Arthur Stone was an English mathematics student attending Princeton in 1939. As he refilled his English style notebooks with American notepaper, he noticed that the paper was an inch too wide. After removing the extra one-inch strips of paper, he started folding them, creating the first flexagon.

He and his friends continued to experiment with other foldings, and the convoluted origami shapes became the rage of the campus. In 1959, Martin Gardner introduced the flexagons to the world in his debut Scientific American article.

Flexagons remind me of the "cootie catchers" we made in childhood, now often called "fortune tellers." Folded and flexed, they would answer questions about the future.

Flexagons are mostly mathematical curiosities, however they are making some inroads into commercial products like drink coasters, greeting cards, and toys. I even found an animal cage that is based on a flexagon.

Someone gave me a book of punch-out and assemble Escher-style kaleidocycles. These are like 3-D flexagons made of multiple tetrahedra hinged together in a donut shape. Rather than fold, they "roll" through their "hole." You can make your own kaleidocycles here.

History of flexagons Tetraflexagon         
Arthur Stone Another tetraflexagon
Interview with Arthur Stone Another tetraflexagon
Gallery of flexagons Tetratetraflexagon
Triangle grid paper Triflexahexagon
Flexing movie Hexaflexagon
Java simulation of flexing Hexahexaflexagon
Crafty Coaster that flexes Pentahexaflexagon
3-D hexaflexagon  

See also Mathematics Teaching, June 2002, pp. 16-18