, mainly because of the general level of the magazine but also because there have been later developments. Acutally Gardner got his ideas from general folklore which had been circulating, mostly in the form of annual demonstrations given during the ceremonies of the Westinghouse National Science Talent Search held in Washington, D. C. each year. Something of a forerunner of the Science Fairs which have been popular in recent years. One of the interesting tasks of a historian is to trace the origins of concepts, of which mathematical puzzles and recreations is one. Chess, checkers, and other board games have a more extensive and documented history, as do card games. From all I know or have been able to find out by reading the scarce literature which exists is that they were indeed recognized for the first time by some students at Princeton University in the late 1930's; mention of which is found in Gardner's article. It seems strange to me that nobody ever noticed them before, although some rudimentary variants such as Jacob's Ladder and the bar room hinge were known.

Even taking the Flexagon Committee's work as a starting point, it doesn't seem that they ever wrote down much, and it is an interesting challenge to figure out exactly how much or how little they knew about flexagons. There are two main ways to regard flexagons - either as friezes or as spiral polygon stacks. The latter is more insightful, the former more historical. That is because flexagons originated from folding leftover strips of paper when someone was trimming notebook paper and got to playing with the debris. It is also true that rolls of adding machine tape or similar objects are a good source of material to play with.

However, using straight strips becomes a limitation when joining polygons into a strip seems to be a better vastage point than making polygons out of a predetermined strip. Also, there comes a moment when using them to make spirals in three dimensions rather than sticking to one or two seems to give a better perspective. The main thing about the spirals is that when they consist of two or more turns, edges where they can be folded become parallel rather than blocking each other, and the spiral can be laid out into two pieces alongside each other to make a plane figure. With triangles, three turns are required; the reason for this can be seen in the requirement for the sum of angles around a common center must sum to 360 degrees if the polygons are all to fit together in a nice rosette. So two of the basic rules of flexagons are:

• any polygon can be flexagonned
• the angles must add to 360 degrees.

Of course, such generalities have to be qualified. The polygon should be convex, otherwise using an edge as a hinge doesn't work too well. And it is an immediate question -- what angles? Obviously the ones at the point where common vertices meet when the flexagon is laid out on a tabletop, but which ones are they on the frieze? Even without knowing the answer to that, it seems to be a good idea to work with polygons whose angles are divisors of 360 degrees, which seems to suggest triangles, squares, and hexagons. But if the polygon is not regular, still smaller angles can be used. For triangles, there is not only the equilateral 60 - 60 - 60 degree triangle (note these are the internal angles), but also the isosceles right 45 - 90 - 45 triangle, or the scalene 30 - 120 - 30. Even tinier angles like 15 degrees can be used, and there is no requirement that the triangle be isosceles. For example, the 30 - 60 - 90 right triangle could be used. Going on to squares, they can be used to build what Martin Gardner called tetraflexagons. Actually, I don't find his nomenclature too satisfying. You get hexaflexagons, hexahexaflexagons and so following in a series which is neither the only one nor necesarily the most interesting one. But anyway, the basic square-based flexagon is not folded up from a straight piece of paper, which is why I have avoided mentioning it until now. Either it runs off at a diagonal, or it has nicks in it depending on the angle at which you want to view it.

To make the basic square flexagon, make a strip of five squares, so that the fifth could be laid over the first and pasted. That is one turn of the spiral, which is never enough. So take two such friezes, join the second to the first to get nine squares, then fanfold the frieze, open it out in the middle, and join the ninth to the first. Faces can now be painted, the flexagon flexed, more faces painted, and one has the basic tetraflexagon.

Besides squares, rhombi could be used, with angles of 60 - 120 degrees rather than just 90 degrees. Two turns will not be enough because of the 60 degree angle but three will work just as it does for the original triangle flexagon. Eventually it turns out that there are far more polygons that can be turned into flexagons than anyone would care to work with. Still, when first learning about flexagons, choosing a reasonable variety serves to illustrate the possibilities when one is still trying to formulate a theory and understand them. In that spirit, trying out some near, but not quite, divisors of 360 degrees helps to show why that choice is a good, if not necessary, one. Which is why building a flexagon out of pentagons is so educational.