Flexagons are paper polygons, folded from straight or crooked strips of paper, which have the fascinating property of changing their faces when they are “flexed”
It all began in the fall of 1939. Arthur H. Stone, a 23-year-old graduate student from England, in residence at Princeton University on a mathematics fellowship, had just trimmed an inch from his American notebook sheets to make them fit his English binder. For amusement he began to fold the trimmed-off strips of paper in various ways, and one of the figures he made turned out to be particularly intriguing. Here’s the video to make a simple hexaflexagon.
At this point Stone found the structure so interesting that he showed his paper models to friends in the graduate school. Soon “flexagons” were appearing in profusion at the lunch and dinner tables. A “Flexagon Committee” was organized to probe further into the mysteries of flexigation. The other members besides Stone were Bryant Tuckerman, a graduate student of mathematics; Richard P. Feynman, a graduate student in physics; and John W. Tukey, a young mathematics instructor. Tuckerman quickly discovered that the simplest way to bring out all the faces of any flexagon was to keep flexing it at the same corner until it refused to open, then to shift to an adjacent corner. This procedure, known as the “Tuckerman traverse,” will bring up the six faces of a hexahexa in a cycle of 12 flexes, but 1,2 and 3 turn up three times as often as 4, 5 and 6. A convenient way to diagram a Tuckerman traverse is shown in Figure 4, the arrows indicating the order in which the faces are brought into view. This type of diagram can be applied usefully to the traversing of any type of flexagon. When the model is turned over, a Tuckerman traverse runs the same cycle in reverse order.
The video below gives a great visual explanation of the above.
So now after trying out the simple ones, if you’d want to achieve pro level…..These are some documents you can refer to!
A mathematical investigation of Flexagons can be found here: Theory of Flexagons
Also details about more complicated structures can be found out in the following link. Also you will find some papers written on flexagon on this website (though written in russian).: http://www.flexagon.net/
Also if you are a foodie this video will be a heaven for you..