We may safely conjecture that such a simple object as the first flexagon must have been discovered long ago, and perhaps many times since. For example, there are reports of such a device existing in elementary schools in pre-war Vienna.
The work on the flexagon came as a result of its discovery at Princeton in 1939. The story has it that Arthur Stone, an English graduate student, was in the practice of doodling with the strips of paper that he cut from around the edges of his notebook paper. American paper was too large for his English binder. One of the constructions arising from this happy misfortune attracted his attention in particular. A committee of graduate students formed to solve the mystery of the ``flexible hexagon'', or, as it soon became know, the ``flexagon''.
The members of this group - Richard P. Feynman, Bryant Tuckerman, John W. Tukey, and, of course, Arthur H. Stone - had laid the groundwork for all consequent study, through developing their yet unpublished theory, by the early 1940's. When the group disbanded, the flexagon was left, nearly forgotten, for ten years. Then, toward the beginning of the fifties, it received slight publicity with several very brief articles in mathematics magazines.
It is, however, chiefly due to three influences that flexagons are as well known as they are today. The first of these is Professor Louis B. Tuckerman, the father of one of the original investigators, who has demonstrated items of flexagon theory to winners of the Westinghouse Science Talent Search in Washington, D. C., each year for several years. The present authors trace their introduction to the flexagon back to this source. The second important influence came through Martin Gardner, who told the story of the flexagon, with instructions for building, to a large audience in a pair of articles in Scientific American during 1956 and 1957. More lately, C. O. Oakley and R. J. Wisner have published an article in the American Mathematical Monthly, and R. F. Wheeler has also contributed an article, in The Mathematical Gazette. These articles, although reaching a smaller audience than the Gardner article, contain more technical material. They are the only articles of a technical nature on the topic of flexagons know to the authors. The problems which concern Oakley and Wisner and something of their method of dealing with them are described herein in the section on the pat structure.
Although flexagons have on several occasions been used as cards or announcements, the commercial possibilities they seem to present apparently have never been exploited.
In this article, the authors hope to bring together much of the information that has been distributed concerning flexagons, along with the results of their own investigations. In this way, it may be possible to present a reasonably comprehensive treatment of the flexagon.